List of Johnson solids: Difference between revisions
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In geometry, [[polyhedra]] is a three-dimensional object with lines meeting at a point that forms [[Polygon|polygons]]. The points, lines, and polygons of polyhedra are respectively known as the [[Vertex (geometry)|vertices]], [[Edge (geometry)|edges]], and [[Face (geometry)|faces]].{{sfnp|Meyer|2006|p=[https://books.google.com/books?id=ez6H5Ho6E3cC&pg=PA418 418]}} A polyhedron is said to be ''[[Convex polyhedron|convex]]'' if, for every two points inside the polyhedron, there is a line connecting them that lies within the polyhedra as well;{{sfnp|Litchenberg|1988}} its faces are not [[Coplanarity|coplanar]] (meaning every face are not in the same plane) and its edges are not [[Colinearity|colinear]] (meaning the edges are not in the same line).{{sfnp|Boissonnat|Yvinec|1989}} A polyhedron is said to be [[Regular polyhedron|regular]] if [[Regular polygon|every polygonal faces are equilateral and equiangular]],{{sfnp|Cromwell|1997|p=[https://books.google.com/books?id=&pg=PA77 77]}} and those with the polyhedron has [[vertex-transitive]] property are called a [[uniform polyhedron]].{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA40 40]}} A [[Johnson solid]] (or Johnson–Zalgaller solid) is a convex polyhedron with its faces are regular polygons. Some authors do not require that the Johnson solid not be uniform, meaning that the Johnson solids may not be [[Platonic solid]], [[Archimedean solid]], [[prism]], or [[antiprism]].{{sfnmp |
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In [[geometry]], a [[Johnson solid]] is a strictly [[Convex polyhedron|convex]] [[polyhedron]], each [[Face (geometry)|face]] of which is a [[regular polygon]], but which is not [[uniform polyhedron|uniform]], i.e., not a [[Platonic solid]], [[Archimedean solid]], [[prism (geometry)|prism]] or [[antiprism]]. In 1966, [[Norman Johnson (mathematician)|Norman Johnson]] published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did [[conjecture]] that there were no others.<ref>{{Cite journal |last=Johnson |first=Norman W. |author-link=Norman Johnson (mathematician) |date=1966 |title=Convex Polyhedra with Regular Faces |url=https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/convex-polyhedra-with-regular-faces/5E3FAE0232E2158F93E0A62BB5B1AD39 |journal=Canadian Journal of Mathematics |language=en |volume=18 |pages=169–200 |doi=10.4153/CJM-1966-021-8 |issn=0008-414X}}</ref> [[Victor Zalgaller]] proved in 1969 that Johnson's list was complete.<ref>{{cite journal |last=Zalgaller |first=V. A. |date=1969 |title=Convex polyhedra with regular faces |url=https://zbmath.org/?format=complete&q=an:0177.24802 |journal=Semin. in Mathematics, V.A. Steklov Math. Inst. |language=English |access-date=January 29, 2024}}</ref> |
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| 1a1 = Todesco | 1y = 2020 | 1p = [https://books.google.com/books?id=wtIBEAAAQBAJ&pg=PA282 282] |
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| 2a1 = Williams | 2a2 = Monteleone | 2y = 2021 | 2p = [https://books.google.com/books?id=w5RBEAAAQBAJ&pg=PA23 23] |
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}} |
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The 92 convex polyhedrons were published by [[Norman Johnson]], conjecturing that there are no other solids. His conjecture was proved by [[Victor Zalgaller]] proved in 1969 that Johnson's list was complete.{{sfnmp |
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Other polyhedra can be constructed that have only approximately regular [[Plane (geometry)|planar]] polygon faces, and are informally called [[near-miss Johnson solid]]s; there can be no definitive count of them.<ref>{{Cite web |title=Craig S. Kaplan · near misses |url=https://cs.uwaterloo.ca/~csk/other/nearmisses/ |access-date=2024-01-29 |website=cs.uwaterloo.ca}}</ref><ref>{{Cite web |title=Johnson Solid Near Misses |url=https://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/jsmn.htm |access-date=2024-01-29 |website=www.orchidpalms.com}}</ref> |
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| 1a1 = Johnson | 1y = 1966 |
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| 2a1 = Zalgaller | 2y = 1969 |
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}} [[Pyramid (geometry)|Pyramids]], [[Cupola (geometry)|cupolae]], and [[Rotunda (geometry)|rotunda]] are the first six Johnson solids that have regular faces and convexity. These solids may be applied to construct another polyhedron that has the same properties, a process known as [[Augmentation (geometry)|augmentation]]; attaching prism or antiprism to those is known as [[Elongation (geometry)|elongation]] or [[gyroelongation]], respectively. Some others may be constructed by [[Diminishment (geometry)|diminishment]], the removal of those from the component of polyhedra, or by [[Snub (geometry)|snubification]], a construction by cutting loose the edges, lifting the faces and rotate in certain angle, after which adding the [[equilateral triangles]] between them.{{sfnmp |
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| 1a1 = Rajwade | 1y = 2001 | 1p = [https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84 84–88] |
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| 2a1 = Slobodan | 2a2 = Obradović | 2a3 = Ðukanović | 2y = 2015 |
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| 3a1 = Berman | 3y = 1971 | 3p = 350 |
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| 4a1 = Holme | 4y = 2010 | 4p = [https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA99 99] |
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}} |
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[[Polyhedron#Characteristics|Every polyhedra has own characteristics]], including [[Symmetry (geometry)|symmetry]] and measurement. An object is said to be symmetrical if there is such [[Transformation (geometry)|transformation]] preserving the immunity to change. All of those transformations may be composed in a concept of [[Group (mathematics)|group]], alongside the number of [[Element (mathematics)|elements]], known as [[Order of a group|order]]. In two-dimensional space, these transformations include [[Rotational symmetry|rotating]] around the center of a polygon and [[reflection symmetry|reflecting an object]] around the [[Bisection#Perpendicular line segment bisector|perpendicular bisector]] of a polygon. A polygon that is rotated symmetrically in <math display="inline"> \frac{360^\circ}{n} </math> is denoted by {{math|1=''C''<sub>''n''</sub>}}, a [[cyclic group]] of order {{math|1=''n''}}; combining with the reflection symmetry results in the symmetry of [[dihedral group]] {{math|1=''D''<sub>''n''</sub>}} of order {{math|1=2''n''}}.{{sfnmp |
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The various sections that follow have tables listing all 92 Johnson solids, and values for some of their most important properties. Each table allows sorting by column so that numerical values, or the names of the solids, can be sorted in order. |
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| 1a1 = Powell | 1y = 2010 | 1p = [https://books.google.com/books?id=ojq5BQAAQBAJ&pg=PA27 27] |
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| 2a1 = Solomon | 2y = 2003 | 2p = [https://books.google.com/books?id=ouvZKQiykf4C&pg=PA40 40] |
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}} In [[Point groups in three dimensions|three-dimensional symmetry point groups]], the transformation of polyhedra's symmetry includes the rotation around the line passing through the base center, known as [[Axial symmetry|axis of symmetry]], and reflection relative to perpendicular planes passing through the bisector of a base; this is known as the [[pyramidal symmetry]] {{math|''C''<sub>''n''v</sub>}} of order {{math|1 = 2''n''}}. Relatedly, polyhedra that preserve their symmetry by rotating it horizontally in <math> 180^\circ </math> are known as [[prismatic symmetry]] {{math|1=''D''<sub>''n''v</sub>}} of order {{math|1=2''n''}}. The [[antiprismatic symmetry]] {{math|1=''D''<sub>''n''d</sub>}} of order {{math|1=4''n''}} preserving the symmetry by rotating its half bottom and reflection across the horizontal plane.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} The symmetry group {{math|1=''C''<sub>''n''h</sub>}} of order {{math|1=2''n''}} preserve the symmetry by rotation around the axis of symmetry and reflection on horizontal plane; one case that preserves the symmetry by one full rotation and one reflection horizontal plane is {{math|1=''C''<sub>1h</sub>}} of order 2, or simply denoted as {{math|1=''C''<sub>s</sub>}}.{{sfnmp |
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| 1a1 = Flusser | 1a2 = Suk | 1a3 = Zitofa | 1y = 2017 | 1p = [https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126] |
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| 2a1 = Hergert | 2a2 = Geilhute | 2y = 2018 | 2p = [https://books.google.com/books?id=p6hWDwAAQBAJ&pg=PA56 56] |
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}} The mensuration of polyhedra includes the [[surface area]] and [[volume]]. An [[area]] is a two-dimensional measurement calculated by the product of length and width, and the surface area is the overall area of all faces of polyhedra that is measured by summing all of them.{{sfnp|Walsh|2014|p=[https://books.google.com/books?id=ZhDdAwAAQBAJ&pg=PA284 284]}} A volume is a measurement of the region in three-dimensional space.{{sfnp|Parker|1997|p=[https://archive.org/details/mcgrawhilldictio00park_0/page/264 264]}} |
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The following table contains the 92 Johnson solids of the edge length {{math|1=''a''}}. Each of the columns includes the enumeration of Johnson solid ({{math|''J''<sub>''n''</sub>}}),{{sfnp|Uehara|2020|p=[https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 62]}} the number of vertices, edges, and faces, symmetry, surface area {{math|1=''A''}} and volume {{math|1=''V''}}. |
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==Vertices, edges, faces, and symmetry== |
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{| class="wikitable sortable" |
{| class="wikitable sortable" |
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|+ Table of all 92 Johnson solids |
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|- |
|- |
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! J<sub>n</sub> |
! {{Math|''J''<sub>''n''</sub>}} |
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! Solid name |
! Solid name |
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! [[Net (polyhedron)|Net]] |
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! Image |
! Image |
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! Vertices |
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! V |
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! |
! Edges |
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! |
! Faces |
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! [[List of spherical symmetry groups|Symmetry group]] and its [[Symmetry number|order]]{{sfnp|Johnson|1966}} |
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! F<sub>3</sub> |
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! Surface area and volume{{sfnp|Berman|1971}} |
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! F<sub>4</sub> |
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! F<sub>5</sub> |
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! F<sub>6</sub> |
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! F<sub>8</sub> |
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! F<sub>10</sub> |
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! [[List of spherical symmetry groups|Symmetry group]] |
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![[Symmetry number|Order]] |
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|- |
|- |
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| 1 |
| 1 |
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| [[ |
| [[Equilateral square pyramid|Equilateral<br>square<br>pyramid]] |
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| [[Image: |
| [[Image:Square pyramid.png|100px]] |
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| [[Image:Square pyramid.png|30px]] |
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| 5 |
| 5 |
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| 8 |
| 8 |
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| 5 |
| 5 |
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| {{math|1=''C''<sub>4v</sub>}} of order 8 |
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| 4 |
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| <math> \begin{align} |
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| 1 |
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A &= \left(1 + \sqrt{3}\right)a^2 \\ |
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| |
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&\approx 2.7321a^2 \\ |
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| |
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V &= \frac{\sqrt{2}}{6}a^3 \\ |
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| |
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&\approx 0.2357a^3 \end{align} </math> |
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| |
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| ''C''<sub>4v</sub>, [4], (*44)||8 |
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|- |
|- |
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| 2 |
| 2 |
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| [[Pentagonal pyramid]] |
| [[Pentagonal pyramid|Pentagonal<br>pyramid]] |
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| [[Image: |
| [[Image:Pentagonal pyramid.png|100px]] |
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| [[Image:Pentagonal pyramid.png|30px]] |
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| 6 |
| 6 |
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| 10 |
| 10 |
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| 6 |
| 6 |
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| {{math|1=''C''<sub>5v</sub>}} of order 10 |
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| 5 |
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| <math> \begin{align} |
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| |
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A &= \frac{a^2}{2}\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)} \\ |
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| 1 |
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&\approx 3.8855a^2 \\ |
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| |
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V &= \left(\frac{5 + \sqrt{5}}{24}\right)a^3 \\ |
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| |
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&\approx 0.3015a^3 \end{align} </math> |
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| |
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| ''C''<sub>5v</sub>, [5], (*55)||10 |
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|- |
|- |
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| 3 |
| 3 |
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| [[Triangular cupola]] |
| [[Triangular cupola|Triangular<br>cupola]] |
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| [[Image: |
| [[Image:Triangular cupola.png|100px]] |
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| [[Image:Triangular cupola.png|30px]] |
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| 9 |
| 9 |
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| 15 |
| 15 |
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| 8 |
| 8 |
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| {{math|1=''C''<sub>3v</sub>}} of order 6 |
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| 4 |
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| <math> \begin{align} |
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| 3 |
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A &= \left(3+\frac{5\sqrt{3}}{2} \right) a^2 \\ |
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| |
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&\approx 7.3301a^2 \\ |
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| 1 |
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V &= \left(\frac{5}{3\sqrt{2}}\right) a^3 \\ |
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| |
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&\approx 1.1785a^3 \end{align} </math> |
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| |
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| ''C''<sub>3v</sub>, [3], (*33)||6 |
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|- |
|- |
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| 4 |
| 4 |
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| [[Square cupola]] |
| [[Square cupola|Square<br>cupola]] |
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| [[Image: |
| [[Image:Square cupola.png|100px]] |
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| [[Image:Square cupola.png|30px]] |
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| 12 |
| 12 |
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| 20 |
| 20 |
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| 10 |
| 10 |
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| {{math|1=''C''<sub>4v</sub>}} of order 8 |
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| 4 |
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| <math> \begin{align} |
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| 5 |
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A &= \left(7+2\sqrt{2}+\sqrt{3}\right)a^2 \\ |
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| |
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&\approx 11.5605a^2 \\ |
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| |
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V &= \left(1+\frac{2\sqrt{2}}{3}\right)a^3 \\ |
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| 1 |
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&\approx 1.9428a^3 \end{align} </math> |
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| |
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| ''C''<sub>4v</sub>, [4], (*44)||8 |
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|- |
|- |
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| 5 |
| 5 |
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| [[Pentagonal cupola]] |
| [[Pentagonal cupola|Pentagonal<br>cupola]] |
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| [[Image: |
| [[Image:Pentagonal cupola.png|100px]] |
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| [[Image:Pentagonal cupola.png|30px]] |
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| 15 |
| 15 |
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| 25 |
| 25 |
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| 12 |
| 12 |
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| {{math|1=''C''<sub>5v</sub>}} of order 10 |
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| 5 |
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| <math> \begin{align} |
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| 5 |
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A &= \left(\frac{1}{4}\left(20+5\sqrt{3}+\sqrt{5\left(145+62\sqrt{5}\right)}\right)\right)a^2 \\ |
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| 1 |
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&\approx 16.5798a^2 \\ |
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| |
