Archimedean solid

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedra composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not 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. 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, as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. The reader may find more information about the rediscovery of the Archimedean solids during the renaissance in the paper by Peter Schreiber et al., published in 2008 (see References below).

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.

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)
r{4,3} or rr{3,3}
or
3.4.3.4

14 8 triangles
6 squares
24 12 2.357023 Oh 0.9049973
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)
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 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".

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 (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form.[further explanation needed] 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 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 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 isohedron can be used as a snub tetrahedron.)