Semi-Regular Polyhedra

Many other Polyhedra may be derived from the truncation of a cube.  Some of these are illustrated below. (Work in progress.)
Cube with 1 edge truncated Typical Edge Truncation

This edge truncating plane passes through the mid-line of the 2 faces adjacent to the edge being truncated.

There are 218 possibilities for truncating Cube Edges in this manner. 74 of these are mirror images of another form in the series. This leaves 144 unique forms.
Rhombic Dodecahedron Rhombic Dodecahedron

If you use the plane described above, to cut off all 12 edges you get a RhomicDodecahedron.  Notice that there is a rhombic face parallel to each of the original edges of the cube. 6 of the Rhombicdodecahedron's vertices lie in the faces of the cube. The remaining 8 vertices lie on the cube's diagonals.

Mark Newbold has another recipe for constructing this figure.  You may also construct the same figure by adding parts of 2 Tetrahedra to an Octahedron. (see Transformations)
6 Edge Truncation 1 Irregular Dodecahedron

Truncating 6 edges of the cube produces this 12 sided figure. The edges removed (in green) are arranged in a Petri Polygon around the cube.

This figure has 6 square faces and 6 rhombic faces, 24 edges and 14 vertices.

It is the reference for Unfolded Cube.
6 Edge Truncation 2 12 Face Polyhedra

Truncating 6 edges not truncated above (in green) produces this 12 sided figure.

This figure has 6 triangular faces, 6 pentagonal faces, 24 edges, and 14 vertices.

It is the reference for 6 Edge Cube Truncation Inside Outside.
Cube Triangulation (6) Cube Triangulation (6)

It is possible to divide a cube into 6 tetrahedra by bisecting the cube 3 times with planes that pass through opposite edges of the cube (shown in transparent yellow).  Only 6 of the cubes edges are involved in this truncation and they are the same 6 that were truncated in the figure above.

The resulting tetrahedra (1 of which is shown in orange) each contain the other cube edges that were unaffected by the truncation.  3 of these tetrahedron are mirror images of the others.
Tetrahedron Cut in Two Tetrahedron Cut in Two

It's possible to cut these tetrahedra once more to produce 12 smaller tetrahedra that are identical (congruent), i.e. not mirror images of one another.

The cutting plane extends from the uncut cube edge to the mid-point of the cube, bisecting the tetrahedron.  This means that the 6 cube edges that were not cut in the cube triangulation above, have now been cut.

These cuts can not extend through the cube to the opposite edge, or 24 tetrahedra would result, instead of the 12 we are looking for.
Cube Triangulation (12) Cube Triangulation (12)

Here 12 of these identical tetrahedra are show being put back together into a cube.

These tetrahedra are interesting building modules.  They can be used to construct any of the 218 forms resulting from the cube edge truncation shown above.  I have also used them as the basic building block for some of my sculpture.
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Last revised: 10/25/2002