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V &= \left(\frac{1}{6}\left(5+4\sqrt{5}\right)\right)a^3 \\ |
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| |
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&\approx 2.3241a^3 \end{align} </math> |
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| 1 |
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| ''C''<sub>5v</sub>, [5], (*55)||10 |
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|- |
|- |
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| 6 |
| 6 |
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| [[Pentagonal rotunda]] |
| [[Pentagonal rotunda|Pentagonal<br>rotunda]] |
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| [[Image: |
| [[Image:Pentagonal rotunda.png|100px]] |
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| [[Image:Pentagonal rotunda.png|30px]] |
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| 20 |
| 20 |
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| 35 |
| 35 |
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| 17 |
| 17 |
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| {{math|1=''C''<sub>5v</sub>}} of order 10 |
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| 10 |
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| <math> \begin{align} |
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| |
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A &= \left(\frac{1}{2}\left(5\sqrt{3}+\sqrt{10\left(65+29\sqrt{5}\right)}\right)\right)a^2 \\ |
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| 6 |
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&\approx 22.3472a^2 \\ |
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| |
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V &= \left(\frac{1}{12}\left(45+17\sqrt{5}\right)\right)a^3 \\ |
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| |
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&\approx 6.9178a^3 \end{align} </math> |
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| 1 |
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| ''C''<sub>5v</sub>, [5], (*55)||10 |
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|- |
|- |
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| 7 |
| 7 |
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| [[Elongated triangular pyramid]] |
| [[Elongated triangular pyramid|Elongated<br>triangular<br>pyramid]] |
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| [[Image: |
| [[Image:Elongated triangular pyramid.png|100px]] |
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| [[Image:Elongated triangular pyramid.png|30px]] |
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| 7 |
| 7 |
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| 12 |
| 12 |
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| 7 |
| 7 |
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| {{math|1=''C''<sub>3v</sub>}} of order 6 |
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| 4 |
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| <math> \begin{align} |
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| 3 |
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A &= \left(3+\sqrt{3}\right)a^2 \\ |
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| |
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&\approx 4.7321a^2 \\ |
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| |
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V &= \left(\frac{1}{12}\left(\sqrt{2}+3\sqrt{3}\right)\right)a^3 \\ |
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| |
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&\approx 0.5509a^3 |
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| |
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\end{align} </math> |
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| ''C''<sub>3v</sub>, [3], (*33)||6 |
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|- |
|- |
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| 8 |
| 8 |
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| [[Elongated square pyramid]] |
| [[Elongated square pyramid|Elongated<br>square<br>pyramid]] |
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| [[Image: |
| [[Image:Elongated square pyramid.png|100px]] |
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| [[Image:Elongated square pyramid.png|30px]] |
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| 9 |
| 9 |
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| 16 |
| 16 |
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| 9 |
| 9 |
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| {{math|1=''C''<sub>4v</sub>}} of order 8 |
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| 4 |
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| <math> \begin{align} |
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| 5 |
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A &= \left( 5 + \sqrt{3} \right)a^2 \\ |
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| |
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&\approx 6.7321a^2 \\ |
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| |
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V &= \left( 1 + \frac{\sqrt{2}}{6}\right)a^3 \\ |
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| |
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&\approx 1.2357a^3 \end{align} </math> |
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| |
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| ''C''<sub>4v</sub>, [4], (*44)||8 |
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|- |
|- |
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| 9 |
| 9 |
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| [[Elongated pentagonal pyramid]] |
| [[Elongated pentagonal pyramid|Elongated<br>pentagonal<br>pyramid]] |
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| [[Image: |
| [[Image:Elongated pentagonal pyramid.png|100px]] |
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| [[Image:Elongated pentagonal pyramid.png|30px]] |
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| 11 |
| 11 |
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| 20 |
| 20 |
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| 11 |
| 11 |
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| {{math|1=''C''<sub>5v</sub>}} of order 10 |
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| 5 |
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| <math> \begin{align} |
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| 5 |
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A &= \frac{20 + 5\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4}a^2 \\ |
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| 1 |
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&\approx 8.8855a^2 \\ |
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| |
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V &= \left(\frac{5 + \sqrt{5} + 6\sqrt{25 + 10\sqrt{5}}}{24} \right)a^3 \\ |
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| |
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&\approx 2.022a^3 \end{align} </math> |
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| |
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| ''C''<sub>5v</sub>, [5], (*55)||10 |
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|- |
|- |
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| 10 |
| 10 |
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| [[Gyroelongated square pyramid]] |
| [[Gyroelongated square pyramid|Gyroelongated<br>square<br>pyramid]] |
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| [[Image: |
| [[Image:Gyroelongated square pyramid.png|100px]] |
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| [[Image:Gyroelongated square pyramid.png|30px]] |
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| 9 |
| 9 |
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| 20 |
| 20 |
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| 13 |
| 13 |
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| {{math|1=''C''<sub>4v</sub>}} of order 8 |
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| 12 |
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| <math> \begin{align} |
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| 1 |
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A &= (1 + 3\sqrt{3})a^2 \\ |
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| |
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&\approx 6.1962a^2 \\ |
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| |
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V &= \frac{1}{6} \left(\sqrt{2}+2 \sqrt{4+3 \sqrt{2}}\right)a^3 \\ |
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| |
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&\approx 1.1927a^3 |
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| |
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\end{align} </math> |
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| ''C''<sub>4v</sub>, [4], (*44)||8 |
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|- |
|- |
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| 11 |
| 11 |
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| [[Gyroelongated pentagonal pyramid]] |
| [[Gyroelongated pentagonal pyramid|Gyroelongated<br>pentagonal<br>pyramid]] |
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| [[Image: |
| [[Image:Gyroelongated pentagonal pyramid.png|100px]] |
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| [[Image:Gyroelongated pentagonal pyramid.png|30px]] |
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| 11 |
| 11 |
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| 25 |
| 25 |
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| 16 |
| 16 |
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| {{math|1=''C''<sub>5v</sub>}} of order 10 |
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| 15 |
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| <math> \begin{align} |
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| |
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A &= \frac{1}{4} \left(15 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
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| 1 |
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&\approx 8.2157a^2 \\ |
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| |
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V &= \frac{1}{24} \left(25+9 \sqrt{5}\right)a^3 \\ |
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| |
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&\approx 1.8802a^3 |
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| |
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\end{align} </math> |
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| ''C''<sub>5v</sub>, [5], (*55)||10 |
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|- |
|- |
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| 12 |
| 12 |
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| [[Triangular bipyramid]] |
| [[Triangular bipyramid|Triangular<br>bipyramid]] |
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| [[Image: |
| [[Image:Triangular dipyramid.png|100px]] |
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| [[Image:Triangular dipyramid.png|30px]] |
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| 5 |
| 5 |
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| 9 |
| 9 |
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| 6 |
| 6 |
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| {{math|1=''D''<sub>3h</sub>}} of order 12 |
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| 6 |
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| <math> \begin{align} |
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| |
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A &= \frac{3\sqrt{3}}{2}a^2 \\ |
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| |
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&\approx 2.5981a^2 \\ |
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| |
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V &= \frac{\sqrt{2}}{6}a^3 \\ |
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| |
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&\approx 0.2358a^3 |
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| |
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\end{align} </math> |
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| ''D''<sub>3h</sub>, [3,2], (*223)||12 |
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|- |
|- |
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| 13 |
| 13 |
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| [[Pentagonal bipyramid]] |
| [[Pentagonal bipyramid|Pentagonal<br>bipyramid]] |
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| [[Image: |
| [[Image:Pentagonal dipyramid.png|100px]] |
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| [[Image:Pentagonal dipyramid.png|30px]] |
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| 7 |
| 7 |
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| 15 |
| 15 |
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| 10 |
| 10 |
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| {{math|1=''D''<sub>5h</sub>}} of order 20 |
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| 10 |
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| <math> \begin{align} |
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| |
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A &= \frac{5 \sqrt{3}}{2}a^2 \\ |
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| |
|||
&\approx 4.3301a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \left(5+\sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 0.603a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5h</sub>, [5,2], (*225)||20 |
|||
|- |
|- |
||
| 14 |
| 14 |
||
| [[Elongated triangular bipyramid]] |
| [[Elongated triangular bipyramid|Elongated<br>triangular<br>bipyramid]] |
||
| [[Image: |
| [[Image:Elongated triangular dipyramid.png|100px]] |
||
| [[Image:Elongated triangular dipyramid.png|30px]] |
|||
| 8 |
| 8 |
||
| 15 |
| 15 |
||
| 9 |
| 9 |
||
| {{math|1=''D''<sub>3h</sub>}} of order 12 |
|||
| 6 |
|||
| <math> \begin{align} |
|||
| 3 |
|||
A &= \frac{3}{2} \left(2+\sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 5.5981a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \left(2 \sqrt{2}+3 \sqrt{3}\right)a^3 \\ |
|||
| |
|||
&\approx 0.6687a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>3h</sub>, [3,2], (*223)||12 |
|||
|- |
|- |
||
| 15 |
| 15 |
||
| [[Elongated square bipyramid]] |
| [[Elongated square bipyramid|Elongated<br>square<br>bipyramid]] |
||
| [[Image: |
| [[Image:Elongated square dipyramid.png|100px]] |
||
| [[Image:Elongated square dipyramid.png|30px]] |
|||
| 10 |
| 10 |
||
| 20 |
| 20 |
||
| 12 |
| 12 |
||
| {{math|1=''D''<sub>4h</sub>}} of order 16 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 4 |
|||
A &= 2 \left(2+\sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 7.4641a^2 \\ |
|||
| |
|||
V &= \frac{1}{3} \left(3+\sqrt{2}\right)a^3 \\ |
|||
| |
|||
&\approx 1.4714a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>4h</sub>, [4,2], (*224)||16 |
|||
|- |
|- |
||
| 16 |
| 16 |
||
| [[Elongated pentagonal bipyramid]] |
| [[Elongated pentagonal bipyramid|Elongated<br>pentagonal<br>bipyramid]] |
||
| [[Image: |
| [[Image:Elongated pentagonal dipyramid.png|100px]] |
||
| [[Image:Elongated pentagonal dipyramid.png|30px]] |
|||
| 12 |
| 12 |
||
| 25 |
| 25 |
||
| 15 |
| 15 |
||
| {{math|1=''D''<sub>5h</sub>}} of order 20 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= \frac{5}{2} \left(2+\sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 9.3301a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \left(5+\sqrt{5}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \\ |
|||
| |
|||
&\approx 2.3235a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5h</sub>, [5,2], (*225)||20 |
|||
|- |
|- |
||
| 17 |
| 17 |
||
| [[Gyroelongated square bipyramid]] |
| [[Gyroelongated square bipyramid|Elongated<br>square<br>bipyramid]] |
||
| [[Image: |
| [[Image:Gyroelongated square dipyramid.png|100px]] |
||
| [[Image:Gyroelongated square dipyramid.png|30px]] |
|||
| 10 |
| 10 |
||
| 24 |
| 24 |
||
| 16 |
| 16 |
||
| {{math|1=''D''<sub>4d</sub>}} of order 16 |
|||
| 16 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= 4 \sqrt{3}a^2 \\ |
|||
| |
|||
&\approx 6.9282a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \left(5+\sqrt{5}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \\ |
|||
| |
|||
&\approx 2.3235a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>4d</sub>, [2<sup>+</sup>,8], (2*4)||16 |
|||
|- |
|- |
||
| 18 |
| 18 |
||
| [[Elongated triangular cupola]] |
| [[Elongated triangular cupola|Elongated<br>triangular<br>cupola]] |
||
| [[Image: |
| [[Image:Elongated triangular cupola.png|100px]] |
||
| [[Image:Elongated triangular cupola.png|30px]] |
|||
| 15 |
| 15 |
||
| 27 |
| 27 |
||
| 14 |
| 14 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 4 |
|||
| <math> \begin{align} |
|||
| 9 |
|||
A &= \frac{1}{2} \left(18+5 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 13.3301a^2 \\ |
|||
| 1 |
|||
V &= \frac{1}{3} \left(\sqrt{2}+\sqrt{4+3 \sqrt{2}}\right)a^3 \\ |
|||
| |
|||
&\approx 1.4284a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 19 |
| 19 |
||
| [[Elongated square cupola]] |
| [[Elongated square cupola|Elongated<br>square<br>cupola]] |
||
| [[Image: |
| [[Image:Elongated square cupola.png|100px]] |
||
| [[Image:Elongated square cupola.png|30px]] |
|||
| 20 |
| 20 |
||
| 36 |
| 36 |
||
| 18 |
| 18 |
||
| {{math|1=''C''<sub>4v</sub>}} of order 8 |
|||
| 4 |
|||
| <math> \begin{align} |
|||
| 13 |
|||
A &= (15+2 \sqrt{2}+\sqrt{3})a^2\\ |
|||
| |
|||
&\approx 19.5605a^2 \\ |
|||
| |
|||
V &= \left(3+\frac{8 \sqrt{2}}{3}\right)a^3 \\ |
|||
| 1 |
|||
&\approx 6.7712a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>4v</sub>, [4], (*44)||8 |
|||
|- |
|- |
||
| 20 |
| 20 |
||
| [[Elongated pentagonal cupola]] |
| [[Elongated pentagonal cupola|Elongated<br>pentagonal<br>cupola]] |
||
| [[Image: |
| [[Image:Elongated pentagonal cupola.svg|100px]] |
||
| [[Image:Elongated pentagonal cupola.svg|30px]] |
|||
| 25 |
| 25 |
||
| 45 |
| 45 |
||
| 22 |
| 22 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 5 |
|||
| <math> \begin{align} |
|||
| 15 |
|||
A &= \frac{1}{4} \left(60+5 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 1 |
|||
&\approx 26.5798a^2 \\ |
|||
| |
|||
V &= \frac{1}{6} \left(5+4 \sqrt{5}+15 \sqrt{5+2 \sqrt{5}}\right)a^3 \\ |
|||
| |
|||
&\approx 10.0183a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 21 |
| 21 |
||
| [[Elongated pentagonal rotunda]] |
| [[Elongated pentagonal rotunda|Elongated<br>pentagonal<br>rotunda]] |
||
| [[Image: |
| [[Image:Elongated pentagonal rotunda.png|100px]] |
||
| [[Image:Elongated pentagonal rotunda.png|30px]] |
|||
| 30 |
| 30 |
||
| 55 |
| 55 |
||
| 27 |
| 27 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \frac{1}{2}a^2 \left(20+5 \sqrt{3}+5 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\ |
|||
| 6 |
|||
&\approx 32.3472a^2 \\ |
|||
| |
|||
V &= \frac{1}{12}a^3 \left(45+17 \sqrt{5}+30 \sqrt{5+2 \sqrt{5}}\right) \\ |
|||
| |
|||
&\approx 14.612a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 22 |
| 22 |
||
| [[Gyroelongated triangular cupola]] |
| [[Gyroelongated triangular cupola|Gyroelongated<br>triangular<br>cupola]] |
||
| [[Image: |
| [[Image:Gyroelongated triangular cupola.png|100px]] |
||
| [[Image:Gyroelongated triangular cupola.png|30px]] |
|||
| 15 |
| 15 |
||
| 33 |
| 33 |
||
| 20 |
| 20 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 16 |
|||
| <math> \begin{align} |
