Polyhedra
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.
(The animations
and figures were made with mathematica© and converted with the help
of LiveGraphics3D.
About how to use the software see help).
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.
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.
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.
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.
