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I am trying to divide octahedron into congruent parts. I found octahedron inside tetrahedron sided by four smaller tetrahedrons. I found some division here to 12 congruent parts. I can divide octahedron to four congruent parts by cutting it by fartest-away vertices.

Does there exist some sort of general theory about dividing the octahedron or other regular polyhedrons such as tetrahedrons?

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There is a general theory, starting from the solution by Dehn of Hilbert's Third Problem. Google should find information about this. –  Mariano Suárez-Alvarez Feb 11 '13 at 23:20
...trying to find some pictures about Scissor-equivalency and Dehn's theorem -- found some proof here. Not yet understanding its implications... –  hhh Feb 11 '13 at 23:30

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