|||
| 3 |
|||
A &= \frac{1}{2} \left(6+11 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 12.5263a^2 \\ |
|||
| 1 |
|||
V &= \frac{1}{3} \sqrt{\frac{61}{2}+18 \sqrt{3}+30 \sqrt{1+\sqrt{3}}}a^3 \\ |
|||
| |
|||
&\approx 3.5161a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 23 |
| 23 |
||
| [[Gyroelongated square cupola]] |
| [[Gyroelongated square cupola|Gyroelongated<br>square<br>cupola]] |
||
| [[Image: |
| [[Image:Gyroelongated square cupola.png|100px]] |
||
| [[Image:Gyroelongated square cupola.png|30px]] |
|||
| 20 |
| 20 |
||
| 44 |
| 44 |
||
| 26 |
| 26 |
||
| {{math|1=''C''<sub>4v</sub>}} of order 8 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= (7+2 \sqrt{2}+5 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 18.4887a^2 \\ |
|||
| |
|||
V &= \left(1+\frac{2}{3}\sqrt{2} + \frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right)a^3 \\ |
|||
| 1 |
|||
&\approx 6.2108a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>4v</sub>, [4], (*44)||8 |
|||
|- |
|- |
||
| 24 |
| 24 |
||
| [[Gyroelongated pentagonal cupola]] |
| [[Gyroelongated pentagonal cupola|Gyroelongated<br>pentagonal<br>cupola]] |
||
| [[Image: |
| [[Image:Gyroelongated pentagonal cupola.png|100px]] |
||
| [[Image:Gyroelongated pentagonal cupola.png|30px]] |
|||
| 25 |
| 25 |
||
| 55 |
| 55 |
||
| 32 |
| 32 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 25 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= \frac{1}{4} \left(20+25 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 1 |
|||
&\approx 25.2400a^2 \\ |
|||
| |
|||
V &= \left(\frac{5}{6}+\frac{2}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ |
|||
| |
|||
&\approx 9.0733a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 25 |
| 25 |
||
| [[Gyroelongated pentagonal rotunda]] |
| [[Gyroelongated pentagonal rotunda|Gyroelongated<br>pentagonal<br>rotunda]] |
||
| [[Image: |
| [[Image:Gyroelongated pentagonal rotunda.png|100px]] |
||
| [[Image:Gyroelongated pentagonal rotunda.png|30px]] |
|||
| 30 |
| 30 |
||
| 65 |
| 65 |
||
| 37 |
| 37 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 30 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{1}{2}\left( 15\sqrt{3}+\left(5+3\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2 \\ |
|||
| 6 |
|||
&\approx 31.0075a^2 \\ |
|||
| |
|||
V &= \left(\frac{45}{12}+\frac{17}{12}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ |
|||
| |
|||
&\approx 13.6671a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 26 |
| 26 |
||
| [[Gyrobifastigium]] |
| [[Gyrobifastigium]] |
||
| [[Image: |
| [[Image:Gyrobifastigium.png|100px]] |
||
| [[Image:Gyrobifastigium.png|30px]] |
|||
| 8 |
| 8 |
||
| 14 |
| 14 |
||
| 8 |
| 8 |
||
| {{math|1=''D''<sub>2d</sub>}} of order 8 |
|||
| 4 |
|||
| <math> \begin{align} |
|||
| 4 |
|||
A &= \left(4+\sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 5.7321a^2 \\ |
|||
| |
|||
V &= \left(\frac{\sqrt{3}}{2}\right)a^3 \\ |
|||
| |
|||
&\approx 0.866a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>2d</sub>, [2<sup>+</sup>,4], (2*2)||8 |
|||
|- |
|- |
||
| 27 |
| 27 |
||
| [[Triangular orthobicupola]] |
| [[Triangular orthobicupola|Triangular<br>orthobicupola]] |
||
| [[Image: |
| [[Image:Triangular orthobicupola.png|100px]] |
||
| [[Image:Triangular orthobicupola.png|30px]] |
|||
| 12 |
| 12 |
||
| 24 |
| 24 |
||
| 14 |
| 14 |
||
| {{math|1=''D''<sub>3h</sub>}} of order 12 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 6 |
|||
A &= 2\left(3+\sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 9.4641a^2 \\ |
|||
| |
|||
V &= \frac{5\sqrt{2}}{3}a^3 \\ |
|||
| |
|||
&\approx 2.357a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>3h</sub>, [3,2], (*223)||12 |
|||
|- |
|- |
||
| 28 |
| 28 |
||
| [[Square orthobicupola]] |
| [[Square orthobicupola|Square<br>orthobicupola]] |
||
| [[Image: |
| [[Image:Square orthobicupola.png|100px]] |
||
| [[Image:Square orthobicupola.png|30px]] |
|||
| 16 |
| 16 |
||
| 32 |
| 32 |
||
| 18 |
| 18 |
||
| {{math|1=''D''<sub>4h</sub>}} of order 16 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= 2(\sqrt{5} + \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 13.4641a^2 \\ |
|||
| |
|||
V &= \left(2 + \frac{4\sqrt{2}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 3.8856a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>4h</sub>, [4,2], (*224)||16 |
|||
|- |
|- |
||
| 29 |
| 29 |
||
| [[Square gyrobicupola]] |
| [[Square gyrobicupola|Square<br>gyrobicupola]] |
||
| [[Image: |
| [[Image:Square gyrobicupola.png|100px]] |
||
| [[Image:Square gyrobicupola.png|30px]] |
|||
| 16 |
| 16 |
||
| 32 |
| 32 |
||
| 18 |
| 18 |
||
| {{math|1=''D''<sub>4d</sub>}} of order 16 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= 2(\sqrt{5} + \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 13.4641a^2 \\ |
|||
| |
|||
V &= \left(2 + \frac{4\sqrt{2}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 3.8856a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>4d</sub>, [2<sup>+</sup>,8], (2*4)||16 |
|||
|- |
|- |
||
| 30 |
| 30 |
||
| [[Pentagonal orthobicupola]] |
| [[Pentagonal orthobicupola|Pentagonal<br>orthobicupola]] |
||
| [[Image: |
| [[Image:Pentagonal orthobicupola.png|100px]] |
||
| [[Image:Pentagonal orthobicupola.png|30px]] |
|||
| 20 |
| 20 |
||
| 40 |
| 40 |
||
| 22 |
| 22 |
||
| {{math|1=''D''<sub>5h</sub>}} of order 20 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \left(10+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 17.7711a^2 \\ |
|||
| |
|||
V &= \frac{1}{3}\left(5+4\sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 4.6481a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5h</sub>, [5,2], (*225)||20 |
|||
|- |
|- |
||
| 31 |
| 31 |
||
| [[Pentagonal gyrobicupola]] |
| [[Pentagonal gyrobicupola|Pentagonal<br>gyrobicupola]] |
||
| [[Image: |
| [[Image:Pentagonal gyrobicupola.png|100px]] |
||
| [[Image:Pentagonal gyrobicupola.png|30px]] |
|||
| 20 |
| 20 |
||
| 40 |
| 40 |
||
| 22 |
| 22 |
||
| {{math|1=''D''<sub>5d</sub>}} of order 20 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \left(10+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 17.7711a^2 \\ |
|||
| |
|||
V &= \frac{1}{3}\left(5+4\sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 4.6481a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)||20 |
|||
|- |
|- |
||
| 32 |
| 32 |
||
| [[Pentagonal orthocupolarotunda]] |
| [[Pentagonal orthocupolarotunda|Pentagonal<br>orthocupolarotunda]] |
||
| [[Image: |
| [[Image:Pentagonal orthocupolarotunda.png|100px]] |
||
| [[Image:Pentagonal orthocupolarotunda.png|30px]] |
|||
| 25 |
| 25 |
||
| 50 |
| 50 |
||
| 27 |
| 27 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= \left(5+\frac{1}{4}\sqrt{1900+490\sqrt{5}+210\sqrt{75+30\sqrt{5}}}\right)a^2 \\ |
|||
| 7 |
|||
&\approx 23.5385a^2 \\ |
|||
| |
|||
V &= \frac{5}{12}\left(11+5\sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 9.2418a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 33 |
| 33 |
||
| [[Pentagonal gyrocupolarotunda]] |
| [[Pentagonal gyrocupolarotunda|Pentagonal<br>gyrocupolarotunda]] |
||
| [[Image: |
| [[Image:Pentagonal gyrocupolarotunda.png|100px]] |
||
| [[Image:Pentagonal gyrocupolarotunda.png|30px]] |
|||
| 25 |
| 25 |
||
| 50 |
| 50 |
||
| 27 |
| 27 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= \left(5+\frac{15}{4}\sqrt{3}+\frac{7}{4}\sqrt{25+10\sqrt{5}}\right)a^2 \\ |
|||
| 7 |
|||
&\approx 23.5385a^2 \\ |
|||
| |
|||
V &= \frac{5}{12}\left(11+5\sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 9.2418a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 34 |
| 34 |
||
| [[Pentagonal orthobirotunda]] |
| [[Pentagonal orthobirotunda|Pentagonal<br>orthobirotunda]] |
||
| [[Image: |
| [[Image:Pentagonal orthobirotunda.png|100px]] |
||
| [[Image:Pentagonal orthobirotunda.png|30px]] |
|||
| 30 |
| 30 |
||
| 60 |
| 60 |
||
| 32 |
| 32 |
||
| {{math|1=''D''<sub>5h</sub>}} of order 20 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \left((5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 12 |
|||
&\approx 29.306a^2 \\ |
|||
| |
|||
V &= \frac{1}{6}(45 + 17\sqrt{5})a^3 \\ |
|||
| |
|||
&\approx 13.8355a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5h</sub>, [5,2], (*225)||20 |
|||
|- |
|- |
||
| 35 |
| 35 |
||
| [[Elongated triangular orthobicupola]] |
| [[Elongated triangular orthobicupola|Elongated<br>triangular<br>orthobicupola]] |
||
| [[Image: |
| [[Image:Elongated triangular orthobicupola.png|100px]] |
||
| [[Image:Elongated triangular orthobicupola.png|30px]] |
|||
| 18 |
| 18 |
||
| 36 |
| 36 |
||
| 20 |
| 20 |
||
| {{math|1=''D''<sub>3h</sub>}} of order 12 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 12 |
|||
A &= 2(6 + \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 15.4641a^2 \\ |
|||
| |
|||
V &= \left(\frac{5 \sqrt{2}}{3} + \frac{3 \sqrt{3}}{2}\right)a^3 \\ |
|||
| |
|||
&\approx 4.9551a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>3h</sub>, [3,2], (*223)||12 |
|||
|- |
|- |
||
| 36 |
| 36 |
||
| [[Elongated triangular gyrobicupola]] |
| [[Elongated triangular gyrobicupola|Elongated<br>triangular<br>gyrobicupola]] |
||
| [[Image: |
| [[Image:Elongated triangular gyrobicupola.png|100px]] |
||
| [[Image:Elongated triangular gyrobicupola.png|30px]] |
|||
| 18 |
| 18 |
||
| 36 |
| 36 |
||
| 20 |
| 20 |
||
| {{math|1=''D''<sub>3d</sub>}} of order 12 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 12 |
|||
A &= 2(6 + \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 15.4641a^2 \\ |
|||
| |
|||
V &= \left(\frac{5 \sqrt{2}}{3} + \frac{3 \sqrt{3}}{2}\right)a^3 \\ |
|||
| |
|||
&\approx 4.9551a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)||12 |
|||
|- |
|- |
||
| 37 |
| 37 |
||
| [[Elongated square gyrobicupola]] |
| [[Elongated square gyrobicupola|Elongated<br>square<br>gyrobicupola]] |
||
| [[Image: |
| [[Image:Elongated square gyrobicupola.png|100px]] |
||
| [[Image:Elongated square gyrobicupola.png|30px]] |
|||
| 24 |
| 24 |
||
| 48 |
| 48 |
||
| 26 |
| 26 |
||
| {{math|1=''D''<sub>4d</sub>}} of order 16 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 18 |
|||
A &= 2(9 + \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 21.4641a^2 \\ |
|||
| |
|||
V &= \left(4 + \frac{10\sqrt{2}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 8.714a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>4d</sub>, [2<sup>+</sup>,8], (2*4)||16 |
|||
|- |
|- |
||
| 38 |
| 38 |
||
| [[Elongated pentagonal orthobicupola]] |
| [[Elongated pentagonal orthobicupola|Elongated<br>pentagonal<br>orthobicupola]] |
||
| [[Image: |
| [[Image:Elongated pentagonal orthobicupola.png|100px]] |
||
| [[Image:Elongated pentagonal orthobicupola.png|30px]] |
|||
| 30 |
| 30 |
||
| 60 |
| 60 |
||
| 32 |
| 32 |
||
| {{math|1=''D''<sub>5h</sub>}} of order 20 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 20 |
|||
A &= \left(20+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 27.7711a^2 \\ |
|||
| |
|||
V &= \frac{1}{6}\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ |
|||
| |
|||
&\approx 12.3423a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5h</sub>, [5,2], (*225)||20 |
|||
|- |
|- |
||
| 39 |
| 39 |
||
| [[Elongated pentagonal gyrobicupola]] |
| [[Elongated pentagonal gyrobicupola|Elongated<br>pentagonal<br>gyrobicupola]] |
||
| [[Image: |
| [[Image:Elongated pentagonal gyrobicupola.png|100px]] |
||
| [[Image:Elongated pentagonal gyrobicupola.png|30px]] |
|||
| 30 |
| 30 |
||
| 60 |
| 60 |
||
| 32 |
| 32 |
||
| {{math|1=''D''<sub>5d</sub>}} of order 20 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 20 |
|||
A &= \left(20+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 27.7711a^2 \\ |
|||
| |
|||
V &= \frac{1}{6}\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ |
|||
| |
|||
&\approx 12.3423a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)||20 |
|||
|- |
|- |
||
| 40 |
| 40 |
||
| [[Elongated pentagonal orthocupolarotunda]] |
| [[Elongated pentagonal orthocupolarotunda|Elongated<br>pentagonal<br>orthocupolarotunda]] |
||
| [[Image: |
| [[Image:Elongated pentagonal orthocupolarotunda.png|100px]] |
||
| [[Image:Elongated pentagonal orthocupolarotunda.png|30px]] |
|||
| 35 |
| 35 |
||
| 70 |
| 70 |
||
| 37 |
| 37 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 15 |
|||
A &= \frac{1}{4}\left(60+\sqrt{10\left(190+49\sqrt{5}+21\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 7 |
|||
&\approx 33.5385a^2 \\ |
|||
| |
|||
V &= \frac{5}{12}\left(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}}\right)a^3 \\ |
|||
| |
|||
&\approx 16.936a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 41 |
| 41 |
||
| [[Elongated pentagonal gyrocupolarotunda]] |
| [[Elongated pentagonal gyrocupolarotunda|Elongated<br>pentagonal<br>gyrocupolarotunda]] |
||
| [[Image: |
| [[Image:Elongated pentagonal gyrocupolarotunda.png|100px]] |
||
| [[Image:Elongated pentagonal gyrocupolarotunda.png|30px]] |
|||
| 35 |
| 35 |
||
| 70 |
| 70 |
||
| 37 |
| 37 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 15 |
|||
A &= \frac{1}{4}\left(60+\sqrt{10\left(190+49\sqrt{5}+21\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 7 |
|||
&\approx 33.5385a^2 \\ |
|||
| |
|||
V &= \frac{5}{12}\left(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}}\right)a^3 \\ |
|||
| |
|||
&\approx 16.936a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 42 |
| 42 |
||
| [[Elongated pentagonal orthobirotunda]] |
| [[Elongated pentagonal orthobirotunda|Elongated<br>pentagonal<br>orthobirotunda]] |
||
| [[Image: |
| [[Image:Elongated pentagonal orthobirotunda.png|100px]] |
||
| [[Image:Elongated pentagonal orthobirotunda.png|30px]] |
|||
| 40 |
| 40 |
||
| 80 |
| 80 |
||
| 42 |
| 42 |
||
| {{math|1=''D''<sub>5h</sub>}} of order 20 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 12 |
|||
&\approx 39.306a^2 \\ |
|||
| |
|||
V &= \frac{1}{6}\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ |
|||
| |
|||
&\approx 21.5297a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5h</sub>, [5,2], (*225)||20 |
|||
|- |
|- |
||
| 43 |
| 43 |
||
| [[Elongated pentagonal gyrobirotunda]] |
| [[Elongated pentagonal gyrobirotunda|Elongated<br>pentaognal<br>gyrobirotunda]] |
||
| [[Image: |
| [[Image:Elongated pentagonal gyrobirotunda.png|100px]] |
||
| [[Image:Elongated pentagonal gyrobirotunda.png|30px]] |
|||
| 40 |
| 40 |
||
| 80 |
| 80 |
||
| 42 |
| 42 |
||
| {{math|1=''D''<sub>5d</sub>}} of order 20 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ |
|||
| 12 |
|||
&\approx 39.306a^2 \\ |
|||
| |
|||
V &= \frac{1}{6}\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ |
|||
| |
|||
&\approx 21.5297a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)||20 |
|||
|- |
|- |
||
| 44 |
| 44 |
||
| [[Gyroelongated triangular bicupola]] |
| [[Gyroelongated triangular bicupola|Gyroelongated<br>triangular<br>bicupola]] |
||
| [[Image: |
| [[Image:Gyroelongated triangular bicupola.png|100px]] |
||
| [[Image:Gyroelongated triangular bicupola.png|30px]] |
|||
| 18 |
| 18 |
||
| 42 |
| 42 |
||
| 26 |
| 26 |
||
| {{math|1=''D''<sub>3</sub>}} of order 6 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 6 |
|||
A &= \left(6+5\sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 14.6603a^2 \\ |
|||
| |
|||
V &= \sqrt{2} \left(\frac{5}{3}+\sqrt{1+\sqrt{3}}\right) a^3 \\ |
|||
| |
|||
&\approx 4.6946a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>3</sub>, [3,2]<sup>+</sup>,(223)||6 |
|||
|- |
|- |
||
| 45 |
| 45 |
||
| [[Gyroelongated square bicupola]] |
| [[Gyroelongated square bicupola|Gyroelongated<br>square<br>bicupola]] |
||
| [[Image: |
| [[Image:Gyroelongated square bicupola.png|100px]] |
||
| [[Image:Gyroelongated square bicupola.png|30px]] |
|||
| 24 |
| 24 |
||
| 56 |
| 56 |
||
| 34 |
| 34 |
||
| {{math|1=''D''<sub>4</sub>}} of order 8 |
|||
| 24 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \left(10+6\sqrt{3}\right) a^2 \\ |
|||
| |
|||
&\approx 20.3923a^2 \\ |
|||
| |
|||
V &= \left(2+\frac{4}{3}\sqrt{2} + \frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right) a^3 \\ |
|||
| |
|||
&\approx 8.1536a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>4</sub>, [4,2]<sup>+</sup>, (224)||8 |
|||
|- |
|- |
||
| 46 |
| 46 |
||
| [[Gyroelongated pentagonal bicupola]] |
| [[Gyroelongated pentagonal bicupola|Gyroelongated<br>pentagonal<br>bicupola]] |
||
| [[Image: |
| [[Image:Gyroelongated pentagonal bicupola.png|100px]] |
||
| [[Image:Gyroelongated pentagonal bicupola.png|30px]] |
|||
| 30 |
| 30 |
||
| 70 |
| 70 |
||
| 42 |
| 42 |
||
| {{math|1=''D''<sub>5</sub>}} of order 10 |
|||
| 30 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \frac{1}{2}\left(20+15\sqrt{3}+\sqrt{25+10\sqrt{5}}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 26.4313a^2 \\ |
|||
| |
|||
V &= \left(\frac{5}{3}+\frac{4}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ |
|||
| |
|||
&\approx 11.3974a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5</sub>, [5,2]<sup>+</sup>, (225)||10 |
|||
|- |
|- |
||
| 47 |
| 47 |
||
| [[Gyroelongated pentagonal cupolarotunda]] |
| [[Gyroelongated pentagonal cupolarotunda|Gyroelongated<br>pentagonal<br>cupolarotunda]] |
||
| [[Image: |
| [[Image:Gyroelongated pentagonal cupolarotunda.png|100px]] |
||
| [[Image:Gyroelongated pentagonal cupolarotunda.png|30px]] |
|||
| 35 |
| 35 |
||
| 80 |
| 80 |
||
| 47 |
| 47 |
||
| {{math|1=''C''<sub>5</sub>}} of order 5 |
|||
| 35 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= \frac{1}{4}\left(20+35\sqrt{3}+7\sqrt{25+10\sqrt{5}}\right)a^2 \\ |
|||
| 7 |
|||
&\approx 32.1988a^2 \\ |
|||
| |
|||
V &= \left(\frac{55}{12}+\frac{25}{12}\sqrt{5}+ \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ |
|||
| |
|||
&\approx 15.9911a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>5</sub>, [5]<sup>+</sup>, (55)||5 |
|||
|- |
|- |
||
| 48 |
| 48 |
||
| [[Gyroelongated pentagonal birotunda]] |
| [[Gyroelongated pentagonal birotunda|Gyroelongated<br>pentagonal<br>birotunda]] |
||
| [[Image: |
| [[Image:Gyroelongated pentagonal birotunda.png|100px]] |
||
| [[Image:Gyroelongated pentagonal birotunda.png|30px]] |
|||
| 40 |
| 40 |
||
| 90 |
| 90 |
||
| 52 |
| 52 |
||
| {{math|1=''D''<sub>5</sub>}} of order 10 |
|||
| 40 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \left(10\sqrt{3} + 3\sqrt{25+10\sqrt{5}}\right) a^2 \\ |
|||
| 12 |
|||
&\approx 37.9662a^2 \\ |
|||
| |
|||
V &= \left(\frac{45}{6}+\frac{17}{6}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ |
|||
| |
|||
&\approx 20.5848a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5</sub>, [5,2]<sup>+</sup>, (225)||10 |
|||
|- |
|- |
||
| 49 |
| 49 |
||
| [[Augmented triangular prism]] |
| [[Augmented triangular prism|Augmented<br>triangular<br>prism]] |
||
| [[Image: |
| [[Image:Augmented triangular prism.png|100px]] |
||
| [[Image:Augmented triangular prism.png|30px]] |
|||
| 7 |
| 7 |
||
| 13 |
| 13 |
||
| 8 |
| 8 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 6 |
|||
| <math> \begin{align} |
|||
| 2 |
|||
A &= \frac{1}{2}(4 + 3\sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 4.5981a^2 \\ |
|||
| |
|||
V &= \frac{1}{12}(2\sqrt{2} + 3\sqrt{3})a^3 \\ |
|||
| |
|||
&\approx 0.6687a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 50 |
| 50 |
||
| [[Biaugmented triangular prism]] |
| [[Biaugmented triangular prism|Biaugmented<br>triangular<br>prism]] |
||
| [[Image: |
| [[Image:Biaugmented triangular prism.png|100px]] |
||
| [[Image:Biaugmented triangular prism.png|30px]] |
|||
| 8 |
| 8 |
||
| 17 |
| 17 |
||
| 11 |
| 11 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 1 |
|||
A &= \frac{1}{2}(2 + 5\sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 5.3301a^2 \\ |
|||
| |
|||
V &= \left(\frac{59}{144} + \frac{1}{\sqrt{6}}\right)a^3 \\ |
|||
| |
|||
&\approx 0.9044a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 51 |
| 51 |
||
| [[Triaugmented triangular prism]] |
