Octahedron in Tetrahedron

The applet below illustrates the fact that a tetrahedron is a union of four smaller tetrahedra and one octahedron. This fact (observed by Matt Henderson) leads to a solution to the problem of comparing the volumes of a regular tetrahedron and a square pyramid, with all 14 edges of the two shapes equal.

Unlike the dual embedding of an octahedron into a cube, now the vertices of the octahedron are located at the midpoint of the edges of the tetrahedron, and the edges of the former are the midlines of the latter.

What if applet does not run?

Drag the mouse to rotate the cube. Use the right button to remove and put back individual faces.

(Acknowledgement: I have learned most of Java details from the implementation by Meiko Rachimow.)

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Copyright © 1996-2018 Alexander Bogomolny

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