The Regular and Semi-regular Polyhedra

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The Regular and Semiregular Polyhedra

You can click the link below to run a Windows program that displays pictures of the five Platonic polyhedra, and allows you to truncate and explode them by any amount:Archimedes.exeAn explanation of what this means follows. We let the program provide the illustrations.

A polyhedron is a solid whose surface is made of polygons, called its faces, meeting edge to edge.

A Platonic (or “regular”) polyhedron is a convex polyhedron whose faces are regular and

(1) the faces are all the same,

(2) the same number of faces meet at each vertex.

It turns out there are only five Platonic polyhedra.

An Archimedean (or “semiregular”) polyhedron is a polyhedron whose faces are regular and

(1) the faces are not all the same,

(2) the vertices are all the same, that is, at each vertex the faces meet the same way.

(3) it is not a prism or antiprism.

It has been said that there are only 13 Archimedean polyhedra but if the above “local” requirement (2) is followed to the letter there are 14. The extra polyhedron can be made by a sort of Rubik cube twist of the rhombicuboctahedron. This new polyhedron isn’t as symmetrical as the others and can be excluded by replacing (2) above with a “global” requirement involving symmetry. The 13 polyhedra thus defined are sometimes called the “uniform” polyhedra.

All but two of the Archimedean polyhedra and what we called the twisted one can be made by “truncating” and “exploding” the Platonic polyhedra.

Truncate means to shave away the polyhedron at each vertex.

Explode means to move the faces of the polyhedron outward. The size of each face remains the same: each edge splits to form a rectangle. , and each vertex splits to form a regular polygon

Two of the Archimedian polyhedra are more regular than the others in that not only are all the corners the same (considering the faces that meet there) the edges are too. These “quasi-regular” polyhedra are the cuboctahedron and the icosadodecahedron. It turns out that their edges can be divided into “great polygons” that encircle them.

Here are two interesting formulas that apply to all polyhedra, not just the Platonic and Archimedean:

Euler’s formula. Let V = number of vertices, E = number of edges, F = number of faces of any polyhedron. Then it’s always true that:

V – E + F = 2

(for example consider a cube: 8 – 12 + 6 = 2)

Descarte’s formula. Let the “angular defect” at a vertex be how much the sum of the polygonal angles there lacks making 360°. It measures how “pointy” the corner is, how much it deviates from a plane. For any polyhedron it’s always true that:

Sum of angular defects at all vertices = 720°

(for example consider a cube: the angular defect at each of the 8 vertices is 90°, and 90° times 8 = 720°)

The analog of Euler’s formula for polygons in the plane is

V – E = 0

This expresses the obvious fact that the number of vertices equals the number of edges. The analog of Descarte’s formula for polygons in the plane is

Sum of supplementary angles = 360°

This expresses the obvious fact that as you travel around the polygon the total of your deviations from straight ahead is a full circle.

If we measure angles in radians and solid angles in steradians the plane and spatial versions of Descarte’s formula become

Polygons: sum of supplementary angles = 2π = circumference of the unit circle.

Polyhedra: sum of angular defects = 4π = area of the unit sphere.

Returning to Euler’s formula, if we count the center of the object, always 1 component, then for any point, line segment, polygon, and polyhedron, respectively:

V = 1V – E = 1V – E + F = 1V – E + F – S = 1

where V, E, F, S count the 0-, 1-, 2-, 3-dimensional components. In the second case. a line segment, V = 2 and E = 1.