A dodecahedron is a
composed of twelve pentagonal faces, with three meeting at each vertex. Its
is the icosahedron. Canonical coordinates for the vertices of a dodecahedron
centered at the origin are (0,±1/τ,±τ), (±1/τ,±τ,0), (±τ,0,±1/τ), (±1,±1,±1),
where τ = (1+√5)/2 is the golden mean. Five cubes can be made from these, with
their edges as diagonals of the dodecahedron's faces, and together these
comprise the regular polyhedral compound of five cubes. The stellations of the
dodecahedron make up three of the four Kepler-Poinsot solids.
Platonic? - From Plato.
Your second question is...What is a Platonic Solid?
A Platonic solid is a convex regular polyhedron all the faces of which share the same regular polygon and having the same number of faces meeting at all its vertices. Compare with the Kepler solids, which are not convex, and the Archimedean and Johnson solids, which while made of regular polygons are not themselves regular.
Examples - Platonic Solids
The leads you to another question -
what is a polyhedron?
A Polyhedron is a shape, made up of faces. 'Poly-' is from the Greek word for 'many' and '-hedron' is a Greek combining form meaning 'base', 'seat', or 'face'. Roughly speaking, a polyhedron is a higher dimension version of a polygon - for instance prisms and pyramids are polyhedra.
If you about to ask just what is a polygon?
A polygon (from the Greek poly, for "many", and gwnos, for "angle") is a closed planar path composed of a finite number of straight line segments. The term polygon sometimes also describes the interior of the polygon (the open area that this path encloses) or to the union of both. Named for the number or sides - pentagon, etc. The pentagon is a building with five sides - easy to remember.
Now back to the dodecahedron.
What did it mean by "It's dual is the
Polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges. So the regular polyhedra - the Platonic solids and Kepler-Poinsot polyhedra - are arranged into dual pairs.
Wait - you told me what Platonic solids were, what is a Kepler-Poinsot polyhedra?
Kepler solid is a regular nonconvex polyhedron, all the faces of which are regular polygons and which has the same number of faces meeting at all its vertices. Named for Johannes Kepler.
Wait...convex? What was that word?
An object is said to be convex if for any pair of points within the object, any point on the line that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not convex.
Dodecahedron In Motion - Easier To See This Way