Octahedron: Difference between revisions
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Abe did this |
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If you are from 141 you know who this is its Abe yes its me ha i ruined u from answering the science homework lolz |
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== Geometric relations == |
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The interior of the [[polyhedral compound|compound]] of two dual [[tetrahedron|tetrahedra]] is an octahedron, and this compound, called the [[stella octangula]], is its first and only [[stellation]]. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. [[Rectification (geometry)|rectifying]] the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the [[cuboctahedron]] and [[icosidodecahedron]] relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the [[golden mean]] to define the vertices of an [[icosahedron]]. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a '''regular compound.''' |
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[[Tetrahedral-octahedral honeycomb|Octahedra and tetrahedra]] can be alternated to form a vertex, edge, and face-uniform [[tessellation of space]], called the [[octet truss]] by [[Buckminster Fuller]]. This is the only such tiling save the regular tessellation of [[cube]]s, and is one of the 28 [[convex uniform honeycomb]]s. Another is a tessellation of octahedra and [[cuboctahedron|cuboctahedra]]. |
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The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces. |
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Using the standard nomenclature for [[Johnson solid]]s, an octahedron would be called a ''square bipyramid''. |
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==Related polyhedra== |
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=== Tetratetrahedron === |
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The octahedron can also be considered a ''[[rectification (geometry)|rectified]] tetrahedron'' - and can be called a '''tetratetrahedron'''. This can be shown by a 2-color face model. With this coloring, the octahedron has [[tetrahedral symmetry]]. |
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Compare this truncation sequence between a tetrahedron and its dual: |
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{| class="prettytable" |
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|[[Image:Uniform polyhedron-33-t0.png|100px]]<BR>[[Tetrahedron]] |
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|[[Image:Uniform polyhedron-33-t01.png|100px]]<BR>[[Truncated tetrahedron]] |
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|[[Image:Uniform polyhedron-33-t1.png|100px]]<BR>'''octahedron''' |
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|[[Image:Uniform polyhedron-33-t12.png|100px]]<BR>[[Truncated tetrahedron]] |
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|[[Image:Uniform polyhedron-33-t2.png|100px]]<BR>[[Tetrahedron]] |
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|} |
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== Octahedra in the physical world == |
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[[Image:Fluorite octahedron.jpg|thumb|Fluorite octahedron.]] |
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*Especially in [[roleplaying game]]s, this solid is known as a [[dice#Non-cubical dice|d8]], one of the more common [[Polyhedral dice]]. |
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*If each edge of an octahedron is replaced by a one [[ohm (unit)|ohm]] [[resistor]], the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.<ref>{{cite journal | last = Klein | first = Douglas J. | year = 2002 | title = Resistance-Distance Sum Rules | journal = Croatica Chemica Acta | volume = 75 | issue = 2 | pages = 633–649 | url = http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf | format = PDF | accessdate = 2006-09-30}}</ref> |
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*Natural crystals of [[diamond]], [[alum]] or [[fluorite]] are commonly octahedral. |
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*The plates of [[kamacite]] alloy in [[octahedrite]] [[meteorites]] are arranged paralleling the eight faces of an octahedron |
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== Octahedra in music == |
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If you place notes on every vertex of an octahedron, you can get a six note just intonation scale with remarkable properties - it is highly symmetrical and has eight consonant triads and twelve consonant diads. See [[hexany]] |
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==Other octahedra == |
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The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; nonregular octahedra may have as many as 12 vertices and 18 edges.[http://www.uwgb.edu/dutchs/symmetry/polynum0.htm] |
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* [[Hexagonal prism]]: 6 squares, 2 hexagons |
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* Heptagonal [[Pyramid (geometry)|pyramid]]: 7 triangles, 1 heptagon |
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* Tetragonal [[bipyramid]]: 8 triangles, usually [[triangles|isosceles]]) |
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** The regular octahedron is a special case with [[triangles|equilateral triangle]]s |
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* [[Truncated tetrahedron]]: 4 triangles, 4 hexagons |
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* [[Tetragonal trapezohedron]] - 8 kites |
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==See also== |
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* [[:Image:Octahedron.gif|Spinning octahedron]] |
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*[[Stella octangula]] |
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*[[Triakis octahedron]] |
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*[[Disdyakis dodecahedron|Hexakis octahedron]] |
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*[[Truncated octahedron]] |
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*[[Octahedral molecular geometry]] |
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==References== |
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<!-- See [[Wikipedia:Footnotes]] for instructions. --> |
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<references /> |
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==External links== |
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*[http://mathworld.wolfram.com/Octahedron.html Octahedron] - Mathworld.com |
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*[http://www.software3d.com/Octahedron.php Paper model of the octahedron] |
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*[http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra] |
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*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra] |
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*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra |
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*[http://www.korthalsaltes.com/ Paper Models of Polyhedra] Many links |
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*[http://www.octahedron.com.au/ Octahedron] - Jewellery Software |
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{{Polyhedra}} |
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[[Category:Deltahedra]] |
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[[Category:Platonic solids]] |
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[[Category:Regular polyhedra]] |
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[[Category:Prismatoid polyhedra]] |
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[[Category:Pyramids and bipyramids]] |
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[[Category:Polyhedra]] |
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[[az:Oktaedr]] |
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[[ca:Octàedre]] |
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[[da:Oktaeder]] |
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[[et:Korrapärane oktaeeder]] |
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[[fr:Octaèdre]] |
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[[he:תמניון]] |
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[[it:Ottaedro]] |
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[[ja:正八面体]] |
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[[pl:Ośmiościan foremny]] |
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[[pt:Octaedro]] |
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[[ru:Октаэдр]] |
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[[simple:Octahedron]] |
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[[sr:Октаедар]] |
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[[ta:எண்முக முக்கோணகம்]] |
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[[th:ทรงแปดหน้า]] |
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[[zh:正八面體]] |
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Revision as of 20:51, 8 September 2008
Abe did this