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) Create: April 6, 1996. Last Modified: May 20, 1997.