| [[Triaugmented triangular prism|Triaugmented<br>triangular<br>prism]] |
||
| [[Image: |
| [[Image:Triaugmented triangular prism.png|100px]] |
||
| [[Image:Triaugmented triangular prism.png|30px]] |
|||
| 9 |
| 9 |
||
| 21 |
| 21 |
||
| 14 |
| 14 |
||
| {{math|1=''D''<sub>3h</sub>}} of order 12 |
|||
| 14 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{7\sqrt{3}}{2}a^2 \\ |
|||
| |
|||
&\approx 6.0622a^2 \\ |
|||
| |
|||
V &= \frac{2\sqrt{2}+\sqrt{3}}{4}a^3 \\ |
|||
| |
|||
&\approx 1.1401a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>3h</sub>, [3,2], (*223)||12 |
|||
|- |
|- |
||
| 52 |
| 52 |
||
| [[Augmented pentagonal prism]] |
| [[Augmented pentagonal prism|Augmented<br>pentagonal<br>prism]] |
||
| [[Image: |
| [[Image:Augmented pentagonal prism.png|100px]] |
||
| [[Image:Augmented pentagonal prism.png|30px]] |
|||
| 11 |
| 11 |
||
| 19 |
| 19 |
||
| 10 |
| 10 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 4 |
|||
| <math> \begin{align} |
|||
| 4 |
|||
A &= \frac{1}{2} \left(8+2 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 9.173a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \sqrt{233+90 \sqrt{5}+12 \sqrt{50+20 \sqrt{5}}}a^3 \\ |
|||
| |
|||
&\approx 1.9562a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 53 |
| 53 |
||
| [[Biaugmented pentagonal prism]] |
| [[Biaugmented pentagonal prism|Biaugmented<br>pentagonal<br>prism]] |
||
| [[Image: |
| [[Image:Biaugmented pentagonal prism.png|100px]] |
||
| [[Image:Biaugmented pentagonal prism.png|30px]] |
|||
| 12 |
| 12 |
||
| 23 |
| 23 |
||
| 13 |
| 13 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 3 |
|||
A &= \frac{1}{2}a^2 \left(6+4 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\ |
|||
| 2 |
|||
&\approx 9.9051a^2 \\ |
|||
| |
|||
V &= \frac{1}{12}a^3 \sqrt{257+90 \sqrt{5}+24 \sqrt{50+20 \sqrt{5}}} \\ |
|||
| |
|||
&\approx 2.1919a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 54 |
| 54 |
||
| [[Augmented hexagonal prism]] |
| [[Augmented hexagonal prism|Augmented<br>hexagonal<br>prism]] |
||
| [[Image: |
| [[Image:Augmented hexagonal prism.png|100px]] |
||
| [[Image:Augmented hexagonal prism.png|30px]] |
|||
| 13 |
| 13 |
||
| 22 |
| 22 |
||
| 11 |
| 11 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 4 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= (5+4 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 11.9282a^2 \\ |
|||
| 2 |
|||
V &= \frac{1}{6} \left(\sqrt{2}+9 \sqrt{3}\right)a^3 \\ |
|||
| |
|||
&\approx 2.8338a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 55 |
| 55 |
||
| [[Parabiaugmented hexagonal prism]] |
| [[Parabiaugmented hexagonal prism|Parabiaugmented<br>hexagonal<br>prism]] |
||
| [[Image: |
| [[Image:Parabiaugmented hexagonal prism.png|100px]] |
||
| [[Image:Parabiaugmented hexagonal prism.png|30px]] |
|||
| 14 |
| 14 |
||
| 26 |
| 26 |
||
| 14 |
| 14 |
||
| {{math|1=''D''<sub>2h</sub>}} of order 8 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 4 |
|||
A &= (4+5 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 12.6603a^2 \\ |
|||
| 2 |
|||
V &= \frac{1}{6} \left(2 \sqrt{2}+9 \sqrt{3}\right)a^3 \\ |
|||
| |
|||
&\approx 3.0695a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>2h</sub>, [2,2], (*222)||8 |
|||
|- |
|- |
||
| 56 |
| 56 |
||
| [[Metabiaugmented hexagonal prism]] |
| [[Metabiaugmented hexagonal prism|Metabiaugmented<br>hexagonal<br>prism]] |
||
| [[Image: |
| [[Image:Metabiaugmented hexagonal prism.png|100px]] |
||
| [[Image:Metabiaugmented hexagonal prism.png|30px]] |
|||
| 14 |
| 14 |
||
| 26 |
| 26 |
||
| 14 |
| 14 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 4 |
|||
A &= (4+5 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 12.6603a^2 \\ |
|||
| 2 |
|||
V &= \frac{1}{6} \left(2 \sqrt{2}+9 \sqrt{3}\right)a^3 \\ |
|||
| |
|||
&\approx 3.0695a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 57 |
| 57 |
||
| [[Triaugmented hexagonal prism]] |
| [[Triaugmented hexagonal prism|Triaugmented<br>hexagonal<br>prism]] |
||
| [[Image: |
| [[Image:Triaugmented hexagonal prism.png|100px]] |
||
| [[Image:Triaugmented hexagonal prism.png|30px]] |
|||
| 15 |
| 15 |
||
| 30 |
| 30 |
||
| 17 |
| 17 |
||
| {{math|1=''D''<sub>3h</sub>}} of order 12 |
|||
| 12 |
|||
| <math> \begin{align} |
|||
| 3 |
|||
A &= 3 \left(1+2 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 13.3923a^2 \\ |
|||
| 2 |
|||
V &= \left(\frac{1}{\sqrt{2}}+\frac{3 \sqrt{3}}{2}\right)a^3 \\ |
|||
| |
|||
&\approx 3.3052a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>3h</sub>, [3,2], (*223)||12 |
|||
|- |
|- |
||
| 58 |
| 58 |
||
| [[Augmented dodecahedron]] |
| [[Augmented dodecahedron|Augmented<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Augmented dodecahedron.png|100px]] |
||
| [[Image:Augmented dodecahedron.png|30px]] |
|||
| 21 |
| 21 |
||
| 35 |
| 35 |
||
| 16 |
| 16 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 5 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{1}{4} \left(5 \sqrt{3}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 11 |
|||
&\approx 21.0903a^2 \\ |
|||
| |
|||
V &= \frac{1}{24} \left(95+43 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 7.9646a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 59 |
| 59 |
||
| [[Parabiaugmented dodecahedron]] |
| [[Parabiaugmented dodecahedron|Parabiaugmented<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Parabiaugmented dodecahedron.png|100px]] |
||
| [[Image:Parabiaugmented dodecahedron.png|30px]] |
|||
| 22 |
| 22 |
||
| 40 |
| 40 |
||
| 20 |
| 20 |
||
| {{math|1=''D''<sub>5d</sub>}} of order 20 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{5}{2} \left(\sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 10 |
|||
&\approx 21.5349a^2 \\ |
|||
| |
|||
V &= \frac{1}{6} \left(25+11 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 8.2661a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)||20 |
|||
|- |
|- |
||
| 60 |
| 60 |
||
| [[Metabiaugmented dodecahedron]] |
| [[Metabiaugmented dodecahedron|Metabiaugmented<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Metabiaugmented dodecahedron.png|100px]] |
||
| [[Image:Metabiaugmented dodecahedron.png|30px]] |
|||
| 22 |
| 22 |
||
| 40 |
| 40 |
||
| 20 |
| 20 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{5}{2} \left(\sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 10 |
|||
&\approx 21.5349a^2 \\ |
|||
| |
|||
V &= \frac{1}{6} \left(25+11 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 8.2661a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 61 |
| 61 |
||
| [[Triaugmented dodecahedron]] |
| [[Triaugmented dodecahedron|Triaugmented<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Triaugmented dodecahedron.png|100px]] |
||
| [[Image:Triaugmented dodecahedron.png|30px]] |
|||
| 23 |
| 23 |
||
| 45 |
| 45 |
||
| 24 |
| 24 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{3}{4} \left(5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 9 |
|||
&\approx 21.9795a^2 \\ |
|||
| |
|||
V &= \frac{5}{8} \left(7+3 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 8.5676a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 62 |
| 62 |
||
| [[Metabidiminished icosahedron]] |
| [[Metabidiminished icosahedron|Metabidiminished<br>icosahedron]] |
||
| [[Image: |
| [[Image:Metabidiminished icosahedron.png|100px]] |
||
| [[Image:Metabidiminished icosahedron.png|30px]] |
|||
| 10 |
| 10 |
||
| 20 |
| 20 |
||
| 12 |
| 12 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{1}{2} \left(5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 7.7711a^2 \\ |
|||
| |
|||
V &= \frac{1}{6} \left(5+2 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 1.5787a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 63 |
| 63 |
||
| [[Tridiminished icosahedron]] |
| [[Tridiminished icosahedron|Tridiminished<br>icosahedron]] |
||
| [[Image: |
| [[Image:Tridiminished icosahedron.png|100px]] |
||
| [[Image:Tridiminished icosahedron.png|30px]] |
|||
| 9 |
| 9 |
||
| 15 |
| 15 |
||
| 8 |
| 8 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 5 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{1}{4} \left(5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \\ |
|||
| 3 |
|||
&\approx 7.3265a^3 \\ |
|||
| |
|||
V &= \left(\frac{5}{8}+\frac{7 \sqrt{5}}{24}\right)a^3 \\ |
|||
| |
|||
&\approx 1.2772a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 64 |
| 64 |
||
| [[Augmented tridiminished icosahedron]] |
| [[Augmented tridiminished icosahedron|Augmented<br>tridiminished<br>icosahedron]] |
||
| [[Image: |
| [[Image:Augmented tridiminished icosahedron.png|100px]] |
||
| [[Image:Augmented tridiminished icosahedron.png|30px]] |
|||
| 10 |
| 10 |
||
| 18 |
| 18 |
||
| 10 |
| 10 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 7 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= \frac{1}{4} \left(7 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 3 |
|||
&\approx 8.1925a^2 \\ |
|||
| |
|||
V &= \frac{1}{24} \left(15+2 \sqrt{2}+7 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 1.395a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 65 |
| 65 |
||
| [[Augmented truncated tetrahedron]] |
| [[Augmented truncated tetrahedron|Augmented<br>truncated<br>tetrahedron]] |
||
| [[Image: |
| [[Image:Augmented truncated tetrahedron.png|100px]] |
||
| [[Image:Augmented truncated tetrahedron.png|30px]] |
|||
| 15 |
| 15 |
||
| 27 |
| 27 |
||
| 14 |
| 14 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 3 |
|||
A &= \frac{1}{2} \left(6+13 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 14.2583a^2 \\ |
|||
| 3 |
|||
V &= \frac{11}{2 \sqrt{2}}a^3 \\ |
|||
| |
|||
&\approx 3.8891a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 66 |
| 66 |
||
| [[Augmented truncated cube]] |
| [[Augmented truncated cube|Augmented<br>truncated<br>cube]] |
||
| [[Image: |
| [[Image:Augmented truncated cube.png|100px]] |
||
| [[Image:Augmented truncated cube.png|30px]] |
|||
| 28 |
| 28 |
||
| 48 |
| 48 |
||
| 22 |
| 22 |
||
| {{math|1=''C''<sub>4v</sub>}} of order 8 |
|||
| 12 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= (15+10 \sqrt{2}+3 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 34.3383a^2 \\ |
|||
| |
|||
V &= \left(8+\frac{16 \sqrt{2}}{3}\right)a^3 \\ |
|||
| 5 |
|||
&\approx 15.5425a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>4v</sub>, [4], (*44)||8 |
|||
|- |
|- |
||
| 67 |
| 67 |
||
| [[Biaugmented truncated cube]] |
| [[Biaugmented truncated cube|Biaugmented<br>truncated<br>cube]] |
||
| [[Image: |
| [[Image:Biaugmented truncated cube.png|100px]] |
||
| [[Image:Biaugmented truncated cube.png|30px]] |
|||
| 32 |
| 32 |
||
| 60 |
| 60 |
||
| 30 |
| 30 |
||
| {{math|1=''D''<sub>4h</sub>}} of order 16 |
|||
| 16 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= 2 \left(9+4 \sqrt{2}+2 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 36.2419a^2 \\ |
|||
| |
|||
V &= (9+6 \sqrt{2})a^3 \\ |
|||
| 4 |
|||
&\approx 17.4853a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>4h</sub>, [4,2], (*224)||16 |
|||
|- |
|- |
||
| 68 |
| 68 |
||
| [[Augmented truncated dodecahedron]] |
| [[Augmented truncated dodecahedron|Augmented<br>truncated<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Augmented truncated dodecahedron.png|100px]] |
||
| [[Image:Augmented truncated dodecahedron.png|30px]] |
|||
| 65 |
| 65 |
||
| 105 |
| 105 |
||
| 42 |
| 42 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 25 |
|||
| <math> \begin{align} |
|||
| 5 |
|||
A &= \frac{1}{4} \left(20+25 \sqrt{3}+110 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 1 |
|||
&\approx 102.1821a^2 \\ |
|||
| |
|||
V &= \left(\frac{505}{12}+\frac{81 \sqrt{5}}{4}\right)a^3 \\ |
|||
| |
|||
&\approx 87.3637a^3 |
|||
| 11 |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 69 |
| 69 |
||
| [[Parabiaugmented truncated dodecahedron]] |
| [[Parabiaugmented truncated dodecahedron|Parabiaugmented<br>truncated<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Parabiaugmented truncated dodecahedron.png|100px]] |
||
| [[Image:Parabiaugmented truncated dodecahedron.png|30px]] |
|||
| 70 |
| 70 |
||
| 120 |
| 120 |
||
| 52 |
| 52 |
||
| {{math|1=''D''<sub>5d</sub>}} of order 20 |
|||
| 30 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \frac{1}{2} \left(20+15 \sqrt{3}+50 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 103.3734a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \left(515+251 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 89.6878a^3 |
|||
| 10 |
|||
\end{align} </math> |
|||
| ''D''<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)||20 |
|||
|- |
|- |
||
| 70 |
| 70 |
||
| [[Metabiaugmented truncated dodecahedron]] |
| [[Metabiaugmented truncated dodecahedron|Metabiaugmented<br>truncated<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Metabiaugmented truncated dodecahedron.png|100px]] |
||
| [[Image:Metabiaugmented truncated dodecahedron.png|30px]] |
|||
| 70 |
| 70 |
||
| 120 |
| 120 |
||
| 52 |
| 52 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 30 |
|||
| <math> \begin{align} |
|||
| 10 |
|||
A &= \frac{1}{2} \left(20+15 \sqrt{3}+50 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 2 |
|||
&\approx 103.3734a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \left(515+251 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 89.6878a^3 |
|||
| 10 |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 71 |
| 71 |
||
| [[Triaugmented truncated dodecahedron]] |
| [[Triaugmented truncated dodecahedron|Triaugmented<br>truncated<br>dodecahedron]] |
||
| [[Image: |
| [[Image:Triaugmented truncated dodecahedron.png|100px]] |
||
| [[Image:Triaugmented truncated dodecahedron.png|30px]] |
|||
| 75 |
| 75 |
||
| 135 |
| 135 |
||
| 62 |
| 62 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 35 |
|||
| <math> \begin{align} |
|||
| 15 |
|||
A &= \frac{1}{4} \left(60+35 \sqrt{3}+90 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 3 |
|||
&\approx 104.5648a^2 \\ |
|||
| |
|||
V &= \frac{7}{12} \left(75+37 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 92.0118a^3 |
|||
| 9 |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 72 |
| 72 |
||
| [[Gyrate rhombicosidodecahedron]] |
| [[Gyrate rhombicosidodecahedron|Gyrate<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Gyrate rhombicosidodecahedron.png|100px]] |
||
| [[Image:Gyrate rhombicosidodecahedron.png|30px]] |
|||
| 60 |
| 60 |
||
| 120 |
| 120 |
||
| 62 |
| 62 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 30 |
|||
A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 12 |
|||
&\approx 59.306a^2 \\ |
|||
| |
|||
V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 41.6153a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 73 |
| 73 |
||
| [[Parabigyrate rhombicosidodecahedron]] |
| [[Parabigyrate rhombicosidodecahedron|Parabigyrate<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Parabigyrate rhombicosidodecahedron.png|100px]] |
||
| [[Image:Parabigyrate rhombicosidodecahedron.png|30px]] |
|||
| 60 |
| 60 |
||
| 120 |
| 120 |
||
| 62 |
| 62 |
||
| {{math|1=''D''<sub>5d</sub>}} of order 20 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 30 |
|||
A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 12 |
|||
&\approx 59.306a^2 \\ |
|||
| |
|||
V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 41.6153a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)||20 |
|||
|- |
|- |
||
| 74 |
| 74 |
||
| [[Metabigyrate rhombicosidodecahedron]] |
| [[Metabigyrate rhombicosidodecahedron|Metabigyrate<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Metabigyrate rhombicosidodecahedron.png|100px]] |
||
| [[Image:Metabigyrate rhombicosidodecahedron.png|30px]] |
|||
| 60 |
| 60 |
||
| 120 |
| 120 |
||
| 62 |
| 62 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 30 |
|||
A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 12 |
|||
&\approx 59.306a^2 \\ |
|||
| |
|||
V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 41.6153a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 75 |
| 75 |
||
| [[Trigyrate rhombicosidodecahedron]] |
| [[Trigyrate rhombicosidodecahedron|Trigyrate<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Trigyrate rhombicosidodecahedron.png|100px]] |
||
| [[Image:Trigyrate rhombicosidodecahedron.png|30px]] |
|||
| 60 |
| 60 |
||
| 120 |
| 120 |
||
| 62 |
| 62 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 30 |
|||
A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 12 |
|||
&\approx 59.306a^2 \\ |
|||
| |
|||
V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 41.6153a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 76 |
| 76 |
||
| [[Diminished rhombicosidodecahedron]] |
| [[Diminished rhombicosidodecahedron|Diminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Diminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Diminished rhombicosidodecahedron.png|30px]] |
|||
| 55 |
| 55 |
||
| 105 |
| 105 |
||
| 52 |
| 52 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 25 |
|||
A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 11 |
|||
&\approx 58.1147a^2 \\ |
|||
| |
|||
V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 39.2913a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 77 |
| 77 |
||
| [[Paragyrate diminished rhombicosidodecahedron]] |
| [[Paragyrate diminished rhombicosidodecahedron|Paragyrate<br>diminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Paragyrate diminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Paragyrate diminished rhombicosidodecahedron.png|30px]] |
|||
| 55 |
| 55 |
||
| 105 |
| 105 |
||
| 52 |
| 52 |
||
| {{math|1=''C''<sub>5v</sub>}} of order 10 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 25 |
|||
A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 11 |
|||
&\approx 58.1147a^2 \\ |
|||
| |
|||
V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 39.2913a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>5v</sub>, [5], (*55)||10 |
|||
|- |
|- |
||
| 78 |
| 78 |
||
| [[Metagyrate diminished rhombicosidodecahedron]] |
| [[Metagyrate diminished rhombicosidodecahedron|Metagyrate<br>diminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Metagyrate diminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Metagyrate diminished rhombicosidodecahedron.png|30px]] |
|||
| 55 |
| 55 |
||
| 105 |
| 105 |
||
| 52 |
| 52 |
||
| {{math|1=''C''<sub>s</sub>}} of order 2 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 25 |
|||
A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 11 |
|||
&\approx 58.1147a^2 \\ |
|||
| |
|||
V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 39.2913a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>s</sub>, [ ], (*11)||2 |
|||
|- |
|- |
||
| 79 |
| 79 |
||
| [[Bigyrate diminished rhombicosidodecahedron]] |
| [[Bigyrate diminished rhombicosidodecahedron|Bigyrate<br>diminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Bigyrate diminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Bigyrate diminished rhombicosidodecahedron.png|30px]] |
|||
| 55 |
| 55 |
||
| 105 |
| 105 |
||
| 52 |
| 52 |
||
| {{math|1=''C''<sub>s</sub>}} of order 2 |
|||
| 15 |
|||
| <math> \begin{align} |
|||
| 25 |
|||
A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 11 |
|||
&\approx 58.1147a^2 \\ |
|||
| |
|||
V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 39.2913a^3 |
|||
| 1 |
|||
\end{align} </math> |
|||
| ''C''<sub>s</sub>, [ ], (*11)||2 |
|||
|- |
|- |
||
| 80 |
| 80 |
||
| [[Parabidiminished rhombicosidodecahedron]] |
| [[Parabidiminished rhombicosidodecahedron|Parabidiminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Parabidiminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Parabidiminished rhombicosidodecahedron.png|30px]] |
|||
| 50 |
| 50 |
||
| 90 |
| 90 |
||
| 42 |
| 42 |
||
| {{math|1=''D''<sub>5d</sub>}} of order 20 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 20 |
|||
