The Platonic Solids

The Platonic Solids

Tetrahedron


The exact projection of a Tetrahedron onto a sphere




Hexadron (Cube)


The exact projection of a Hexahedron (Cube) onto a sphere



Octahedron


The exact projection of a Octahedron onto a sphere




Icosahedron


The exact projection of a Icosahedron onto a sphere




Dodecahedredon


The exact projection of a Dodecahedron onto a sphere


Higher Dimensions of the Platonic Solids
In the mid-19th century the Swiss mathematician Ludwig Schläfli discovered the four-dimensional analogues of the Platonic solids, called convex regular 4-polytopes. There are exactly six of these figures; five are analogous to the Platonic solids, while the sixth one, the 24-cell, has no lower-dimensional analogue.

In dimensions higher than four, there are only three convex regular polytopes: the simplex, the hypercube, and the cross-polytope. In three dimensions, these coincide with the tetrahedron, the cube, and the octahedron.