<p>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", <a href="http://arxiv.org/abs/1006.1807" rel="nofollow">arXiv:1006.1807</a>.</p>

CC BY-SA 3.0
2011-09-27T22:01:45.543Z
2011-09-27T22:01:45.543Z
David Eppstein
<blockquote>
  <p>(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 ?</p>
</blockquote>

<p>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?</p>

<blockquote>
  <p>(ii) What if we require the tetrahedra in the subdivision to have the
  same size? (Again, I know $3 \leq k \leq 8$)</p>
</blockquote>

<cg.comp-geom>
Dissecting tetrahedra into similar tetrahedra
Herman Haverkort
# Dissecting tetrahedra into similar tetrahedra

(Asked by: username_0 on 2011-09-27)

> (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 $3 \leq k \leq 8$)

**Tags:** cg.comp-geom


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## 1 answers

### Answer by username_1 on 2011-09-27. Score: 4

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](http://arxiv.org/abs/1006.1807).


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