### 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