#2             80. TWO EXTRA VERTICES             

I have just received another nuisance call from Geo. Metter. He was prattling on that he had heard it was possible to cut a regular tetrahedron into two identical pieces such that each piece had six vertices.

He couldn’t do it, and would I be able to help him since I was the fount of all knowledge and ... .

I told him I would ring him back. I found it easy to obtain two identical pieces each having four vertices (see left). Six was more difficult, but after some time I managed to solve Geoff’s problem. I then wondered whether the problem generalized.

Thus, it is easy to cut a cube into two identical pieces each having eight vertices. But could I obtain two identical pieces each having ten vertices?

Can you solve these two problems?
HINT 1

These are three-dimensional problems. It might be helpful to handle a model of each solid.
HINT 2

In the tetrahedron case, you are told that each half had six vertices. Where could these vertices lie? Remember that they must include the actual vertices of the tetrahedron.
SOLUTION

EXTENSIONS

1. If you are good with your hands, for each polyhedron you might try making the two pieces from wood or plastic. I have a commercially built version of the tetrahedral dissection. People find it amazingly difficult to assemble the two pieces correctly.

2. Explore the other regular solids. Are their analogous dissections for them?