A &= \frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 10 |
|||
&\approx 56.9233a^2 \\ |
|||
| |
|||
V &= \frac{5}{3} \left(11+5 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 36.9672a^3 |
|||
| 2 |
|||
\end{align} </math> |
|||
| ''D''<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)||20 |
|||
|- |
|- |
||
| 81 |
| 81 |
||
| [[Metabidiminished rhombicosidodecahedron]] |
| [[Metabidiminished rhombicosidodecahedron|Metabidiminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Metabidiminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Metabidiminished rhombicosidodecahedron.png|30px]] |
|||
| 50 |
| 50 |
||
| 90 |
| 90 |
||
| 42 |
| 42 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 20 |
|||
A &= \frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 10 |
|||
&\approx 56.9233a^2 \\ |
|||
| |
|||
V &= \frac{5}{3} \left(11+5 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 36.9672a^3 |
|||
| 2 |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 82 |
| 82 |
||
| [[Gyrate bidiminished rhombicosidodecahedron]] |
| [[Gyrate bidiminished rhombicosidodecahedron|Gyrate<br>bidiminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Gyrate bidiminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Gyrate bidiminished rhombicosidodecahedron.png|30px]] |
|||
| 50 |
| 50 |
||
| 90 |
| 90 |
||
| 42 |
| 42 |
||
| {{math|1=''C''<sub>s</sub>}} of order 2 |
|||
| 10 |
|||
| <math> \begin{align} |
|||
| 20 |
|||
A &= \frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 10 |
|||
&\approx 56.9233a^2 \\ |
|||
| |
|||
V &= \frac{5}{3} \left(11+5 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 36.9672a^3 |
|||
| 2 |
|||
\end{align} </math> |
|||
| ''C''<sub>s</sub>, [ ], (*11)||2 |
|||
|- |
|- |
||
| 83 |
| 83 |
||
| [[Tridiminished rhombicosidodecahedron]] |
| [[Tridiminished rhombicosidodecahedron|Tridiminished<br>rhombicosidodecahedron]] |
||
| [[Image: |
| [[Image:Tridiminished rhombicosidodecahedron.png|100px]] |
||
| [[Image:Tridiminished rhombicosidodecahedron.png|30px]] |
|||
| 45 |
| 45 |
||
| 75 |
| 75 |
||
| 32 |
| 32 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 5 |
|||
| <math> \begin{align} |
|||
| 15 |
|||
A &= \frac{1}{4} \left(60+5 \sqrt{3}+30 \sqrt{5+2 \sqrt{5}}+9 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 9 |
|||
&\approx 55.732a^2 \\ |
|||
| |
|||
V &= \left(\frac{35}{2}+\frac{23 \sqrt{5}}{3}\right)a^3 \\ |
|||
| |
|||
&\approx 34.6432a^3 |
|||
| 3 |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
|||
|- |
|- |
||
| 84 |
| 84 |
||
| [[Snub disphenoid]] |
| [[Snub disphenoid|Snub<br>disphenoid]] |
||
| [[Image: |
| [[Image:Snub disphenoid.png|100px]] |
||
| [[Image:Snub disphenoid.png|30px]] |
|||
| 8 |
| 8 |
||
| 18 |
| 18 |
||
| 12 |
| 12 |
||
| {{math|1=''D''<sub>2d</sub>}} of order 8 |
|||
| 12 |
|||
| <math> \begin{align} |
|||
| |
|||
A &= 2 \left(1+3 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 12.3923a^2 \\ |
|||
| |
|||
V &\approx 0.8595a^3 |
|||
| |
|||
\end{align} </math> |
|||
| |
|||
| ''D''<sub>2d</sub>, [2<sup>+</sup>,4], (2*2)||8 |
|||
|- |
|- |
||
| 85 |
| 85 |
||
| [[Snub square antiprism]] |
| [[Snub square antiprism|Snub<br>square<br>antiprism]] |
||
| [[Image: |
| [[Image:Snub square antiprism.png|100px]] |
||
| [[Image:Snub square antiprism.png|30px]] |
|||
| 16 |
| 16 |
||
| 40 |
| 40 |
||
| 26 |
| 26 |
||
| {{math|1=''D''<sub>4d</sub>}} of order 16 |
|||
| 24 |
|||
| <math> \begin{align} |
|||
| 2 |
|||
A &= 2 \left(1+3 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 12.3923a^2 \\ |
|||
| |
|||
V &\approx 3.6012a^3 |
|||
| |
|||
\end{align} </math> |
|||
| |
|||
| ''D''<sub>4d</sub>, [2<sup>+</sup>,8], (2*4)||16 |
|||
|- |
|- |
||
| 86 |
| 86 |
||
| [[Sphenocorona]] |
| [[Sphenocorona]] |
||
| [[Image: |
| [[Image:Sphenocorona.png|100px]] |
||
| [[Image:Sphenocorona.png|30px]] |
|||
| 10 |
| 10 |
||
| 22 |
| 22 |
||
| 14 |
| 14 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 12 |
|||
| <math> \begin{align} |
|||
| 2 |
|||
A &= (2+3 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 7.1962a^2 \\ |
|||
| |
|||
V &= \frac{1}{2}a^3 \sqrt{1+3 \sqrt{\frac{3}{2}}+\sqrt{13+3 \sqrt{6}}} \\ |
|||
| |
|||
&\approx 1.5154a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 87 |
| 87 |
||
| [[Augmented sphenocorona]] |
| [[Augmented sphenocorona|Augmented<br>sphenocorona]] |
||
| [[Image: |
| [[Image:Augmented sphenocorona.png|100px]] |
||
| [[Image:Augmented sphenocorona.png|30px]] |
|||
| 11 |
| 11 |
||
| 26 |
| 26 |
||
| 17 |
| 17 |
||
| {{math|1=''C''<sub>s</sub>}} of order 2 |
|||
| 16 |
|||
| <math> \begin{align} |
|||
| 1 |
|||
A &= (1+4 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 7.9282a^2 \\ |
|||
| |
|||
V &= \frac{1}{2}a^3\sqrt{1 + 3 \sqrt{\frac{3}{2}} + \sqrt{13 + 3 \sqrt{6}}}+\frac{1}{3\sqrt{2}} \\ |
|||
| |
|||
&\approx 1.7511a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>s</sub>, [ ], (*11)||2 |
|||
|- |
|- |
||
| 88 |
| 88 |
||
| [[Sphenomegacorona]] |
| [[Sphenomegacorona]] |
||
| [[Image: |
| [[Image:Sphenomegacorona.png|100px]] |
||
| [[Image:Sphenomegacorona.png|30px]] |
|||
| 12 |
| 12 |
||
| 28 |
| 28 |
||
| 18 |
| 18 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 16 |
|||
| <math> \begin{align} |
|||
| 2 |
|||
A &= 2 \left(1+2 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 8.9282a^2 \\ |
|||
| |
|||
V &\approx 1.9481a^3 |
|||
| |
|||
\end{align} </math> |
|||
| |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 89 |
| 89 |
||
| [[Hebesphenomegacorona]] |
| [[Hebesphenomegacorona]] |
||
| [[Image: |
| [[Image:Hebesphenomegacorona.png|100px]] |
||
| [[Image:Hebesphenomegacorona.png|30px]] |
|||
| 14 |
| 14 |
||
| 33 |
| 33 |
||
| 21 |
| 21 |
||
| {{math|1=''C''<sub>2v</sub>}} of order 4 |
|||
| 18 |
|||
| <math> \begin{align} |
|||
| 3 |
|||
A &= \frac{3}{2} \left(2+3 \sqrt{3}\right)a^2 \\ |
|||
| |
|||
&\approx 10.7942a^2 \\ |
|||
| |
|||
V &\approx 2.9129a^3 |
|||
| |
|||
\end{align} </math> |
|||
| |
|||
| ''C''<sub>2v</sub>, [2], (*22)||4 |
|||
|- |
|- |
||
| 90 |
| 90 |
||
| [[Disphenocingulum]] |
| [[Disphenocingulum]] |
||
| [[Image: |
| [[Image:Disphenocingulum.png|100px]] |
||
| [[Image:Disphenocingulum.png|30px]] |
|||
| 16 |
| 16 |
||
| 38 |
| 38 |
||
| 24 |
| 24 |
||
| {{math|1=''D''<sub>2d</sub>}} of order 8 |
|||
| 20 |
|||
| <math> \begin{align} |
|||
| 4 |
|||
A &= (4+5 \sqrt{3})a^2 \\ |
|||
| |
|||
&\approx 12.6603a^2 \\ |
|||
| |
|||
V &\approx 3.7776a^3 |
|||
| |
|||
\end{align} </math> |
|||
| |
|||
| ''D''<sub>2d</sub>, [2<sup>+</sup>,4], (2*2)||8 |
|||
|- |
|- |
||
| 91 |
| 91 |
||
| [[Bilunabirotunda]] |
| [[Bilunabirotunda]] |
||
| [[Image: |
| [[Image:Bilunabirotunda.png|100px]] |
||
| [[Image:Bilunabirotunda.png|30px]] |
|||
| 14 |
| 14 |
||
| 26 |
| 26 |
||
| 14 |
| 14 |
||
| {{math|1=''D''<sub>2h</sub>}} of order 8 |
|||
| 8 |
|||
| <math> \begin{align} |
|||
| 2 |
|||
A &= \left(2+2 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 4 |
|||
&\approx 12.346a^2 \\ |
|||
| |
|||
V &= \frac{1}{12} \left(17+9 \sqrt{5}\right)a^3 \\ |
|||
| |
|||
&\approx 3.0937a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''D''<sub>2h</sub>, [2,2], (*222)||8 |
|||
|- |
|- |
||
| 92 |
| 92 |
||
| [[Triangular hebesphenorotunda]] |
| [[Triangular hebesphenorotunda|Triangular<br>hebespenorotunda]] |
||
| [[Image: |
| [[Image:Triangular hebesphenorotunda.png|100px]] |
||
| [[Image:Triangular hebesphenorotunda.png|30px]] |
|||
| 18 |
| 18 |
||
| 36 |
| 36 |
||
| 20 |
| 20 |
||
| {{math|1=''C''<sub>3v</sub>}} of order 6 |
|||
| 13 |
|||
| <math> \begin{align} |
|||
| 3 |
|||
A &= \frac{1}{4} \left(12+19 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ |
|||
| 3 |
|||
&\approx 16.3887a^2 \\ |
|||
| 1 |
|||
V &= \left(\frac{5}{2}+\frac{7 \sqrt{5}}{6}\right)a^3 \\ |
|||
| |
|||
&\approx 5.1087a^3 |
|||
| |
|||
\end{align} </math> |
|||
| ''C''<sub>3v</sub>, [3], (*33)||6 |
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|} |
|} |
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{{right|[[#top|back to top]] [[File:WWC arrow up.png|link=#top]]}} |
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{{Back to top2}} |
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Legend: |
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* J<sub>n</sub> – Johnson solid number |
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* [[Net (polyhedron)|Net]] – Flattened (unfolded) image |
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* V – Number of [[Vertex (geometry)|vertices]] |
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* E – Number of [[Edge (geometry)|edges]] |
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* F – Number of [[Face (geometry)|faces]] (total) |
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* F<sub>3</sub>–F<sub>10</sub> – Number of faces by side counts |
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== Notes == |
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The square pyramid {{math|''J''<sub>1</sub>}} has the fewest vertices (5), the fewest edges (8), and the fewest faces (5). |
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{{reflist}} |
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The triaugmented truncated dodecahedron {{math|''J''<sub>71</sub>}} has the most vertices (75) and the most edges (135). It also has the highest number of faces (62), along with the gyrate rhombicosidodecahedron {{math|''J''<sub>72</sub>}}, the parabigyrate rhombicosidodecahedron {{math|''J''<sub>73</sub>}}, the metabigyrate rhombicosidodecahedron {{math|''J''<sub>74</sub>}}, and the trigyrate rhombicosidodecahedron {{math|''J''<sub>75</sub>}}. |
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==Surface area== |
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Since all [[Face (geometry)|faces]] of Johnson solids are [[regular polygon]]s with 3, 4, 5, 6, 8, or 10 sides, and since all these polygons have the same [[Edge (geometry)|edge]] length {{mvar|a}}, the [[surface area]] of a Johnson solid can be calculated as |
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<math display="block">A=\sum_{n=3,4,5,6,8,10}F_n A_n</math> |
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where the {{math|''F''<sub>''n''</sub>}} are the polygonal face counts in the previous table and |
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<math display="block">A_n = \left( \frac{n}{4} \cot{\frac{\pi}{n}} \right) a^2</math> |
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is the area of a regular polygon with {{mvar|n}} sides of length {{mvar|a}}. In terms of [[nth root|radicals]], one has |
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<math display="block">A_3 = \frac{1}{4} \sqrt{3}\,a^2</math> |
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<math display="block">A_4 = a^2</math> |
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<math display="block">A_5 = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})}\,a^2</math> |
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<math display="block">A_6 = \frac{3}{2} \sqrt{3}\,a^2</math> |
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<math display="block">A_8 = 2(1 + \sqrt{2})\,a^2</math> |
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<math display="block">A_{10} = \frac{5}{2} \sqrt{5 + 2\sqrt{5}}\,a^2,</math> |
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resulting in the following table of surface areas. |
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{| class="wikitable sortable" |
|||
|- |
|||
! J<sub>n</sub> |
|||
! Solid name |
|||
! A/a<sup>2</sup> (approx.) |
|||
! A/a<sup>2</sup> (exact) |
|||
|- |
|||
| 1 |
|||
| [[Square pyramid]] |
|||
| 2.732050808 |
|||
| <math>1+\sqrt{3}</math> |
|||
|- |
|||
| 2 |
|||
| [[Pentagonal pyramid]] |
|||
| 3.885540910 |
|||
| <math>\frac{1}{4} \left(5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 3 |
|||
| [[Triangular cupola]] |
|||
| 7.330127019 |
|||
| <math>\frac{1}{2} \left(6+5 \sqrt{3}\right)</math> |
|||
|- |
|||
| 4 |
|||
| [[Square cupola]] |
|||
| 11.560477932 |
|||
| <math>7+2 \sqrt{2}+\sqrt{3}</math> |
|||
|- |
|||
| 5 |
|||
| [[Pentagonal cupola]] |
|||
| 16.579749753 |
|||
| <math>\frac{1}{4} \left(20+5 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 6 |
|||
| [[Pentagonal rotunda]] |
|||
| 22.347200265 |
|||
| <math>\frac{1}{2} \left(5 \sqrt{3}+5 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 7 |
|||
| [[Elongated triangular pyramid]] |
|||
| 4.732050808 |
|||
| <math>3+\sqrt{3}</math> |
|||
|- |
|||
| 8 |
|||
| [[Elongated square pyramid]] |
|||
| 6.732050808 |
|||
| <math>5+\sqrt{3}</math> |
|||
|- |
|||
| 9 |
|||
| [[Elongated pentagonal pyramid]] |
|||
| 8.885540910 |
|||
| <math>\frac{1}{4} \left(20+5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 10 |
|||
| [[Gyroelongated square pyramid]] |
|||
| 6.196152423 |
|||
| <math>1+3 \sqrt{3}</math> |
|||
|- |
|||
| 11 |
|||
| [[Gyroelongated pentagonal pyramid]] |
|||
| 8.215667929 |
|||
| <math>\frac{1}{4} \left(15 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 12 |
|||
| [[Triangular dipyramid]] |
|||
| 2.598076211 |
|||
| <math>\frac{3 \sqrt{3}}{2}</math> |
|||
|- |
|||
| 13 |
|||
| [[Pentagonal dipyramid]] |
|||
| 4.330127019 |
|||
| <math>\frac{5 \sqrt{3}}{2}</math> |
|||
|- |
|||
| 14 |
|||
| [[Elongated triangular dipyramid]] |
|||
| 5.598076211 |
|||
| <math>\frac{3}{2} \left(2+\sqrt{3}\right)</math> |
|||
|- |
|||
| 15 |
|||
| [[Elongated square dipyramid]] |
|||
| 7.464101615 |
|||
| <math>2 \left(2+\sqrt{3}\right)</math> |
|||
|- |
|||
| 16 |
|||
| [[Elongated pentagonal dipyramid]] |
|||
| 9.330127019 |
|||
| <math>\frac{5}{2} \left(2+\sqrt{3}\right)</math> |
|||
|- |
|||
| 17 |
|||
| [[Gyroelongated square dipyramid]] |
|||
| 6.928203230 |
|||
| <math>4 \sqrt{3}</math> |
|||
|- |
|||
| 18 |
|||
| [[Elongated triangular cupola]] |
|||
| 13.330127019 |
|||
| <math>\frac{1}{2} \left(18+5 \sqrt{3}\right)</math> |
|||
|- |
|||
| 19 |
|||
| [[Elongated square cupola]] |
|||
| 19.560477932 |
|||
| <math>15+2 \sqrt{2}+\sqrt{3}</math> |
|||
|- |
|||
| 20 |
|||
| [[Elongated pentagonal cupola]] |
|||
| 26.579749753 |
|||
| <math>\frac{1}{4} \left(60+5 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 21 |
|||
| [[Elongated pentagonal rotunda]] |
|||
| 32.347200265 |
|||
| <math>\frac{1}{2} \left(20+5 \sqrt{3}+5 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 22 |
|||
| [[Gyroelongated triangular cupola]] |
|||
| 12.526279442 |
|||
| <math>\frac{1}{2} \left(6+11 \sqrt{3}\right)</math> |
|||
|- |
|||
| 23 |
|||
| [[Gyroelongated square cupola]] |
|||
| 18.488681163 |
|||
| <math>7+2 \sqrt{2}+5 \sqrt{3}</math> |
|||
|- |
|||
| 24 |
|||
| [[Gyroelongated pentagonal cupola]] |
|||
| 25.240003791 |
|||
| <math>\frac{1}{4} \left(20+25 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 25 |
|||
| [[Gyroelongated pentagonal rotunda]] |
|||
| 31.007454303 |
|||
| <math>\frac{1}{2} \left(15 \sqrt{3}+5 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 26 |
|||
| [[Gyrobifastigium]] |
|||
| 5.732050808 |
|||
| <math>4+\sqrt{3}</math> |
|||
|- |
|||
| 27 |
|||
| [[Triangular orthobicupola]] |
|||
| 9.464101615 |
|||
| <math>2 \left(3+\sqrt{3}\right)</math> |
|||
|- |
|||
| 28 |
|||
| [[Square orthobicupola]] |
|||
| 13.464101615 |
|||
| <math>2 \left(5+\sqrt{3}\right)</math> |
|||
|- |
|||
| 29 |
|||
| [[Square gyrobicupola]] |
|||
| 13.464101615 |
|||
| <math>2 \left(5+\sqrt{3}\right)</math> |
|||
|- |
|||
| 30 |
|||
| [[Pentagonal orthobicupola]] |
|||
| 17.771081820 |
|||
| <math>\frac{1}{2} \left(20+5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 31 |
|||
| [[Pentagonal gyrobicupola]] |
|||
| 17.771081820 |
|||
| <math>\frac{1}{2} \left(20+5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 32 |
|||
| [[Pentagonal orthocupolarotunda]] |
|||
| 23.538532333 |
|||
| <math>\frac{1}{4} \left(20+15 \sqrt{3}+7 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 33 |
|||
| [[Pentagonal gyrocupolarotunda]] |
|||
| 23.538532333 |
|||
| <math>\frac{1}{4} \left(20+15 \sqrt{3}+7 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 34 |
|||
| [[Pentagonal orthobirotunda]] |
|||
| 29.305982845 |
|||
| <math>5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 35 |
|||
| [[Elongated triangular orthobicupola]] |
|||
| 15.464101615 |
|||
| <math>2 \left(6+\sqrt{3}\right)</math> |
|||
|- |
|||
| 36 |
|||
| [[Elongated triangular gyrobicupola]] |
|||
| 15.464101615 |
|||
| <math>2 \left(6+\sqrt{3}\right)</math> |
|||
|- |
|||
| 37 |
|||
| [[Elongated square gyrobicupola]] |
|||
| 21.464101615 |
|||
| <math>2 \left(9+\sqrt{3}\right)</math> |
|||
|- |
|||
| 38 |
|||
| [[Elongated pentagonal orthobicupola]] |
|||
| 27.771081820 |
|||
| <math>\frac{1}{2} \left(40+5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 39 |
|||
| [[Elongated pentagonal gyrobicupola]] |
|||
| 27.771081820 |
|||
| <math>\frac{1}{2} \left(40+5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 40 |
|||
| [[Elongated pentagonal orthocupolarotunda]] |
|||
| 33.538532333 |
|||
| <math>\frac{1}{4} \left(60+15 \sqrt{3}+7 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 41 |
|||
| [[Elongated pentagonal gyrocupolarotunda]] |
|||
| 33.538532333 |
|||
| <math>\frac{1}{4} \left(60+15 \sqrt{3}+7 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 42 |
|||
| [[Elongated pentagonal orthobirotunda]] |
|||
| 39.305982845 |
|||
| <math>10+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 43 |
|||
| [[Elongated pentagonal gyrobirotunda]] |
|||
| 39.305982845 |
|||
| <math>10+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 44 |
|||
| [[Gyroelongated triangular bicupola]] |
|||
| 14.660254038 |
|||
| <math>6+5 \sqrt{3}</math> |
|||
|- |
|||
| 45 |
|||
| [[Gyroelongated square bicupola]] |
|||
| 20.392304845 |
|||
| <math>2 \left(5+3 \sqrt{3}\right)</math> |
|||
|- |
|||
| 46 |
|||
| [[Gyroelongated pentagonal bicupola]] |
|||
| 26.431335858 |
|||
| <math>\frac{1}{2} \left(20+15 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 47 |
|||
| [[Gyroelongated pentagonal cupolarotunda]] |
|||
| 32.198786370 |
|||
| <math>\frac{1}{4} \left(20+35 \sqrt{3}+7 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 48 |
|||
| [[Gyroelongated pentagonal birotunda]] |
|||
| 37.966236883 |
|||
| <math>10 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 49 |
|||
| [[Augmented triangular prism]] |
|||
| 4.598076211 |
|||
| <math>\frac{1}{2} \left(4+3 \sqrt{3}\right)</math> |
|||
|- |
|||
| 50 |
|||
| [[Biaugmented triangular prism]] |
|||
| 5.330127019 |
|||
| <math>\frac{1}{2} \left(2+5 \sqrt{3}\right)</math> |
|||
|- |
|||
| 51 |
|||
| [[Triaugmented triangular prism]] |
|||
| 6.062177826 |
|||
| <math>\frac{7 \sqrt{3}}{2}</math> |
|||
|- |
|||
| 52 |
|||
| [[Augmented pentagonal prism]] |
|||
| 9.173005609 |
|||
| <math>\frac{1}{2} \left(8+2 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 53 |
|||
| [[Biaugmented pentagonal prism]] |
|||
| 9.905056416 |
|||
| <math>\frac{1}{2} \left(6+4 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 54 |
|||
| [[Augmented hexagonal prism]] |
|||
| 11.928203230 |
|||
| <math>5+4 \sqrt{3}</math> |
|||
|- |
|||
| 55 |
|||
| [[Parabiaugmented hexagonal prism]] |
|||
| 12.660254038 |
|||
| <math>4+5 \sqrt{3}</math> |
|||
|- |
|||
| 56 |
|||
| [[Metabiaugmented hexagonal prism]] |
|||
| 12.660254038 |
|||
| <math>4+5 \sqrt{3}</math> |
|||
|- |
|||
| 57 |
|||
| [[Triaugmented hexagonal prism]] |
|||
| 13.392304845 |
|||
| <math>3 \left(1+2 \sqrt{3}\right)</math> |
|||
|- |
|||
| 58 |
|||
| [[Augmented dodecahedron]] |
|||
| 21.090314916 |
|||
| <math>\frac{1}{4} \left(5 \sqrt{3}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 59 |
|||
| [[Parabiaugmented dodecahedron]] |
|||
| 21.534901025 |
|||
| <math>\frac{5}{2} \left(\sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 60 |
