Cube Containing a Tetrahedron and an Octahedron

This is the image of a cube after parallel projections (affine transformations) so what you see is a parallelopiped. Parallel lines and midpoints remain in this view, but angles and distances are not preserved.

Nonetheless, this shows pretty well how a regular tetrahedron can be constructed from 4 of the 8 vertices of the cube. The midpoints of the sides of the tetrahedron are the centers of the faces of the cube, so these midpoints are the 6 vertices of a regular octahedron. The 4 faces of the octahedron that are colored at the faces that are midpoint triangles of the faces of the tetrahedron.


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