Regular Polyhedra

Cube as Base
Platonic Solids	I find the relationship of the cube to the other Platonic solids very beautiful. The figures below, illustrate this relationship by truncating the cube's edges and vertices.
Archimedean	There are similar relations between the cube and the Archimedean solids.
Semi-Regular Solids	The Polyhedra shown on this page are also produced by truncating a cube.

Platonic Solids

		Typical Vertex Truncation This simple truncation of the cube's vertex can be used to produce the Tetrahedra and the Octahedra. A variation of this cut in combination with edge truncations can result in the Icosahedron. On this page the purple faces are the original faces of the cube, the red surfaces represent vertex truncations and the yellow faces result from cube edge truncation.
		Tetrahedron In this figure, 4 vertices (half those of the original cube), have been removed from the cube by making cuts like the one in the first figure above, leaving the Tetrahedron. If you join the 4 pieces that were cut off such that all the original cube vertices join at one point you will have half of an Octahedron.
		Octahedron If the remaining 4 cube vertices are removed in the same manner as above, an Octahedron is revealed. The Octahedron and Cube are duals of one another. That is they each have a vertex for every face of the other figure, those vertices are surrounded by the same number of faces as there are edges surrounding the corresponding face in the other figure. They each have the same number of edges. If you truncate all the vertices of a cube with a cut that only reaches the midpoint of the edges joining at that vertex a Cubeoctahedron is formed.
		Edge Truncations for a Dodecahedron The Dodecahedron can be formed by truncating all the edges from a cube in a particular way. Start by drawing the mid-line on each face of the cube that it is parallel and equidistant from the opposite sides of that face. The mid-lines (shown in green) are chosen so that none touch each other. Truncate the edges opposite the mid-line with 2 planes that form an angle with each other, equal to the dihedral angle of the Dodecahedron.
		Dodecahedron Here you see the Dodecahedron, resulting from the truncation of all edges of the cube in the manner described in the figure above. Six of the edges of the Dodecahedron lie on the faces of the cube. 12 of its 20 vertices also lie in the faces of the cube. The remaining 8 vertices lie on the diagonals of the cube. If you were to enlarge the Dodecahedron, you could make these 8 vertices exactly coincide with the original 8 vertices of the cube. There is another semi-regular Dodecahedron Rhombicdodecahedron that can be produced by truncating the cube's edges. View an animation of this transformation. (140kb)
		Edge and Vertex Truncations for the Icosahedron The Icosahedron can be constructed by truncating the edges in a manner similar to the edge truncation of the Dodecahedron, and the vertices similar to the Octahedron. The dihedral angle of the truncating edge planes is equal to the dihedral angle of the Icosahedron. The vertex truncating plane, although not as deep as the one above, is still perpendicular to the diagonal of the cube.
		Icosahedron Completing the edge and vertex removal as described above, produces the Icosahedron. All 12 vertices of the Icosahedron lie in the faces of the cube. Only 6 of the edges lie in the faces of the cube. The remaining 24 edges bound the faces resulting from the vertex truncations. The faces colored yellow are the result of truncating the edges, and the red faces are the result of the vertex truncations. The Icosahedron and the Dodecahedron are duals of each other. View an animation of this transformation. (140kb)

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Cube as Base

Archimedean Solids

Last revised: 9/25/2002