|||
| [[Metabiaugmented dodecahedron]] |
|||
| 21.534901025 |
|||
| <math>\frac{5}{2} \left(\sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 61 |
|||
| [[Triaugmented dodecahedron]] |
|||
| 21.979487134 |
|||
| <math>\frac{3}{4} \left(5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 62 |
|||
| [[Metabidiminished icosahedron]] |
|||
| 7.771081820 |
|||
| <math>\frac{1}{2} \left(5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 63 |
|||
| [[Tridiminished icosahedron]] |
|||
| 7.326495711 |
|||
| <math>\frac{1}{4} \left(5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 64 |
|||
| [[Augmented tridiminished icosahedron]] |
|||
| 8.192521115 |
|||
| <math>\frac{1}{4} \left(7 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 65 |
|||
| [[Augmented truncated tetrahedron]] |
|||
| 14.258330249 |
|||
| <math>\frac{1}{2} \left(6+13 \sqrt{3}\right)</math> |
|||
|- |
|||
| 66 |
|||
| [[Augmented truncated cube]] |
|||
| 34.338288046 |
|||
| <math>15+10 \sqrt{2}+3 \sqrt{3}</math> |
|||
|- |
|||
| 67 |
|||
| [[Biaugmented truncated cube]] |
|||
| 36.241911729 |
|||
| <math>2 \left(9+4 \sqrt{2}+2 \sqrt{3}\right)</math> |
|||
|- |
|||
| 68 |
|||
| [[Augmented truncated dodecahedron]] |
|||
| 102.182092220 |
|||
| <math>\frac{1}{4} \left(20+25 \sqrt{3}+110 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 69 |
|||
| [[Parabiaugmented truncated dodecahedron]] |
|||
| 103.373424287 |
|||
| <math>\frac{1}{2} \left(20+15 \sqrt{3}+50 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 70 |
|||
| [[Metabiaugmented truncated dodecahedron]] |
|||
| 103.373424287 |
|||
| <math>\frac{1}{2} \left(20+15 \sqrt{3}+50 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 71 |
|||
| [[Triaugmented truncated dodecahedron]] |
|||
| 104.564756354 |
|||
| <math>\frac{1}{4} \left(60+35 \sqrt{3}+90 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 72 |
|||
| [[Gyrate rhombicosidodecahedron]] |
|||
| 59.305982845 |
|||
| <math>30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 73 |
|||
| [[Parabigyrate rhombicosidodecahedron]] |
|||
| 59.305982845 |
|||
| <math>30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 74 |
|||
| [[Metabigyrate rhombicosidodecahedron]] |
|||
| 59.305982845 |
|||
| <math>30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 75 |
|||
| [[Trigyrate rhombicosidodecahedron]] |
|||
| 59.305982845 |
|||
| <math>30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 76 |
|||
| [[Diminished rhombicosidodecahedron]] |
|||
| 58.114650778 |
|||
| <math>\frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 77 |
|||
| [[Paragyrate diminished rhombicosidodecahedron]] |
|||
| 58.114650778 |
|||
| <math>\frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 78 |
|||
| [[Metagyrate diminished rhombicosidodecahedron]] |
|||
| 58.114650778 |
|||
| <math>\frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 79 |
|||
| [[Bigyrate diminished rhombicosidodecahedron]] |
|||
| 58.114650778 |
|||
| <math>\frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 80 |
|||
| [[Parabidiminished rhombicosidodecahedron]] |
|||
| 56.923318711 |
|||
| <math>\frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 81 |
|||
| [[Metabidiminished rhombicosidodecahedron]] |
|||
| 56.923318711 |
|||
| <math>\frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 82 |
|||
| [[Gyrate bidiminished rhombicosidodecahedron]] |
|||
| 56.923318711 |
|||
| <math>\frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 83 |
|||
| [[Tridiminished rhombicosidodecahedron]] |
|||
| 55.731986644 |
|||
| <math>\frac{1}{4} \left(60+5 \sqrt{3}+30 \sqrt{5+2 \sqrt{5}}+9 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 84 |
|||
| [[Snub disphenoid]] |
|||
| 5.196152423 |
|||
| <math>3 \sqrt{3}</math> |
|||
|- |
|||
| 85 |
|||
| [[Snub square antiprism]] |
|||
| 12.392304845 |
|||
| <math>2 \left(1+3 \sqrt{3}\right)</math> |
|||
|- |
|||
| 86 |
|||
| [[Sphenocorona]] |
|||
| 7.196152423 |
|||
| <math>2+3 \sqrt{3}</math> |
|||
|- |
|||
| 87 |
|||
| [[Augmented sphenocorona]] |
|||
| 7.928203230 |
|||
| <math>1+4 \sqrt{3}</math> |
|||
|- |
|||
| 88 |
|||
| [[Sphenomegacorona]] |
|||
| 8.928203230 |
|||
| <math>2 \left(1+2 \sqrt{3}\right)</math> |
|||
|- |
|||
| 89 |
|||
| [[Hebesphenomegacorona]] |
|||
| 10.794228634 |
|||
| <math>\frac{3}{2} \left(2+3 \sqrt{3}\right)</math> |
|||
|- |
|||
| 90 |
|||
| [[Disphenocingulum]] |
|||
| 12.660254038 |
|||
| <math>4+5 \sqrt{3}</math> |
|||
|- |
|||
| 91 |
|||
| [[Bilunabirotunda]] |
|||
| 12.346011217 |
|||
| <math>2+2 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}</math> |
|||
|- |
|||
| 92 |
|||
| [[Triangular hebesphenorotunda]] |
|||
| 16.388673538 |
|||
| <math>\frac{1}{4} \left(12+19 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|} |
|||
{{Back to top2}} |
|||
For a fixed edge length, the triangular dipyramid {{math|''J''<sub>12</sub>}} has the smallest surface area and the triaugmented truncated dodecahedron {{math|''J''<sub>71</sub>}} has the largest, more than 40 times larger. |
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==Volume== |
|||
The following table lists the [[volume]] of each Johnson solid. Here {{mvar|V}} is the volume (not the number of vertices, as in the first table) and {{mvar|a}} is the [[Edge (geometry)|edge]] length. |
|||
The source for this table is the '''PolyhedronData[..., "Volume"]''' command in Wolfram Research's [[Wolfram Mathematica|''Mathematica'']]. |
|||
<!--Many polynomials of degrees 5 and up have zeroes that are NOT related to the integers by radicals; this gives a reason some of the cells in the "exact volume" column are simply mentioned as such.--> |
|||
These volumes can be calculated from a set of vertex coordinates; such coordinates are known for all 92 Johnson solids. A conceptually simple approach is to [[Triangulation (geometry)|triangulate]] the surface of the solid (for example, by adding an extra point in the center of each non-triangular face) and choose some interior point as an "origin" so that the interior can be subdivided into irregular [[Tetrahedron|tetrahedra]]. Each tetrahedron has one vertex at the origin inside and three vertices on the surface. The volume of the solid is then the sum of the volumes of these tetrahedra. There is a simple formula for the [[Tetrahedron#Volume|volume of an irregular tetrahedron]]. |
|||
{| class="wikitable sortable" |
|||
|- |
|||
! J<sub>n</sub> |
|||
! Solid name |
|||
! V/a<sup>3</sup> (approx.) |
|||
! V/a<sup>3</sup> (exact) |
|||
|- |
|||
| 1 |
|||
| [[Square pyramid]] |
|||
| 0.235702260 |
|||
| <math>\frac{1}{3 \sqrt{2}}</math> |
|||
|- |
|||
| 2 |
|||
| [[Pentagonal pyramid]] |
|||
| 0.301502832 |
|||
| <math>\frac{1}{24} \left(5+\sqrt{5}\right)</math> |
|||
|- |
|||
| 3 |
|||
| [[Triangular cupola]] |
|||
| 1.178511302 |
|||
| <math>\frac{5}{3 \sqrt{2}}</math> |
|||
|- |
|||
| 4 |
|||
| [[Square cupola]] |
|||
| 1.942809042 |
|||
| <math>1+\frac{2 \sqrt{2}}{3}</math> |
|||
|- |
|||
| 5 |
|||
| [[Pentagonal cupola]] |
|||
| 2.324045318 |
|||
| <math>\frac{1}{6} \left(5+4 \sqrt{5}\right)</math> |
|||
|- |
|||
| 6 |
|||
| [[Pentagonal rotunda]] |
|||
| 6.917762968 |
|||
| <math>\frac{1}{12} \left(45+17 \sqrt{5}\right)</math> |
|||
|- |
|||
| 7 |
|||
| [[Elongated triangular pyramid]] |
|||
| 0.550863832 |
|||
| <math>\frac{1}{12} \left(\sqrt{2}+3 \sqrt{3}\right)</math> |
|||
|- |
|||
| 8 |
|||
| [[Elongated square pyramid]] |
|||
| 1.235702260 |
|||
| <math>\frac{1}{6} \left(6+\sqrt{2}\right)</math> |
|||
|- |
|||
| 9 |
|||
| [[Elongated pentagonal pyramid]] |
|||
| 2.021980233 |
|||
| <math>\frac{1}{24} \left(5+\sqrt{5}+6 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 10 |
|||
| [[Gyroelongated square pyramid]] |
|||
| 1.192702242 |
|||
| <math>\frac{1}{6} \left(\sqrt{2}+2 \sqrt{4+3 \sqrt{2}}\right)</math> |
|||
|- |
|||
| 11 |
|||
| [[Gyroelongated pentagonal pyramid]] |
|||
| 1.880192158 |
|||
| <math>\frac{1}{24} \left(25+9 \sqrt{5}\right)</math> |
|||
|- |
|||
| 12 |
|||
| [[Triangular dipyramid]] |
|||
| 0.235702260 |
|||
| <math>\frac{1}{3 \sqrt{2}}</math> |
|||
|- |
|||
| 13 |
|||
| [[Pentagonal dipyramid]] |
|||
| 0.603005665 |
|||
| <math>\frac{1}{12} \left(5+\sqrt{5}\right)</math> |
|||
|- |
|||
| 14 |
|||
| [[Elongated triangular dipyramid]] |
|||
| 0.668714962 |
|||
| <math>\frac{1}{12} \left(2 \sqrt{2}+3 \sqrt{3}\right)</math> |
|||
|- |
|||
| 15 |
|||
| [[Elongated square dipyramid]] |
|||
| 1.471404521 |
|||
| <math>\frac{1}{3} \left(3+\sqrt{2}\right)</math> |
|||
|- |
|||
| 16 |
|||
| [[Elongated pentagonal dipyramid]] |
|||
| 2.323483065 |
|||
| <math>\frac{1}{12} \left(5+\sqrt{5}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)</math> |
|||
|- |
|||
| 17 |
|||
| [[Gyroelongated square dipyramid]] |
|||
| 1.428404503 |
|||
| <math>\frac{1}{3} \left(\sqrt{2}+\sqrt{4+3 \sqrt{2}}\right)</math> |
|||
|- |
|||
| 18 |
|||
| [[Elongated triangular cupola]] |
|||
| 3.776587513 |
|||
| <math>\frac{1}{6} \left(5 \sqrt{2}+9 \sqrt{3}\right)</math> |
|||
|- |
|||
| 19 |
|||
| [[Elongated square cupola]] |
|||
| 6.771236166 |
|||
| <math>3+\frac{8 \sqrt{2}}{3}</math> |
|||
|- |
|||
| 20 |
|||
| [[Elongated pentagonal cupola]] |
|||
| 10.018254161 |
|||
| <math>\frac{1}{6} \left(5+4 \sqrt{5}+15 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 21 |
|||
| [[Elongated pentagonal rotunda]] |
|||
| 14.611971811 |
|||
| <math>\frac{1}{12} \left(45+17 \sqrt{5}+30 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 22 |
|||
| [[Gyroelongated triangular cupola]] |
|||
| 3.516053091 |
|||
| <math>\frac{1}{3} \sqrt{\frac{61}{2}+18 \sqrt{3}+30 \sqrt{1+\sqrt{3}}}</math> |
|||
|- |
|||
| 23 |
|||
| [[Gyroelongated square cupola]] |
|||
| 6.210765792 |
|||
| root of polynomial of degree 8 |
|||
|- |
|||
| 24 |
|||
| [[Gyroelongated pentagonal cupola]] |
|||
| 9.073333194 |
|||
| root of polynomial of degree 8 |
|||
|- |
|||
| 25 |
|||
| [[Gyroelongated pentagonal rotunda]] |
|||
| 13.667050844 |
|||
| root of polynomial of degree 8 |
|||
|- |
|||
| 26 |
|||
| [[Gyrobifastigium]] |
|||
| 0.866025404 |
|||
| <math>\frac{\sqrt{3}}{2}</math> |
|||
|- |
|||
| 27 |
|||
| [[Triangular orthobicupola]] |
|||
| 2.357022604 |
|||
| <math>\frac{5 \sqrt{2}}{3}</math> |
|||
|- |
|||
| 28 |
|||
| [[Square orthobicupola]] |
|||
| 3.885618083 |
|||
| <math>2+\frac{4 \sqrt{2}}{3}</math> |
|||
|- |
|||
| 29 |
|||
| [[Square gyrobicupola]] |
|||
| 3.885618083 |
|||
| <math>2+\frac{4 \sqrt{2}}{3}</math> |
|||
|- |
|||
| 30 |
|||
| [[Pentagonal orthobicupola]] |
|||
| 4.648090637 |
|||
| <math>\frac{1}{3} \left(5+4 \sqrt{5}\right)</math> |
|||
|- |
|||
| 31 |
|||
| [[Pentagonal gyrobicupola]] |
|||
| 4.648090637 |
|||
| <math>\frac{1}{3} \left(5+4 \sqrt{5}\right)</math> |
|||
|- |
|||
| 32 |
|||
| [[Pentagonal orthocupolarotunda]] |
|||
| 9.241808286 |
|||
| <math>\frac{5}{12} \left(11+5 \sqrt{5}\right)</math> |
|||
|- |
|||
| 33 |
|||
| [[Pentagonal gyrocupolarotunda]] |
|||
| 9.241808286 |
|||
| <math>\frac{5}{12} \left(11+5 \sqrt{5}\right)</math> |
|||
|- |
|||
| 34 |
|||
| [[Pentagonal orthobirotunda]] |
|||
| 13.835525936 |
|||
| <math>\frac{1}{6} \left(45+17 \sqrt{5}\right)</math> |
|||
|- |
|||
| 35 |
|||
| [[Elongated triangular orthobicupola]] |
|||
| 4.955098815 |
|||
| <math>\frac{5 \sqrt{2}}{3}+\frac{3 \sqrt{3}}{2}</math> |
|||
|- |
|||
| 36 |
|||
| [[Elongated triangular gyrobicupola]] |
|||
| 4.955098815 |
|||
| <math>\frac{5 \sqrt{2}}{3}+\frac{3 \sqrt{3}}{2}</math> |
|||
|- |
|||
| 37 |
|||
| [[Elongated square gyrobicupola]] |
|||
| 8.714045208 |
|||
| <math>4+\frac{10 \sqrt{2}}{3}</math> |
|||
|- |
|||
| 38 |
|||
| [[Elongated pentagonal orthobicupola]] |
|||
| 12.342299480 |
|||
| <math>\frac{1}{6} \left(10+8 \sqrt{5}+15 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 39 |
|||
| [[Elongated pentagonal gyrobicupola]] |
|||
| 12.342299480 |
|||
| <math>\frac{1}{6} \left(10+8 \sqrt{5}+15 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 40 |
|||
| [[Elongated pentagonal orthocupolarotunda]] |
|||
| 16.936017129 |
|||
| <math>\frac{5}{12} \left(11+5 \sqrt{5}+6 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 41 |
|||
| [[Elongated pentagonal gyrocupolarotunda]] |
|||
| 16.936017129 |
|||
| <math>\frac{5}{12} \left(11+5 \sqrt{5}+6 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 42 |
|||
| [[Elongated pentagonal orthobirotunda]] |
|||
| 21.529734779 |
|||
| <math>\frac{1}{6} \left(45+17 \sqrt{5}+15 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 43 |
|||
| [[Elongated pentagonal gyrobirotunda]] |
|||
| 21.529734779 |
|||
| <math>\frac{1}{6} \left(45+17 \sqrt{5}+15 \sqrt{5+2 \sqrt{5}}\right)</math> |
|||
|- |
|||
| 44 |
|||
| [[Gyroelongated triangular bicupola]] |
|||
| 4.694564393 |
|||
| <math>\frac{1}{3} \sqrt{68+18 \sqrt{3}+60 \sqrt{1+\sqrt{3}}}</math> |
|||
|- |
|||
| 45 |
|||
| [[Gyroelongated square bicupola]] |
|||
| 8.153574834 |
|||
| root of polynomial of degree 8 |
|||
|- |
|||
| 46 |
|||
| [[Gyroelongated pentagonal bicupola]] |
|||
| 11.397378512 |
|||
| root of polynomial of degree 8 |
|||
|- |
|||
| 47 |
|||
| [[Gyroelongated pentagonal cupolarotunda]] |
|||
| 15.991096162 |
|||
| root of polynomial of degree 8 |
|||
|- |
|||
| 48 |
|||
| [[Gyroelongated pentagonal birotunda]] |
|||
| 20.584813812 |
|||
| root of polynomial of degree 8 |
|||
|- |
|||
| 49 |
|||
| [[Augmented triangular prism]] |
|||
| 0.668714962 |
|||
| <math>\frac{1}{12} \left(2 \sqrt{2}+3 \sqrt{3}\right)</math> |
|||
|- |
|||
| 50 |
|||
| [[Biaugmented triangular prism]] |
|||
| 0.904417223 |
|||
| <math>\sqrt{\frac{59}{144}+\frac{1}{\sqrt{6}}}</math> |
|||
|- |
|||
| 51 |
|||
| [[Triaugmented triangular prism]] |
|||
| 1.140119483 |
|||
| <math>\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{4}</math> |
|||
|- |
|||
| 52 |
|||
| [[Augmented pentagonal prism]] |
|||
| 1.956179661 |
|||
| <math>\frac{1}{12} \sqrt{233+90 \sqrt{5}+12 \sqrt{50+20 \sqrt{5}}}</math> |
|||
|- |
|||
| 53 |
|||
| [[Biaugmented pentagonal prism]] |
|||
| 2.191881921 |
|||
| <math>\frac{1}{12} \sqrt{257+90 \sqrt{5}+24 \sqrt{50+20 \sqrt{5}}}</math> |
|||
|- |
|||
| 54 |
|||
| [[Augmented hexagonal prism]] |
|||
| 2.833778472 |
|||
| <math>\frac{1}{6} \left(\sqrt{2}+9 \sqrt{3}\right)</math> |
|||
|- |
|||
| 55 |
|||
| [[Parabiaugmented hexagonal prism]] |
|||
| 3.069480732 |
|||
| <math>\frac{1}{6} \left(2 \sqrt{2}+9 \sqrt{3}\right)</math> |
|||
|- |
|||
| 56 |
|||
| [[Metabiaugmented hexagonal prism]] |
|||
| 3.069480732 |
|||
| <math>\frac{1}{6} \left(2 \sqrt{2}+9 \sqrt{3}\right)</math> |
|||
|- |
|||
| 57 |
|||
| [[Triaugmented hexagonal prism]] |
|||
| 3.305182993 |
|||
| <math>\frac{1}{\sqrt{2}}+\frac{3 \sqrt{3}}{2}</math> |
|||
|- |
|||
| 58 |
|||
| [[Augmented dodecahedron]] |
|||
| 7.964621793 |
|||
| <math>\frac{1}{24} \left(95+43 \sqrt{5}\right)</math> |
|||
|- |
|||
| 59 |
|||
| [[Parabiaugmented dodecahedron]] |
|||
| 8.266124625 |
|||
| <math>\frac{1}{6} \left(25+11 \sqrt{5}\right)</math> |
|||
|- |
|||
| 60 |
|||
| [[Metabiaugmented dodecahedron]] |
|||
| 8.266124625 |
|||
| <math>\frac{1}{6} \left(25+11 \sqrt{5}\right)</math> |
|||
|- |
|||
| 61 |
|||
| [[Triaugmented dodecahedron]] |
|||
| 8.567627458 |
|||
| <math>\frac{5}{8} \left(7+3 \sqrt{5}\right)</math> |
|||
|- |
|||
| 62 |
|||
| [[Metabidiminished icosahedron]] |
|||
| 1.578689326 |
|||
| <math>\frac{1}{6} \left(5+2 \sqrt{5}\right)</math> |
|||
|- |
|||
| 63 |
|||
| [[Tridiminished icosahedron]] |
|||
| 1.277186493 |
|||
| <math>\frac{5}{8}+\frac{7 \sqrt{5}}{24}</math> |
|||
|- |
|||
| 64 |
|||
| [[Augmented tridiminished icosahedron]] |
|||
| 1.395037624 |
|||
| <math>\frac{1}{24} \left(15+2 \sqrt{2}+7 \sqrt{5}\right)</math> |
|||
|- |
|||
| 65 |
|||
| [[Augmented truncated tetrahedron]] |
|||
| 3.889087297 |
|||
| <math>\frac{11}{2 \sqrt{2}}</math> |
|||
|- |
|||
| 66 |
|||
| [[Augmented truncated cube]] |
|||
| 15.542472333 |
|||
| <math>8+\frac{16 \sqrt{2}}{3}</math> |
|||
|- |
|||
| 67 |
|||
| [[Biaugmented truncated cube]] |
|||
| 17.485281374 |
|||
| <math>9+6 \sqrt{2}</math> |
|||
|- |
|||
| 68 |
|||
| [[Augmented truncated dodecahedron]] |
|||
| 87.363709878 |
|||
| <math>\frac{505}{12}+\frac{81 \sqrt{5}}{4}</math> |
|||
|- |
|||
| 69 |
|||
| [[Parabiaugmented truncated dodecahedron]] |
|||
| 89.687755196 |
|||
| <math>\frac{1}{12} \left(515+251 \sqrt{5}\right)</math> |
|||
|- |
|||
| 70 |
|||
| [[Metabiaugmented truncated dodecahedron]] |
|||
| 89.687755196 |
|||
| <math>\frac{1}{12} \left(515+251 \sqrt{5}\right)</math> |
|||
|- |
|||
| 71 |
|||
| [[Triaugmented truncated dodecahedron]] |
|||
| 92.011800514 |
|||
| <math>\frac{7}{12} \left(75+37 \sqrt{5}\right)</math> |
|||
|- |
|||
| 72 |
|||
| [[Gyrate rhombicosidodecahedron]] |
|||
| 41.615323782 |
|||
| <math>20+\frac{29 \sqrt{5}}{3}</math> |
|||
|- |
|||
| 73 |
|||
| [[Parabigyrate rhombicosidodecahedron]] |
|||
| 41.615323782 |
|||
| <math>20+\frac{29 \sqrt{5}}{3}</math> |
|||
|- |
|||
| 74 |
|||
| [[Metabigyrate rhombicosidodecahedron]] |
|||
| 41.615323782 |
|||
| <math>20+\frac{29 \sqrt{5}}{3}</math> |
|||
|- |
|||
| 75 |
|||
| [[Trigyrate rhombicosidodecahedron]] |
|||
| 41.615323782 |
|||
| <math>20+\frac{29 \sqrt{5}}{3}</math> |
|||
|- |
|||
| 76 |
|||
| [[Diminished rhombicosidodecahedron]] |
|||
| 39.291278464 |
|||
| <math>\frac{115}{6}+9 \sqrt{5}</math> |
|||
|- |
|||
| 77 |
|||
| [[Paragyrate diminished rhombicosidodecahedron]] |
|||
| 39.291278464 |
|||
| <math>\frac{115}{6}+9 \sqrt{5}</math> |
|||
|- |
|||
| 78 |
|||
| [[Metagyrate diminished rhombicosidodecahedron]] |
|||
| 39.291278464 |
|||
| <math>\frac{115}{6}+9 \sqrt{5}</math> |
|||
|- |
|||
| 79 |
|||
| [[Bigyrate diminished rhombicosidodecahedron]] |
|||
| 39.291278464 |
|||
| <math>\frac{115}{6}+9 \sqrt{5}</math> |
|||
|- |
|||
| 80 |
|||
| [[Parabidiminished rhombicosidodecahedron]] |
|||
| 36.967233146 |
|||
| <math>\frac{5}{3} \left(11+5 \sqrt{5}\right)</math> |
|||
|- |
|||
| 81 |
|||
| [[Metabidiminished rhombicosidodecahedron]] |
|||
| 36.967233146 |
|||
| <math>\frac{5}{3} \left(11+5 \sqrt{5}\right)</math> |
|||
|- |
|||
| 82 |
|||
| [[Gyrate bidiminished rhombicosidodecahedron]] |
|||
| 36.967233146 |
|||
| <math>\frac{5}{3} \left(11+5 \sqrt{5}\right)</math> |
|||
|- |
|||
| 83 |
|||
| [[Tridiminished rhombicosidodecahedron]] |
|||
| 34.643187827 |
|||
| <math>\frac{35}{2}+\frac{23 \sqrt{5}}{3}</math> |
|||
|- |
|||
| 84 |
|||
| [[Snub disphenoid]] |
|||
| 0.859493646 |
|||
| root of polynomial of degree 6 |
|||
|- |
|||
| 85 |
|||
| [[Snub square antiprism]] |
|||
| 3.601222010 |
|||
| root of polynomial of degree 12 |
|||
|- |
|||
| 86 |
|||
| [[Sphenocorona]] |
|||
| 1.515351640 |
|||
| <math>\frac{1}{2} \sqrt{1+3 \sqrt{\frac{3}{2}}+\sqrt{13+3 \sqrt{6}}}</math> |
|||
|- |
|||
| 87 |
|||
| [[Augmented sphenocorona]] |
|||
| 1.751053900 |
|||
| root of polynomial of degree 16 |
|||
|- |
|||
| 88 |
|||
| [[Sphenomegacorona]] |
|||
| 1.948108229 |
|||
| root of polynomial of degree 32 |
|||
|- |
|||
| 89 |
|||
| [[Hebesphenomegacorona]] |
|||
| 2.912910415 |
|||
| root of polynomial of degree 20 |
|||
|- |
|||
| 90 |
|||
| [[Disphenocingulum]] |
|||
| 3.777645342 |
|||
| root of polynomial of degree 24 |
|||
|- |
|||
| 91 |
|||
| [[Bilunabirotunda]] |
|||
| 3.093717650 |
|||
| <math>\frac{1}{12} \left(17+9 \sqrt{5}\right)</math> |
|||
|- |
|||
| 92 |
|||
| [[Triangular hebesphenorotunda]] |
|||
| 5.108745974 |
|||
| <math>\frac{5}{2}+\frac{7 \sqrt{5}}{6}</math> |
|||
|} |
|||
{{Back to top2}} |
|||
For a fixed edge length, the square pyramid {{math|''J''<sub>1</sub>}} and the triangular dipyramid {{math|''J''<sub>12</sub>}} have the smallest volume and the triaugmented truncated dodecahedron {{math|''J''<sub>71</sub>}} has the largest, more than 390 times larger. |
|||
Thirteen of the 92 Johnson solids have volumes for which {{math|''V''/''a''<sup>3</sup>}} is not a number expressible using [[Nth root|radicals]]. |
