Polyhedra

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1. The Five Convex Regular Polyhedra   (``Platonic Solids'')

Regular polyhedra are those whose faces are all congruent regular polygons, and whose vertices are identical. A little analysis will convince you that there can only be five. (But what about all those geodesic shapes, made up of lots of equilateral triangles???) The angles of the polygons which meet at one vertex of the polyhedron must be less than 360°, since the polyhedron is convex; therefore the vertex of a regular convex polyhedron can be the intersection of only

2. Four more Regular Polyhedra

If the polygonal faces need not be convex, nor the vertices at which they meet, then there are 4 more possible regular polyhedra. Each face of these four are generalized regular polygons, which means that 5-pointed stars (5/2-gons) are allowed. The four are

3. SemiRegular Polyhedra

Every face, vertex, side and angle of a regular polyhedra is identical. Once we allow multiple kinds of regular polygons to mixed in a single polyhedron (though we still insist that all vertices be congruent) there are many more possibilities. In addition to two infinite families of prisms and antiprisms, there are 75 semiregular polyhedra, as proved by Coxeter et al. in 1953. (They are also called ``uniform'' polyhedra.)

Prisms

Antiprisms

Archimedean Solids

Archimdean Duals

The Johnson Solids (all Convex Polyhedra With Regular Faces)

Others