Are any Platonic solids space-filling?
Similarly, neither will the icosahedron nor the dodecahedron, as it is analogous to trying to tile a plane with pentagons – they leave odd gaps that are not pentagonal and both the dodecahedron and the icosahedron exhibit five-fold symmetry too. To see this, look back at the sections above on the Icosahedron and Dodecahedron and you will find that, in the “other views” each has a view with five-fold symmetry. These views correspond to looking along an axis through the centre of the solids which have five-fold symmetry.
