Archimedean solid

Truncated tetrahedron, cuboctahedron and truncated icosidodecahedron. The first and the last one can be described as the smallest and the largest Archimedean solid, respectively.

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed of only one type of polygon), excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron.[1] They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

"Identical vertices" means that each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position. Branko Grünbaum (2009) observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Grünbaum pointed out a frequent error in which authors define Archimedean solids using this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedean solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

Origin of name

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra.[2] During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra,[3] as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See Schreiber, Fischer & Sternath 2008 for more information about the rediscovery of the Archimedean solids during the renaissance.)

Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.[2]

Classification

There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 15 if the mirror images of two enantiomorphs, the snub cube and snub dodecahedron, are counted separately).

Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of 4.6.8 means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

Name/
(alternative name)
Schläfli
Coxeter
Transparent Solid Net Vertex
conf./fig.
Faces Edges Vert. Volume
(unit edges)
Point
group
Sphericity
Truncated tetrahedron t{3,3}
3.6.6
8 4 triangles
4 hexagons
18 12 2.710576 Td 0.7754132
Cuboctahedron
(rhombitetratetrahedron, triangular gyrobicupola)
r{4,3} or rr{3,3}
or
3.4.3.4
14 8 triangles
6 squares
24 12 2.357023 Oh 0.9049972
Truncated cube t{4,3}
3.8.8
14 8 triangles
6 octagons
36 24 13.599663 Oh 0.8494937
Truncated octahedron
(truncated tetratetrahedron)
t{3,4} or tr{3,3}
or
4.6.6
14 6 squares
8 hexagons
36 24 11.313709 Oh 0.9099178
Rhombicuboctahedron
(small rhombicuboctahedron, elongated square orthobicupola)
rr{4,3}
3.4.4.4
26 8 triangles
18 squares
48 24 8.714045 Oh 0.9540796
Truncated cuboctahedron
(great rhombicuboctahedron)
tr{4,3}
4.6.8
26 12 squares
8 hexagons
6 octagons
72 48 41.798990 Oh 0.9431657
Snub cube
(snub cuboctahedron)
sr{4,3}
3.3.3.3.4
38 32 triangles
6 squares
60 24 7.889295 O 0.9651814
Icosidodecahedron
(pentagonal gyrobirotunda)
r{5,3}
3.5.3.5
32 20 triangles
12 pentagons
60 30 13.835526 Ih 0.9510243
Truncated dodecahedron t{5,3}
3.10.10
32 20 triangles
12 decagons
90 60 85.039665 Ih 0.9260125
Truncated icosahedron t{3,5}
5.6.6
32 12 pentagons
20 hexagons
90 60 55.287731 Ih 0.9666219
Rhombicosidodecahedron
(small rhombicosidodecahedron)
rr{5,3}
3.4.5.4
62 20 triangles
30 squares
12 pentagons
120 60 41.615324 Ih 0.9792370
Truncated icosidodecahedron
(great rhombicosidodecahedron)
tr{5,3}
4.6.10
62 30 squares
20 hexagons
12 decagons
180 120 206.803399 Ih 0.9703127
Snub dodecahedron
(snub icosidodecahedron)
sr{5,3}
3.3.3.3.5
92 80 triangles
12 pentagons
150 60 37.616650 I 0.9820114

Some definitions of Semiregular polyhedron include one more figure, the Elongated square gyrobicupola or "pseudo-rhombicuboctahedron".[4]

Properties

The number of vertices is 720° divided by the vertex angle defect.

The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular.

The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.

Chirality

The snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form (Latin: levomorph or laevomorph) and right-handed form (Latin: dextromorph). When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compounds.)

Construction of Archimedean solids

The Archimedean solids can be constructed as generator positions in a kaleidoscope.

The different Archimedean and Platonic solids can be related to each other using a handful of general constructions. Starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated (see table below), different Platonic and Archimedean (and other) solids can be created. If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification. An expansion, or cantellation, involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull. Expansion with twisting also involves rotating the faces, thus splitting each rectangle corresponding to an edge into two triangles by one of the diagonals of the rectangle. The last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as the rectification of the rectification. Likewise, the cantitruncation can be viewed as the truncation of the rectification.

Construction of Archimedean Solids
Symmetry Tetrahedral
Octahedral
Icosahedral
Starting solid
Operation
Symbol
{p,q}
Tetrahedron
{3,3}
Cube
{4,3}
Octahedron
{3,4}
Dodecahedron
{5,3}
Icosahedron
{3,5}
Truncation (t) t{p,q}
truncated tetrahedron
truncated cube
truncated octahedron
truncated dodecahedron
truncated icosahedron
Rectification (r)
Ambo (a)
r{p,q}
tetratetrahedron
(octahedron)
cuboctahedron
icosidodecahedron
Bitruncation (2t)
Dual kis (dk)
2t{p,q}
truncated tetrahedron
truncated octahedron
truncated cube
truncated icosahedron
truncated dodecahedron
Birectification (2r)
Dual (d)
2r{p,q}
tetrahedron
octahedron
cube
icosahedron
dodecahedron
cantellation (rr)
Expansion (e)
rr{p,q}
rhombitetratetrahedron
(cuboctahedron)
rhombicuboctahedron
rhombicosidodecahedron
Snub rectified (sr)
Snub (s)
sr{p,q}
snub tetratetrahedron
(icosahedron)
snub cuboctahedron
snub icosidodecahedron
Cantitruncation (tr)
Bevel (b)
tr{p,q}
truncated tetratetrahedron
(truncated octahedron)
truncated cuboctahedron
truncated icosidodecahedron

Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, partially because the tetrahedron is self-dual, only one Archimedean solid that has at most tetrahedral symmetry. (All Platonic solids have at least tetrahedral symmetry, as tetrahedral symmetry is a symmetry operation of (i.e. is included in) octahedral and isohedral symmetries, which is demonstrated by the fact that an octahedron can be viewed as a rectified tetrahedron, and an icosahedron can be used as a snub tetrahedron.)

Stereographic projection

truncated tetrahedron truncated cube truncated octahedron truncated dodecahedron truncated icosahedron

triangle-centered

hexagon-centered

octagon-centered

triangle-centered

square-centered

hexagon-centered

Decagon-centered

Triangle-centered

pentagon-centered

hexagon-centered
cuboctahedron icosidodecahedron rhombicuboctahedron rhombicosidodecahedron

square-centered

triangle-centered

vertex-centered

pentagon-centered

triangle-centered

square-centered

square-centered

triangle-centered

Pentagon-centered

Triangle-centered

Square-centered
truncated cuboctahedron truncated icosidodecahedron snub cube

square-centered

hexagon-centered

octagon-centered

decagon-centered

hexagon-centered

square-centered

square-centered

Citations

1. ^ Steckles, Katie. "The Unwanted Shape". YouTube. Retrieved 20 January 2022.
2. ^ a b
3. ^ Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, 50, 1997, 227
4. ^ Malkevitch (1988), p. 85

Works cited

• Grünbaum, Branko (2009), "An enduring error", Elemente der Mathematik, 64 (3): 89–101, doi:10.4171/EM/120, MR 2520469. Reprinted in Pitici, Mircea, ed. (2011), The Best Writing on Mathematics 2010, Princeton University Press, pp. 18–31.
• Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in Senechal, M.; Fleck, G. (eds.), Shaping Space: A Polyhedral Approach, Boston: Birkhäuser, pp. 80–92.