The Thirteen Archimedean Solids: (VRML)

A polyhedron is said to be uniform if it has regular faces and admits symmetries which will transform a given vertex into every other vertex in turn. The Platonic, Kepler-Poinsot solids are uniform, so are the right regular prisms and antiprisms of suitable height - namely, when their lateral faces are squares and equilaterals, respectively. L. Lines in Solid Geometry proves that apart from these, there are just thirteen finite, convex uniform polyhedra. These are called the Archimedean solids.

Plato is said to have known at least one, the cuboctahedron, and Archimedes wrote about the entire set, though his book on them is lost. Durer gives the nets for some Archimedean solids in his Underweysung , but they were first treated systematically by Kepler.

The Archimedean solids can be broken down into various subsets.

There are first of all the five derived by the process of truncation from each vertex along with the vertex itself. This can be done to the Platonic solids in such a way that the new faces are again regular polygons. For example, on cutting off the corners of a cube, byplanes parallel to the faces of the reciprocal octahedron, we have small rectangles, and replace the square faces to octagons. For suitable positions of the cutting planes these octagons will be regular, and we have an Archimedean solids, namely the truncated cube t{4,3} .

 Truncated Tetrahedron VRML Truncated Octahedron VRML Truncated Hexahedron VRML
 Truncated Icosahedron VRML Truncated Dodecahedron VRML

Another subset, containing only two members, is that known as the quasi-regular polyhedra. In the case when two regular polyhedra, {p,q} and {q,p} , are reciprocal with respect to their common mid-sphere, the solid region interior to both polyhedra forms another polyhedron, say {p/q} , which has N1 vertices, namely the mid-edge points of either {p,q} or {q,p} . Its faces consist of N0 {q} 's and N2 {p} 's, which are the vertex figures of {p,q} and {q,p} respectively.

When p = q = 3 , we have the octahedron; hence, {3/3} = {3,4} .

When {p} 's and {q}'s are different, we have {3/4} which is the cuboctahedron and {3/5} whichis the icosidodecahedron.

Note that {p/q}={q/p}

 Cuboctahedron VRML Icosidodecahedron VRML

Then there are two called the small rhombicuboctahedron and the small rhombiicosidodecahedron. The faces of the icosidodecahedron consist of 20 triangles and 12 pentagons (corresponding to the faces of the two parent regulars). Its 60 edges are perpendicularly bisected by those of the reciprocal triacontahedron. The 60 points where these pairs of edges cross one another are the vertices of a polyhedron whose faces consist of 20 triangles, 12 pentagons and 30 rectangles. By slightly displacing the points towards the mid-points of the edges of the triacontahedron, the rectangles can be distorted into squares, and we have the small rhombiicosidodecahedron. An analogous construction leads to the rhombicuboctahedron whose faces consist of 8 triangles and 6+12 squares.

 Small Rhombicuboctahedron VRML Small Rhombiicosidodecahedron VRML

By applying the truncation method to the cuboctahedron and the icosidodecahedron in addition to a distortion to convert rectangles into squares, we obtain the great rhombicuboctahedron and the great rhombiicosidodecahedron.

 Great Rhombicuboctahedron VRML Great Rhombiicosidodecahedron VRML

All the Archimedean solids so far discussed are reflexible (by reflection in the plane that perpendicularly bisects the edge). The remaining two, however, is not reflexible: the snub cube and the snub dodecahedron. Each of them occurs in two forms, and the two forms of each are related to one another like a left-hand and a right-hand glove: they are enantiomorphic. See Line's Solid Geometry (pp.175-pp.184) for discussions about constructions of these two snub polyhedra.

 Snub Cube VRML Snub Dodecahedron VRML

In Maple, one can define an Archimedean solid by using the command Archimedean(gon,sch,o,r); where gon is the name of the polyhedron to be defined, sch the Schläfli symbol (Maple's Schläfli), o the center of the polyhedron, and r the radius of the circum-sphere.

The Schläfli symbol can be one of the following:

 Maple's Schläfli symbol Polyhedron type _t([3,3]) truncated tetrahedron _t([3,4]) truncated octahedron _t([4,3]) truncated cube _t([3,5]) truncated icosahedron _t([5,3]) truncated dodecahedron [[3],[4]] cuboctahedron [[3],[5]] icosidodecahedron _r([[3],[4]]) small rhombicuboctahedron _r([[3],[5]]) small rhombiicosidodecahedron _t([[3],[4]]) great rhombicuboctahedron _t([[3],[5]]) great rhombiicosidodecahedron _s([[3],[4]]) snub cube _s([[3],[5]]) snub dodecahedron

Another way to define an Archimedean solid is to use the command Polyhedron_Name(gon,o,r); where Polyhedron_Name is one of TruncatedTetrahedron, TruncatedOctahedron, TruncatedHexahedron, TruncatedIcosahedron, TruncatedDodecahedron, SmallRhombicuboctahedron, SmallRhombiicosidodecahedron, GreatRhombicuboctahedron, GreatRhombiicosidodecahedron, SnubCube, SnubDodecahedron, cuboctahedron, icosidodecahedron . For example, to define a great rhombicuboctahedron with center (1,2,3), radius of the circum-sphere 2, one can either use:

```> with(geom3d):
> Archimedean(gr1,_t([[3],[4]]),point(o,1,2,3),2);
gr1
```

or

```> with(geom3d):
> GreatRhombicuboctahedron(gr2,o,2);
gr2
```

To access information relating to an Archimedean solid pgon , use the following function calls:

 center(pgon); returns the center of pgon faces(pgon); returns the faces of pgon form(pgon); returns the form of pgon MidRadius(pgon); returns the mid-radius of pgon radius(pgon); returns the circum-radius of pgon schlafli(pgon); returns the Schläfli symbol of pgon sides(pgon); returns the side of pgon vertices(pgon); returns the vertices of pgon

Author: Ha Le (hle@cecm.sfu.ca)