|||
These values are the greatest [[Real number|real]] [[Zero of a function|root]] of the following [[polynomial]]s. |
|||
{| class="wikitable sortable" |
|||
|- |
|||
! J<sub>n</sub> |
|||
! Polynomial |
|||
|- |
|||
| 23 |
|||
| |
|||
{{math| |
|||
6561 ''x''<sup>8</sup> |
|||
− 52488 ''x''<sup>7</sup> |
|||
+ 113724 ''x''<sup>6</sup> |
|||
− 9720 ''x''<sup>5</sup> |
|||
<br> |
|||
− 1616922 ''x''<sup>4</sup> |
|||
+ 396360 ''x''<sup>3</sup> |
|||
+ 1537020 ''x''<sup>2</sup> |
|||
− 178632 ''x'' |
|||
− 3391}} |
|||
|- |
|||
| 24 |
|||
| |
|||
{{math| |
|||
1679616 ''x''<sup>8</sup> |
|||
− 11197440 ''x''<sup>7</sup> |
|||
+ 27060480 ''x''<sup>6</sup> |
|||
+ 35769600 ''x''<sup>5</sup> |
|||
<br> |
|||
− 4456749600 ''x''<sup>4</sup> |
|||
− 10714248000 ''x''<sup>3</sup> |
|||
+ 3828402000 ''x''<sup>2</sup> |
|||
+ 13859430000 ''x'' |
|||
+ 5340175625}} |
|||
|- |
|||
| 25 |
|||
| |
|||
{{math| |
|||
1679616 ''x''<sup>8</sup> |
|||
− 50388480 ''x''<sup>7</sup> |
|||
+ 603262080 ''x''<sup>6</sup> |
|||
− 3520972800 ''x''<sup>5</sup> |
|||
<br> |
|||
+ 5215460400 ''x''<sup>4</sup> |
|||
+ 4128624000 ''x''<sup>3</sup> |
|||
− 8894943000 ''x''<sup>2</sup> |
|||
+ 3881385000 ''x'' |
|||
− 424924375}} |
|||
|- |
|||
| 45 |
|||
| |
|||
{{math| |
|||
6561 ''x''<sup>8</sup> |
|||
− 104976 ''x''<sup>7</sup> |
|||
+ 594864 ''x''<sup>6</sup> |
|||
− 1384128 ''x''<sup>5</sup> |
|||
<br> |
|||
− 552096 ''x''<sup>4</sup> |
|||
+ 1569024 ''x''<sup>3</sup> |
|||
+ 246528 ''x''<sup>2</sup> |
|||
− 119808 ''x'' |
|||
+ 4352}} |
|||
|- |
|||
| 46 |
|||
| |
|||
{{math| |
|||
6561 ''x''<sup>8</sup> |
|||
− 87480 ''x''<sup>7</sup> |
|||
+ 313470 ''x''<sup>6</sup> |
|||
+ 753300 ''x''<sup>5</sup> |
|||
<br> |
|||
− 22424850 ''x''<sup>4</sup> |
|||
− 84591000 ''x''<sup>3</sup> |
|||
− 85909500 ''x''<sup>2</sup> |
|||
+ 8715000 ''x'' |
|||
+ 35547500}} |
|||
|- |
|||
| 47 |
|||
| |
|||
{{math| |
|||
1679616 ''x''<sup>8</sup> |
|||
− 61585920 ''x''<sup>7</sup> |
|||
+ 851472000 ''x''<sup>6</sup> |
|||
− 5108832000 ''x''<sup>5</sup> |
|||
<br> |
|||
+ 4745790000 ''x''<sup>4</sup> |
|||
+ 21346200000 ''x''<sup>3</sup> |
|||
− 29019375000 ''x''<sup>2</sup> |
|||
− 4576875000 ''x'' |
|||
− 405859375}} |
|||
|- |
|||
| 48 |
|||
| |
|||
{{math| |
|||
6561 ''x''<sup>8</sup> |
|||
− 393660 ''x''<sup>7</sup> |
|||
+ 9316620 ''x''<sup>6</sup> |
|||
− 108207900 ''x''<sup>5</sup> |
|||
<br> |
|||
+ 601832025 ''x''<sup>4</sup> |
|||
− 1417189500 ''x''<sup>3</sup> |
|||
+ 965841750 ''x''<sup>2</sup> |
|||
+ 597667500 ''x'' |
|||
− 668786875}} |
|||
|- |
|||
| 84 |
|||
| |
|||
{{math| |
|||
5832 ''x''<sup>6</sup> |
|||
− 1377 ''x''<sup>4</sup> |
|||
− 2160 ''x''<sup>2</sup> |
|||
− 4}} |
|||
|- |
|||
| 85 |
|||
| |
|||
{{math| |
|||
531441 ''x''<sup>12</sup> |
|||
− 85726026 ''x''<sup>8</sup> |
|||
− 48347280 ''x''<sup>6</sup> |
|||
<br> |
|||
+ 11588832 ''x''<sup>4</sup> |
|||
+ 4759488 ''x''<sup>2</sup> |
|||
− 892448}} |
|||
|- |
|||
| 87 |
|||
| |
|||
{{math| |
|||
45137758519296 ''x''<sup>16</sup> |
|||
− 110336743047168 ''x''<sup>14</sup> |
|||
− 191069246324736 ''x''<sup>12</sup> |
|||
+ 209269081571328 ''x''<sup>10</sup> |
|||
<br> |
|||
+ 364547659290624 ''x''<sup>8</sup> |
|||
− 58793017190400 ''x''<sup>6</sup> |
|||
+ 3306865979520 ''x''<sup>4</sup> |
|||
− 1275399855936 ''x''<sup>2</sup> |
|||
+ 1439671249}} |
|||
|- |
|||
| 88 |
|||
| |
|||
{{math| |
|||
521578814501447328359509917696 ''x''<sup>32</sup> |
|||
− 985204427391622731345740955648 ''x''<sup>30</sup> |
|||
<br> |
|||
− 16645447351681991898880656015360 ''x''<sup>28</sup> |
|||
+ 79710816694053483249372512649216 ''x''<sup>26</sup> |
|||
<br> |
|||
− 152195045391070538203422101864448 ''x''<sup>24</sup> |
|||
+ 156280253448056209478031589244928 ''x''<sup>22</sup> |
|||
<br> |
|||
− 96188116617075838858708654227456 ''x''<sup>20</sup> |
|||
+ 30636368373570166303441645731840 ''x''<sup>18</sup> |
|||
<br> |
|||
+ 5828527077458909552923002273792 ''x''<sup>16</sup> |
|||
− 8060049780765551057159394951168 ''x''<sup>14</sup> |
|||
<br> |
|||
+ 1018074792115156107372011716608 ''x''<sup>12</sup> |
|||
+ 35220131544370794950945931264 ''x''<sup>10</sup> |
|||
<br> |
|||
+ 327511698517355918956755959808 ''x''<sup>8</sup> |
|||
− 116978732884218191486738706432 ''x''<sup>6</sup> |
|||
<br> |
|||
+ 10231563774949176791703149568 ''x''<sup>4</sup> |
|||
− 366323949299263261553952192 ''x''<sup>2</sup> |
|||
<br> |
|||
+ 3071435678740442112675625}} |
|||
|- |
|||
| 89 |
|||
| |
|||
{{math| |
|||
47330370277129322496 ''x''<sup>20</sup> |
|||
− 722445512980071186432 ''x''<sup>18</sup> |
|||
<br> |
|||
+ 3596480447590271287296 ''x''<sup>16</sup> |
|||
− 8432333285523990773760 ''x''<sup>14</sup> |
|||
<br> |
|||
+ 8973584611317745975296 ''x''<sup>12</sup> |
|||
− 3065290664181478981632 ''x''<sup>10</sup> |
|||
<br> |
|||
+ 366229890219212144640 ''x''<sup>8</sup> |
|||
− 8337259437908852736 ''x''<sup>6</sup> |
|||
<br> |
|||
− 22211277300912896 ''x''<sup>4</sup> |
|||
+ 132615435213216 ''x''<sup>2</sup> |
|||
<br> |
|||
+ 2693461945329}} |
|||
|- |
|||
| 90 |
|||
| |
|||
{{math| |
|||
1213025622610333925376 ''x''<sup>24</sup> |
|||
+ 54451372392730545094656 ''x''<sup>22</sup> |
|||
<br> |
|||
− 796837093078664749252608 ''x''<sup>20</sup> |
|||
− 4133410366404688544268288 ''x''<sup>18</sup> |
|||
<br> |
|||
+ 20902529024429842816303104 ''x''<sup>16</sup> |
|||
− 133907540390420673677230080 ''x''<sup>14</sup> |
|||
<br> |
|||
+ 246234688242991598853881856 ''x''<sup>12</sup> |
|||
− 63327534106871321714442240 ''x''<sup>10</sup> |
|||
<br> |
|||
+ 14389309497459555704164608 ''x''<sup>8</sup> |
|||
+ 48042947402464500749392128 ''x''<sup>6</sup> |
|||
<br> |
|||
− 5891096640600351061013664 ''x''<sup>4</sup> |
|||
− 3212114716816853362953264 ''x''<sup>2</sup> |
|||
+ 479556973248657693884401}} |
|||
|} |
|||
{{Back to top2}} |
|||
==Inradius, midradius, and circumradius== |
|||
The following table lists the [[radius]] {{math|''R''<sub>i</sub>}} of the [[inscribed sphere|insphere]], the radius {{math|''R''<sub>m</sub>}} of the [[midsphere]], and the radius {{math|''R''<sub>c</sub>}} of the [[circumsphere]], each divided by the [[Edge (geometry)|edge]] length {{mvar|a}}, when these [[sphere]]s exist. |
|||
A [[polyhedron]] does not necessarily have an insphere, or a midsphere, or a circumsphere. For example, it does not have a circumsphere unless all its [[Vertex (geometry)|vertices]] lie on some sphere. The Johnson solids, having less [[Symmetry (geometry)|symmetry]] than, say, the [[Platonic solid]]s, lack many of these spheres. Only {{math|''J''<sub>1</sub>}} and {{math|''J''<sub>2</sub>}} possess all three of these spheres. |
|||
The source for this table is the '''PolyhedronData[..., "Inradius"]''', '''PolyhedronData[..., "Midradius"]''', and '''PolyhedronData[..., "Circumradius"]''' commands in Wolfram Research's [[Wolfram Mathematica|''Mathematica'']]. The output has been simplified to a consistent form in terms of [[Nth root|radicals]]. |
|||
{| class="wikitable sortable" |
|||
|- |
|||
! J<sub>n</sub> |
|||
! R<sub>i</sub>/a (approx.) |
|||
! R<sub>i</sub>/a (exact) |
|||
! R<sub>m</sub>/a (approx.) |
|||
! R<sub>m</sub>/a (exact) |
|||
! R<sub>c</sub>/a (approx.) |
|||
! R<sub>c</sub>/a (exact) |
|||
|- |
|||
| 1 |
|||
| 0.258819045 |
|||
| <math>\frac{\sqrt{2-\sqrt{3}}}{2}</math> |
|||
| 0.500000000 |
|||
| <math>\frac{1}{2}</math> |
|||
| 0.707106781 |
|||
| <math>\frac{1}{\sqrt{2}}</math> |
|||
|- |
|||
| 2 |
|||
| 0.232788309 |
|||
| <math>\frac{1}{\sqrt{25-7 \sqrt{5}+\sqrt{30 \left(5-\sqrt{5}\right)}}}</math> |
|||
| 0.809016994 |
|||
| <math>\frac{1}{4} \left(1+\sqrt{5}\right)</math> |
|||
| 0.951056516 |
|||
| <math>\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math> |
|||
|- |
|||
| 3 |
|||
| - |
|||
| - |
|||
| 0.866025404 |
|||
| <math>\frac{\sqrt{3}}{2}</math> |
|||
| 1.000000000 |
|||
| <math>1</math> |
|||
|- |
|||
| 4 |
|||
| - |
|||
| - |
|||
| 1.306562965 |
|||
| <math>\sqrt{1+\frac{1}{\sqrt{2}}}</math> |
|||
| 1.398966326 |
|||
| <math>\sqrt{\frac{5}{4}+\frac{1}{\sqrt{2}}}</math> |
|||
|- |
|||
| 5 |
|||
| - |
|||
| - |
|||
| 2.176250899 |
|||
| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
|||
| 2.232950509 |
|||
| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
|||
|- |
|||
| 6 |
|||
| - |
|||
| - |
|||
| 1.538841769 |
|||
| <math>\frac{1}{2} \sqrt{5+2 \sqrt{5}}</math> |
|||
| 1.618033989 |
|||
| <math>\frac{1}{2} \left(1+\sqrt{5}\right)</math> |
|||
|- |
|||
| 7 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 8 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 9 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 10 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 11 |
|||
| - |
|||
| - |
|||
| 0.809016994 |
|||
| <math>\frac{1}{4} \left(1+\sqrt{5}\right)</math> |
|||
| 0.951056516 |
|||
| <math>\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math> |
|||
|- |
|||
| 12 |
|||
| 0.272165527 |
|||
| <math>\frac{\sqrt{\frac{2}{3}}}{3}</math> |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 13 |
|||
| 0.417774579 |
|||
| <math>\sqrt{\frac{1}{30} \left(3+\sqrt{5}\right)}</math> |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 14 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 15 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 16 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 17 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 18 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 19 |
|||
| - |
|||
| - |
|||
| 1.306562965 |
|||
| <math>\sqrt{1+\frac{1}{\sqrt{2}}}</math> |
|||
| 1.398966326 |
|||
| <math>\sqrt{\frac{5}{4}+\frac{1}{\sqrt{2}}}</math> |
|||
|- |
|||
| 20 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 21 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 22 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 23 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 24 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 25 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 26 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 27 |
|||
| - |
|||
| - |
|||
| 0.866025404 |
|||
| <math>\frac{\sqrt{3}}{2}</math> |
|||
| 1.000000000 |
|||
| <math>1</math> |
|||
|- |
|||
| 28 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 29 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 30 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 31 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 32 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 33 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 34 |
|||
| - |
|||
| - |
|||
| 1.538841769 |
|||
| <math>\frac{1}{2} \sqrt{5+2 \sqrt{5}}</math> |
|||
| 1.618033989 |
|||
| <math>\frac{1}{2} \left(1+\sqrt{5}\right)</math> |
|||
|- |
|||
| 35 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 36 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 37 |
|||
| - |
|||
| - |
|||
| 1.306562965 |
|||
| <math>\sqrt{1+\frac{1}{\sqrt{2}}}</math> |
|||
| 1.398966326 |
|||
| <math>\sqrt{\frac{5}{4}+\frac{1}{\sqrt{2}}}</math> |
|||
|- |
|||
| 38 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
|||
| 39 |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
| - |
|||
|- |
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| <math>\frac{1}{4} \left(1+\sqrt{5}\right)</math> |
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| <math>\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math> |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 73 |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 76 |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 77 |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 78 |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 79 |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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| 80 |
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| - |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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|- |
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| 81 |
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| - |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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|- |
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| 82 |
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| - |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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|- |
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| 83 |
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| 2.176250899 |
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| <math>\sqrt{\frac{5}{2}+\sqrt{5}}</math> |
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| 2.232950509 |
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| <math>\frac{1}{2} \sqrt{11+4 \sqrt{5}}</math> |
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{{Back to top2}} |
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==References== |
== References == |
||
{{ |
{{refbegin|30em}} |
||
* {{cite journal |
|||
* [[Norman Johnson (mathematician)|Norman W. Johnson]], "Convex Solids with Regular Faces", Canadian Journal of Mathematics, '''18''', 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. |
|||
| last = Berman | first = Martin |
|||
* {{cite book|author=Victor A. Zalgaller|author-link=Victor Zalgaller|title=Convex Polyhedra with Regular Faces|publisher=Consultants Bureau|year=1969|id=No ISBN}} The first proof that there are only 92 Johnson solids. |
|||
| doi = 10.1016/0016-0032(71)90071-8 |
|||
| journal = Journal of the Franklin Institute |
|||
| mr = 290245 |
|||
| pages = 329–352 |
|||
| title = Regular-faced convex polyhedra |
|||
| volume = 291 |
|||
| year = 1971| issue = 5 |
|||
}} |
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* {{cite conference |
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| last1 = Boissonnat | first1= J. D. |
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| last2 = Yvinec | first2 = M. |
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| date = June 1989 |
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| title = Probing a scene of non convex polyhedra |
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| journal = Proceedings of the fifth annual symposium on Computational geometry |
|||
| pages = 237–246 |
|||
| doi = 10.1145/73833.73860 |
|||
}} |
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* {{cite book |
|||
| last = Cromwell | first = Peter R. |
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| title = Polyhedra |
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| year = 1997 |
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| url = https://archive.org/details/polyhedra0000crom |
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| publisher = [[Cambridge University Press]] |
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}} |
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* {{cite book |
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| last = Diudea | first = M. V. |
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| year = 2018 |
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| title = Multi-shell Polyhedral Clusters |
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| publisher = Springer |
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| isbn = 978-3-319-64123-2 |
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| doi = 10.1007/978-3-319-64123-2 |
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}} |
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* {{cite book |
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| last1 = Flusser | first1 = Jan |
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| last2 = Suk | first2 = Tomas |
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| last3 = Zitofa | first3 = Barbara |
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| year = 2017 |
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| title = 2D and 3D Image Analysis by Moments |
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| publisher = John & Sons Wiley |
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}} |
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* {{cite book |
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| last1 = Helbert | first1 = Wolfram |
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| last2 = Geilhufe | first2 = Matthias |
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| year = 2018 |
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| title = Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica |
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| publisher = John & Sons Wiley |
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| isbn = 978-3-527-41300-3 |
|||
}} |
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* {{cite book |
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| last = Holme | first = Audun |
|||
| year = 2010 |
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| title = Geometry: Our Cultural Heritage |
|||
| publisher = Springer |
|||
| url = https://books.google.com/books?id=zXwQGo8jyHUC |
|||
| isbn = 978-3-642-14441-7 |
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| doi = 10.1007/978-3-642-14441-7 |
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}} |
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* {{cite journal |
|||
| last = Johnson | first = Norman | authorlink = Norman Johnson (mathematician) |
|||
| title = Convex Solids with Regular Faces |
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| journal = Canadian Journal of Mathematics |
|||
| volume = 18 |
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| year = 1966 |
|||
| pages = 169–200 |
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| doi = 10.4153/CJM-1966-021-8 |
|||
}} |
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* {{cite journal |
|||
| last = Litchenberg | first = Dorovan R. |
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| year = 1988 |
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| title = Pyramids, Prisms, Antiprisms, and Deltahedra |
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| journal = The Mathematics Teacher |
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| volume = 81 |
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| issue = 4 |
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| pages = 261-265 |
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| jstor = 27965792 |
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}} |
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* {{cite book |
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| last = Meyer | first = W. |
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| year = 2006 |
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| title = Geometry and Its Applications |
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| publisher = Academic Press |
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| url = https://books.google.com/books?id=ez6H5Ho6E3cC |
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| isbn = 978-0-12-369427-0 |
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}} |
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* {{cite book |
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| last = Parker | first = Sybil P. |
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| year = 1997 |
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| title = Dictionary of Mathematics |
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| publisher = McGraw-Hill |
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}} |
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* {{cite book |
