Polyhedra Gallery

For each integer n > 2 there is an n-sided analog of the 3-sided equilateral triangle and 4-sided square---namely the regular n-gon (or n-sided regular polygon). It has n vertices, n equal sides, and successive sides all meet at the same angle. Given one such regular n-gon, one can form infinitely many others by scaling, rotating, and translating it. In fact, one gets all others in this way, and since it is natural to consider figures that can be carried into each other by these simple transformations as being "the same", one says that there is only one regular n-gon. Moreover, it is about as symmetric as it could possiby be---rotating the regular n-gon about its center by an integer multiple of 360/n degrees carries it into itself and can move any vertex to any other.

Can we generalize this to three dimensions? At first glance things look promising. Generalizing the equilateral triangle and square are respectively the regular tetrahedron and cube (or regular hexahedron). The former has four vertices, six edges of equal length and four congruent equilateral triangles as faces, and the pattern of edges meeting at each vertex is the same. Indeed we can rotate it into itself in such a way that any vertex is carried into any other. Similarly, the cube has eight vertices, twelve edges of equal length, and six congruent squares as faces, and again the pattern of edges meeting at each vertex is the same and we can even rotate the cube into itself, carrying any vertex to any other. Let us define a polyhedron to be regular if it is convex, with all of its faces being congruent regular polygons and with the same number of faces meeting at each vertex. From what we have seen so far one might guess that there was one regular polyhedon having any regular n-gon for its faces. In fact however, it was known already in antiquity that aside from the regular tetrahedron and regular hexahedron (cube) there are (up to scaling, rotation, and translation) only three more regular polyhedra namely the regular octahedron, regular dodecahedron, and regular icosahedron, and these five so-called Platonic Solids are the main Exhibits of this Gallery. (There are also two additional two non-regular Exhibits, the Rhombohedron and Rhombic Dodecahedron, both having rhombic faces.)

For each Exhibit, in addition to its standard form, the Exhibit can also be viewed in various "truncated" and "stellated" forms. For a truncated form, an identical piece is cut from each vertex. For two special values of the "size" of truncation, the Archimedean and the Midpoint truncations, all the faces of the resulting polyhedron are again regular polygons. (If you choose Morph from the Animation menu, it is the size of the truncation that gets morphed.) For the stellated form, a "pyramid" is built on each face of the polyhedron.

References and Links

An excellent place to get started going deeper into the theory of Polyhedra is: Polyhedron, From Wikipedia, the free encyclopedia.

One of the Grand Masters of Polyhedra is George W. Hart, who has devoted much of his life to both playing with and studying them. He has two marvelous websites devoted to polyhedra at: The Encyclopedia of Polyhedra and The Pavilion of Polyhedreality

Another fine website devoted to polyhedra is Paper Models of Polyhedra

A book that should be in the library of everyone who loves Mathematics is the classic "Introduction to Geometry", by H.S.M. Coxeter, one of the great mathematicians of the twentieth century. Chapter 10, "The Five Platonic Solids" is an excellent introduction to the subject.

A great place to look for anything related to geometry is David Eppstein's "The Geometry Junkyard". In particular you can find there many further links to Polyhedra and Polytopes