## Polyhedra (*) ### Introduction

In this page not only some known polyhedra but also animations of dual polyhedra are presented.
Of the various polyhedra, which are simultaneously convex and regular?
These polyhedra, already known in ancient Greece are called platonic solids. There are only five platonic solids - tetrahedron, cube, octahedron, icosahedron and dodecahedron.

Platonic Solids

Once all the regular convex polyhedra are known, it is natural to ask:
Are all regular polyhedra convex?
Johannes Kepler, in 1619, found two polyhedra which are simultaneously regular and not convex - the small stellated dodecahedron and the big stellated dodecahedron. Two centuries later it woud be proved that there are only nine polyhedra in these conditions: the five platonic solids and four nonconvex regular polyhedra - the Kepler-Poinsot polyhedra.

Kepler-Poinsot Polyhedra

If we consider any platonic solid and "join" the center ponts of sides, we get a new platonic solid (see the lower table). These two solids are said to be duals of one another.

Duality

The previous table points to a certain distribution of the 5 regular polyhedra in 3 classes: Tetrahedron (dual of itself), Cube and Octahedron, Dodecahedron and Icosahedron.
Consider the pair - octahedron/cube - count the number of faces, vertices and edges of each of these solids. Now consider the pair - dodecahedron/icosahedron - and do the same. To finish, count the number of faces, vertices and edges of the tetrahedron. What do you conclude?

The animations presented in the following table show that it is possible to construct the dual of a given platonic solid by truncating it successively.

Table of Animations

In the model formed by a tetrahedron and its dual (which is also a tetrahedron) presented in the duality table, if we enlarge the interior tetrahedron so that the edges of both tetrahedron are at the same distance of the common center, we obtain a composed polyhedron - the stella octangula.

Composed polyhedron

Stella octangula

(*) The interactive applications included here were carried out under a grant by FCT - Fundação para a Ciência e a Tecnologia.

Translated for Atractor by Gabriela Chaves, from its original version in Portuguese. Atractor is grateful for her cooperation.