Polyhedron A Projected into Polyhedron B

Affiliation: National Tsing Hua University, Taiwan

The basic task of a spy is to take pictures. The basic task of mathematics is to study functions. Here we are to study the central projections of interesting polyhedra among each other. Here are the main results:

- The regular icosahedron may be decomposed into 12 congruent parts with each part the projection of a face of a rhombic dodecahedron.
- The small rhombicosidodecahedron may be decomposed into 6 congrent parts with each part the projection of a face of a cube. Moreover, there are infinitely many such decompositions.
- The regular tetrahedron may be decomposed into four congruent parts non-trivially. In fact, there is a one-parameter family of such decompositions.
- A regular octahedron may be decomposed into six congruent parts with each part the projection of a face of a cube.
- The cube may be decomposed into six congruent parts non-trivially. Moreover, we show two essentially different families of such decompositions.

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