Semi-regular Polyhedron

Archimedean Polyhedra

All of the Archimedean Polyhedra may also be derived from the truncation of a cube's edges and vertices. This is illustrated below. The purple faces are what remains of the original cube faces. The red faces result from vertex truncations, and the yellow faces result from edge truncations.

	Truncated Cube Truncating all 8 vertices with a plane that reduces the original cube faces to regular octagons results in the Truncated Cube.
	Cubeoctahedron Truncating all 8 vertices with a plane that bisects the edges that meet at that vertex, results in the Cubeoctahedron. The 8 pieces cut off, may be reassembled into an Octahedron. The Rhombic Dodecahedron (which can be produced by truncating all the edges of a cube) and this figure are duals of each other. The linked transformation shows a nice relationship between the Cubeoctahedron, the Icosahedron and the Octahedron (152kb).
	Truncated Octahedron Again truncating all the vertices with a deeper cut than in the figure above produces a Truncated Octahedron. This figure is bounded by 6 squares, representing the original cube faces, and 8 hexagons produced by the truncation of the vertices of the cube.
	Truncated Tetrahedron The Truncated Tetrahedron is produced by truncating all the vertices of a cube, but varying the depths of the cuts. 4 of the vertices are truncated to the same depth as the tetrahedron and 4 cuts are much shallower. This results in a figure bounded by 4 hexagons and 4 triangles. The vertex first cross-section of a hypercube 1/4 and 3/4 through, produces this same figure.
	Rhombicuboctahedron When you truncate both edges and vertices, one of the possible results is the Rhombicuboctahedron. It is bounded by 6 square faces (purple), the remnants of the original cube faces, 12 square faces (yellow) resulting from the edge truncations and 8 triangular faces (red) resulting from the vertex truncations. All 18 squares faces are the same, I made the color distinction to call attention to this figure's relationship to the cube.
	Truncated Cubeoctahedron Hear again all the edges and vertices are being truncated, resulting in a Truncated Cubeoctahedron. This figure is bounded by 6 octagons representing the original cube faces, 8 hexagons produced by truncating the vertices and 12 squares produced by truncating the cube's edges. Even though this figure is called a Truncated Cubeoctahedron, it cannot be produced by truncating the vertices of a Cubeoctahedron. Doing so produces rectangles, not squares. To produce this figure you must make deeper vertex cuts than those used to make a Cubeoctahedron.
	Snub Cube The Snub Cube retains portions of the original 8 faces of the cube. These faces are square but smaller and appear twisted in the plane of the cube face. Of the 32 triangular faces the 8 red ones are the result of truncating the cube vertices and the 24 yellow ones are the result of truncating the 12 cube edges twice at different angles. The Snub Cube has some interesting characteristics. There is a right and left handed version. In one the square faces are twisted clockwise, in the other they are twisted counter-clockwise. Although all the triangular faces are equilateral and of the same size, the 8 produced by vertex truncations can be paired with a triangle on the opposite side of the figure that is in a parallel plane. For all the remaining triangular faces there is a vertex on the opposite side of the figure.
	Truncated Dodecahedron As you saw on the Cube As Base page, the dodecahedron can be produced by truncating edges of a cube. The Truncated Dodecahedron is simply and extension of that process. To produce the yellow triangle faces all 12 cube edges are truncated again by planes parallel to the edge but at a different angle and depth that the truncations that produced the pentagon faces. The 8 red triangle faces are the result of truncating the vertices of the cube with a plane perpendicular to the diagonals of the cube. This figure has 12 decagon faces and 20 triangle faces.
	Icosidodecahedron This figure is produced by pushing the truncating planes that produced the triangular faces of the Truncated Icosahedron deeper into the cube, until the decagon faces become pentagons. The 12 pentagon faces are the result of the first truncation of all cube edges that produced the regular Dodecahedron. All the triangular faces are congruent, I have colored the 8 that are normal to the cube's diagonals red.
	Truncated Icosahedron Again as illustrated on the Cube as Base page the Icosahedron can be produced by truncating the cube. Truncating the 12 edges a second time at a different angle produces this solid. All of the hexagonal faces are what is left of the original triangle faces of the Icosahedron, while the pentagon faces are the result of a 2nd truncation of the cube edges with a plane parallel to the edge being truncated but at a different angle and depth than the first cut that produced the triangle faces. This solid has 20 hexagonal faces and 12 pentagonal faces.
	Rhombicosidodecahedron This solid may be produced by truncating all the cube edges as if you were going to make a dodecahedron, but with a shallower cut. This will leave the 6 purple squares from the original cube and produce the yellow pentagons. Then truncate the 8 cube vertices as if you were going to produce an octahedron, but with a shallower cut. This will produce the 8 red triangle faces. Next truncate the same vertices 3 more times. This time the cutting planes are not perpendicular to the diagonal of the cube, but angled toward each of the edges that meet at that vertex. These cuts are represented by the orange squares in this figure. Finally truncate the edges again with a plane parallel to the edge but at a different angle than the dodecahedron cuts. These cuts create the yellow triangles. This solid is bounded by 30 square faces, 12 pentagonal faces and 20 triangular faces.
	Truncated Icosidodecahedron This solid is produced by making the same cuts as used for the Rhombicosidodecahedron, except the depth of the cuts is changed. The Dodecahedral cuts are deeper transforming the pentagons to decagons. The Icosahedral cuts are deeper changing the triangles to hexagons. The Octahedral cuts are also deeper which changes the red triangles to hexagons. The 6 original cube faces are represented by the purple squares. This figure is bounded by 30 square faces, 12 decagons, and 20 hexagons.
	Snub Dodecahedron The 12 pentagonal faces of this solid are formed by the dodecahedral cuts we have seen above, only the cuts are shallower. 6 edges of this figure lie on the cube faces (shown in purple). One way to understand the 80 triangular faces is to group them into 20 sets of 4 triangles each, one Triangle surrounded by 3 others which share an edge with the center triangle. These groups are represented here by the gray and the red/orange groups. The center triangle is darker in color. The 8 red triangle faces are normal to the 4 diagonals of the cube. The 8 red and 12 dark gray triangles can be produced by icosahedral cuts. These 20 (red and dark gray) triangles are the only faces that share a vertex with 3 different pentagons. They are also the only faces that share an edge with 3 other triangles and do not share an edge with a pentagon. They are also the only triangular faces that have a parallel face on the opposite side of the figure. Like the Snub Cube this figure also has a right-hand and left-hand version.