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| last = Powell | first = Richard C. |
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| year = 2010 |
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| title = Symmetry, Group Theory, and the Physical Properties of Crystals |
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| publisher = Springer |
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| isbn = 978-1-4419-7598-0 |
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| doi = 10.1007/978-1-4419-7598-0 |
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}} |
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* {{cite book |
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| last = Rajwade | first = A. R. |
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| title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem |
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| series = Texts and Readings in Mathematics |
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| year = 2001 |
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| publisher = Hindustan Book Agency |
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| isbn = 978-93-86279-06-4 |
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| doi = 10.1007/978-93-86279-06-4 |
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}} |
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* {{cite book |
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| last = Solomon | first = Ronald |
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| year = 2003 |
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| title = Abstract Algebra |
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| publisher = American Mathematical Society |
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| url = https://books.google.com/books?id=ouvZKQiykf4C |
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| isbn = 978-0-8218-4795-4 |
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}} |
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* {{cite journal |
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| last1 = Slobodan | first1 = Mišić |
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| last2 = Obradović | first2 = Marija |
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| last3 = Ðukanović | first3 = Gordana |
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| title = Composite Concave Cupolae as Geometric and Architectural Forms |
|||
| year = 2015 |
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| journal = Journal for Geometry and Graphics |
|||
| volume = 19 |
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| issue = 1 |
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| pages = 79–91 |
|||
| url = https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf |
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}} |
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* {{cite book |
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| last = Todesco | first = Gian Marco |
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| editor-last1 = Emmer | editor-first1 = Michele |
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| editor-last2 = Abate | editor-first2 = Marco |
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| year = 2020 |
|||
| contribution = Hyperbolic Honeycomb |
|||
| title = Imagine Math 7: Between Culture and Mathematics |
|||
| publisher = Springer |
|||
| doi = 10.1007/978-3-030-42653-8 |
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| isbn = 978-3-030-42653-8 |
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}} |
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* {{cite book |
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| last = Uehara | first = Ryuhei |
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| year = 2020 |
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| title = Introduction to Computational Origami: The World of New Computational Geometry |
|||
| url = https://books.google.com/books?id=51juDwAAQBAJ |
|||
| publisher = Springer |
|||
| isbn = 978-981-15-4470-5 |
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| doi = 10.1007/978-981-15-4470-5 |
|||
}} |
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* {{cite book |
|||
| last = Walsh | first = Edward T. |
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| year = 2014 |
|||
| title = A First Course in Geometry |
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| publisher = Dover Publications |
|||
| url = https://books.google.com/books?id=ZhDdAwAAQBAJ |
|||
| isbn = 978-0-486-78020-7 |
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}} |
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* {{cite book |
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| last1 = Williams | first1 = Kim |
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| last2 = Monteleone | first2 = Cosino |
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| year = 2021 |
|||
| title = Daniele Barbaro’s Perspective of 1568 |
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| publisher = Springer |
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| isbn = 978-3-030-76687-0 |
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| doi = 10.1007/978-3-030-76687-0 |
|||
}} |
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* {{cite book |
|||
| last = Zalgaller | first = Victor A. | author-link = Victor Zalgaller |
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| title = Convex Polyhedra with Regular Faces |
|||
| publisher = Consultants Bureau |
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| year = 1969 |
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}} |
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{{refend}} |
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==External links== |
==External links== |
Revision as of 02:17, 29 January 2024
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (January 2024) |
In geometry, polyhedra is a three-dimensional object with lines meeting at a point that forms polygons. The points, lines, and polygons of polyhedra are respectively known as the vertices, edges, and faces.[1] A polyhedron is said to be convex if, for every two points inside the polyhedron, there is a line connecting them that lies within the polyhedra as well;[2] its faces are not coplanar (meaning every face are not in the same plane) and its edges are not colinear (meaning the edges are not in the same line).[3] A polyhedron is said to be regular if every polygonal faces are equilateral and equiangular,[4] and those with the polyhedron has vertex-transitive property are called a uniform polyhedron.[5] A Johnson solid (or Johnson–Zalgaller solid) is a convex polyhedron with its faces are regular polygons. Some authors do not require that the Johnson solid not be uniform, meaning that the Johnson solids may not be Platonic solid, Archimedean solid, prism, or antiprism.[6]
The 92 convex polyhedrons were published by Norman Johnson, conjecturing that there are no other solids. His conjecture was proved by Victor Zalgaller proved in 1969 that Johnson's list was complete.[7] Pyramids, cupolae, and rotunda are the first six Johnson solids that have regular faces and convexity. These solids may be applied to construct another polyhedron that has the same properties, a process known as augmentation; attaching prism or antiprism to those is known as elongation or gyroelongation, respectively. Some others may be constructed by diminishment, the removal of those from the component of polyhedra, or by snubification, a construction by cutting loose the edges, lifting the faces and rotate in certain angle, after which adding the equilateral triangles between them.[8]
Every polyhedra has own characteristics, including symmetry and measurement. An object is said to be symmetrical if there is such transformation preserving the immunity to change. All of those transformations may be composed in a concept of group, alongside the number of elements, known as order. In two-dimensional space, these transformations include rotating around the center of a polygon and reflecting an object around the perpendicular bisector of a polygon. A polygon that is rotated symmetrically in is denoted by Cn, a cyclic group of order n; combining with the reflection symmetry results in the symmetry of dihedral group Dn of order 2n.[9] In three-dimensional symmetry point groups, the transformation of polyhedra's symmetry includes the rotation around the line passing through the base center, known as axis of symmetry, and reflection relative to perpendicular planes passing through the bisector of a base; this is known as the pyramidal symmetry Cnv of order 2n. Relatedly, polyhedra that preserve their symmetry by rotating it horizontally in are known as prismatic symmetry Dnv of order 2n. The antiprismatic symmetry Dnd of order 4n preserving the symmetry by rotating its half bottom and reflection across the horizontal plane.[10] The symmetry group Cnh of order 2n preserve the symmetry by rotation around the axis of symmetry and reflection on horizontal plane; one case that preserves the symmetry by one full rotation and one reflection horizontal plane is C1h of order 2, or simply denoted as Cs.[11] The mensuration of polyhedra includes the surface area and volume. An area is a two-dimensional measurement calculated by the product of length and width, and the surface area is the overall area of all faces of polyhedra that is measured by summing all of them.[12] A volume is a measurement of the region in three-dimensional space.[13]
The following table contains the 92 Johnson solids of the edge length a. Each of the columns includes the enumeration of Johnson solid (Jn),[14] the number of vertices, edges, and faces, symmetry, surface area A and volume V.
Jn | Solid name | Image | Vertices | Edges | Faces | Symmetry group and its order[15] | Surface area and volume[16] |
---|---|---|---|---|---|---|---|
1 | Equilateral square pyramid |
5 | 8 | 5 | C4v of order 8 | ||
2 | Pentagonal pyramid |
6 | 10 | 6 | C5v of order 10 | ||
3 | Triangular cupola |
9 | 15 | 8 | C3v of order 6 | ||
4 | Square cupola |
12 | 20 | 10 | C4v of order 8 | ||
5 | Pentagonal cupola |
15 | 25 | 12 | C5v of order 10 | ||
6 | Pentagonal rotunda |
20 | 35 | 17 | C5v of order 10 | ||
7 | Elongated triangular pyramid |
7 | 12 | 7 | C3v of order 6 | ||
8 | Elongated square pyramid |
9 | 16 | 9 | C4v of order 8 | ||
9 | Elongated pentagonal pyramid |
11 | 20 | 11 | C5v of order 10 | ||
10 | Gyroelongated square pyramid |
9 | 20 | 13 | C4v of order 8 | ||
11 | Gyroelongated pentagonal pyramid |
11 | 25 | 16 | C5v of order 10 | ||
12 | Triangular bipyramid |
5 | 9 | 6 | D3h of order 12 | ||
13 | Pentagonal bipyramid |
7 | 15 | 10 | D5h of order 20 | ||
14 | Elongated triangular bipyramid |
8 | 15 | 9 | D3h of order 12 | ||
15 | Elongated square bipyramid |
10 | 20 | 12 | D4h of order 16 | ||
16 | Elongated pentagonal bipyramid |
12 | 25 | 15 | D5h of order 20 | ||
17 | Elongated square bipyramid |
10 | 24 | 16 | D4d of order 16 | ||
18 | Elongated triangular cupola |
15 | 27 | 14 | C3v of order 6 | ||
19 | Elongated square cupola |
20 | 36 | 18 | C4v of order 8 | ||
20 | Elongated pentagonal cupola |
25 | 45 | 22 | C5v of order 10 | ||
21 | Elongated pentagonal rotunda |
30 | 55 | 27 | C5v of order 10 | ||
22 | Gyroelongated triangular cupola |
15 | 33 | 20 | C3v of order 6 | ||
23 | Gyroelongated square cupola |
20 | 44 | 26 | C4v of order 8 | ||
24 | Gyroelongated pentagonal cupola |
25 | 55 | 32 | C5v of order 10 | ||
25 | Gyroelongated pentagonal rotunda |
30 | 65 | 37 | C5v of order 10 | ||
26 | Gyrobifastigium | 8 | 14 | 8 | D2d of order 8 | ||
27 | Triangular orthobicupola |
12 | 24 | 14 | D3h of order 12 | ||
28 | Square orthobicupola |
16 | 32 | 18 | D4h of order 16 | ||
29 | Square gyrobicupola |
16 | 32 | 18 | D4d of order 16 | ||
30 | Pentagonal orthobicupola |
20 | 40 | 22 | D5h of order 20 | ||
31 | Pentagonal gyrobicupola |
20 | 40 | 22 | D5d of order 20 | ||
32 | Pentagonal orthocupolarotunda |
25 | 50 | 27 | C5v of order 10 | ||
33 | Pentagonal gyrocupolarotunda |
25 | 50 | 27 | C5v of order 10 | ||
34 | Pentagonal orthobirotunda |
30 | 60 | 32 | D5h of order 20 | ||
35 | Elongated triangular orthobicupola |
18 | 36 | 20 | D3h of order 12 | ||
36 | Elongated triangular gyrobicupola |
18 | 36 | 20 | D3d of order 12 | ||
37 | Elongated square gyrobicupola |
24 | 48 | 26 | D4d of order 16 | ||
38 | Elongated pentagonal orthobicupola |
30 | 60 | 32 | D5h of order 20 | ||
39 | Elongated pentagonal gyrobicupola |
30 | 60 | 32 | D5d of order 20 | ||
40 | Elongated pentagonal orthocupolarotunda |
35 | 70 | 37 | C5v of order 10 | ||
41 | Elongated pentagonal gyrocupolarotunda |
35 | 70 | 37 | C5v of order 10 | ||
42 | Elongated pentagonal orthobirotunda |
40 | 80 | 42 | D5h of order 20 | ||
43 | Elongated pentaognal gyrobirotunda |
40 | 80 | 42 | D5d of order 20 | ||
44 | Gyroelongated triangular bicupola |
18 | 42 | 26 | D3 of order 6 | ||
45 | Gyroelongated square bicupola |
24 | 56 | 34 | D4 of order 8 | ||
46 | Gyroelongated pentagonal bicupola |
30 | 70 | 42 | D5 of order 10 | ||
47 | Gyroelongated pentagonal cupolarotunda |
35 | 80 | 47 | C5 of order 5 | ||
48 | Gyroelongated pentagonal birotunda |
40 | 90 | 52 | D5 of order 10 | ||
49 | Augmented triangular prism |
7 | 13 | 8 | C2v of order 4 | ||
50 | Biaugmented triangular prism |
8 | 17 | 11 | C2v of order 4 | ||
51 | Triaugmented triangular prism |
9 | 21 | 14 | D3h of order 12 | ||
52 | Augmented pentagonal prism |
11 | 19 | 10 | C2v of order 4 | ||
53 | Biaugmented pentagonal prism |
12 | 23 | 13 | C2v of order 4 | ||
54 | Augmented hexagonal prism |
13 | 22 | 11 | C2v of order 4 | ||
55 | Parabiaugmented hexagonal prism |
14 | 26 | 14 | D2h of order 8 | ||
56 | Metabiaugmented hexagonal prism |
14 | 26 | 14 | C2v of order 4 | ||
57 | Triaugmented hexagonal prism |
15 | 30 | 17 | D3h of order 12 | ||
58 | Augmented dodecahedron |
21 | 35 | 16 | C5v of order 10 | ||
59 | Parabiaugmented dodecahedron |
22 | 40 | 20 | D5d of order 20 | ||
60 | Metabiaugmented dodecahedron |
22 | 40 | 20 | C2v of order 4 | ||
61 | Triaugmented dodecahedron |
23 | 45 | 24 | C3v of order 6 | ||
62 | Metabidiminished icosahedron |
10 | 20 | 12 | C2v of order 4 | ||
63 | Tridiminished icosahedron |
9 | 15 | 8 | C3v of order 6 | ||
64 | Augmented tridiminished icosahedron |
10 | 18 | 10 | C3v of order 6 | ||
65 | Augmented truncated tetrahedron |
15 | 27 | 14 | C3v of order 6 | ||
66 | Augmented truncated cube |
28 | 48 | 22 | C4v of order 8 | ||
67 | Biaugmented truncated cube |
32 | 60 | 30 | D4h of order 16 | ||
68 | Augmented truncated dodecahedron |
65 | 105 | 42 | C5v of order 10 | ||
69 | Parabiaugmented truncated dodecahedron |
70 | 120 | 52 | D5d of order 20 | ||
70 | Metabiaugmented truncated dodecahedron |
70 | 120 | 52 | C2v of order 4 | ||
71 | Triaugmented truncated dodecahedron |
75 | 135 | 62 | C3v of order 6 | ||
72 | Gyrate rhombicosidodecahedron |
60 | 120 | 62 | C5v of order 10 | ||
73 | Parabigyrate rhombicosidodecahedron |
60 | 120 | 62 | D5d of order 20 | ||
74 | Metabigyrate rhombicosidodecahedron |
60 | 120 | 62 | C2v of order 4 | ||
75 | Trigyrate rhombicosidodecahedron |
60 | 120 | 62 | C3v of order 6 | ||
76 | Diminished rhombicosidodecahedron |
55 | 105 | 52 | C5v of order 10 | ||
77 | Paragyrate diminished rhombicosidodecahedron |
55 | 105 | 52 | C5v of order 10 | ||
78 | Metagyrate diminished rhombicosidodecahedron |
55 | 105 | 52 | Cs of order 2 | ||
79 | Bigyrate diminished rhombicosidodecahedron |
55 | 105 | 52 | Cs of order 2 | ||
80 | Parabidiminished rhombicosidodecahedron |
50 | 90 | 42 | D5d of order 20 | ||
81 | Metabidiminished rhombicosidodecahedron |
50 | 90 | 42 | C2v of order 4 | ||
82 | Gyrate bidiminished rhombicosidodecahedron |
50 | 90 | 42 | Cs of order 2 | ||
83 | Tridiminished rhombicosidodecahedron |
45 | 75 | 32 | C3v of order 6 | ||
84 | Snub disphenoid |
8 | 18 | 12 | D2d of order 8 | ||
85 | Snub square antiprism |
16 | 40 | 26 | D4d of order 16 | ||
86 | Sphenocorona | 10 | 22 | 14 | C2v of order 4 | ||
87 | Augmented sphenocorona |
11 | 26 | 17 | Cs of order 2 | ||
88 | Sphenomegacorona | 12 | 28 | 18 | C2v of order 4 | ||
89 | Hebesphenomegacorona | 14 | 33 | 21 | C2v of order 4 | ||
90 | Disphenocingulum | 16 | 38 | 24 | D2d of order 8 | ||
91 | Bilunabirotunda | 14 | 26 | 14 | D2h of order 8 | ||
92 | Triangular hebespenorotunda |
18 | 36 | 20 | C3v of order 6 |
Notes
- ^ Meyer (2006), p. 418.
- ^ Litchenberg (1988).
- ^ Boissonnat & Yvinec (1989).
- ^ Cromwell (1997), p. 77.
- ^ Diudea (2018), p. 40.
- ^ Todesco (2020), p. 282; Williams & Monteleone (2021), p. 23.
- ^ Johnson (1966); Zalgaller (1969).
- ^ Rajwade (2001), p. 84–88; Slobodan, Obradović & Ðukanović (2015); Berman (1971), p. 350; Holme (2010), p. 99.
- ^ Powell (2010), p. 27; Solomon (2003), p. 40.
- ^ Flusser, Suk & Zitofa (2017), p. 126.
- ^ Flusser, Suk & Zitofa (2017), p. 126; Hergert & Geilhute (2018), p. 56 .
- ^ Walsh (2014), p. 284.
- ^ Parker (1997), p. 264.
- ^ Uehara (2020), p. 62.
- ^ Johnson (1966).
- ^ Berman (1971).
References
- Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
- Boissonnat, J. D.; Yvinec, M. (June 1989). Probing a scene of non convex polyhedra. Proceedings of the fifth annual symposium on Computational geometry. pp. 237–246. doi:10.1145/73833.73860.
- Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press.
- Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Springer. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
- Flusser, Jan; Suk, Tomas; Zitofa, Barbara (2017). 2D and 3D Image Analysis by Moments. John & Sons Wiley.
- Helbert, Wolfram; Geilhufe, Matthias (2018). Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica. John & Sons Wiley. ISBN 978-3-527-41300-3.
- Holme, Audun (2010). Geometry: Our Cultural Heritage. Springer. doi:10.1007/978-3-642-14441-7. ISBN 978-3-642-14441-7.
- Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
- Litchenberg, Dorovan R. (1988). "Pyramids, Prisms, Antiprisms, and Deltahedra". The Mathematics Teacher. 81 (4): 261–265. JSTOR 27965792.
- Meyer, W. (2006). Geometry and Its Applications. Academic Press. ISBN 978-0-12-369427-0.
- Parker, Sybil P. (1997). Dictionary of Mathematics. McGraw-Hill.
- Powell, Richard C. (2010). Symmetry, Group Theory, and the Physical Properties of Crystals. Springer. doi:10.1007/978-1-4419-7598-0. ISBN 978-1-4419-7598-0.
- Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
- Solomon, Ronald (2003). Abstract Algebra. American Mathematical Society. ISBN 978-0-8218-4795-4.
- Slobodan, Mišić; Obradović, Marija; Ðukanović, Gordana (2015). "Composite Concave Cupolae as Geometric and Architectural Forms" (PDF). Journal for Geometry and Graphics. 19 (1): 79–91.
- Todesco, Gian Marco (2020). "Hyperbolic Honeycomb". In Emmer, Michele; Abate, Marco (eds.). Imagine Math 7: Between Culture and Mathematics. Springer. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8.
- Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5.
- Walsh, Edward T. (2014). A First Course in Geometry. Dover Publications. ISBN 978-0-486-78020-7.
- Williams, Kim; Monteleone, Cosino (2021). Daniele Barbaro’s Perspective of 1568. Springer. doi:10.1007/978-3-030-76687-0. ISBN 978-3-030-76687-0.
- Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.
External links
- Sylvain Gagnon, "Convex polyhedra with regular faces[permanent dead link]", Structural Topology, No. 6, 1982, 83-95.
- Johnson Solids by George W. Hart.
- Images of all 92 solids, categorized, on one page
- Weisstein, Eric W. "Johnson Solid". MathWorld.
- VRML models of Johnson Solids by Jim McNeill
- VRML models of Johnson Solids by Vladimir Bulatov