Dissecting tetrahedra into similar tetrahedra (i) Does anybody know what is the smallest number $k$ such that there
  is a tetrahedron T that can be subdivided into $k$ tetrahedra that are
  similar to T ?


I know $k$ is at most 8 (I know a tetrahedron that can be subdivided into 8 similar tetrahedra), and at least 3 (I can prove no tetrahedron can be subdivided into 2 tetrahedra that are similar to the whole). Does anybody know better lower or upper bounds?


  (ii) What if we require the tetrahedra in the subdivision to have the
  same size? (Again, I know $3 \leq k \leq 8$)
Hi Herman — since you asked me this at GD, I've found an answer to version (ii) of your question. Jiří Matoušek and Zuzana Safernová have shown that subdivision of a tetrahedron into $k$ tetrahedra that are similar to the original and congruent to each other is possible only when $k$ is a cube (and it is conjectured more strongly that this is possible exactly when $k$ is a power of eight, the case you already know about). See their paper "On the nonexistence of k-reptile tetrahedra", arXiv:1006.1807.