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Simple observations: A regular tetrahedron can be cut into 2 mutually congruent pieces (in 3 obvious ways which are all basically the same way, giving one and same pair of congruent pieces). The regular tet can be cut into 3 mutually congruent pieces (infinitely many sets of 3 resultant pieces) and into 6 pieces (1 obvious set of resultant pieces). It also allows congruent partition into 4 pieces (only one obvious set of pieces) and also into 12 pieces (infinitely many different sets of pieces based on first cutting into 4), 8 pieces (1 obvious set of pieces) and 24 pieces (1 obvious set).

  • Is there any tetrahedron (regular or otherwise) that can be cut into n mutually congruent connected pieces (no other constraint on the pieces, not even convexity) for any other value of n, say, 16 or 20...?
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    $\begingroup$ Do you know whether this is possible for triangles? $\endgroup$
    – M. Winter
    Commented May 20 at 11:41
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    $\begingroup$ @M.Winter Isosceles triangle admits a filling by $kn^2$ congruent pieces for $k = 1, 2, 3, 6$. For any number which is a sum of 2 squares there's a right-angled triangle which can be cut into that many congruent triangles. As far as I know, the question whether there's a dissection of an isosceles triangle into 7 congruent pieces is still open (they cannot be triangles, which was proved by Beeson quite recently). $\endgroup$
    – Denis T
    Commented May 20 at 12:11
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    $\begingroup$ One simple note: the regular tetrahedron cannot be cut into any number of regular tetrahedra, and generically 'most' tetrahedra can't be cut into any number of tetrahedra congruent to the original, by non-zero Dehn invariant. (Of course, this does nothing to solve OP's question since this involves a strong additional constraint, but I figured it was worth noting) $\endgroup$
    – Steven Stadnicki
    Commented May 20 at 16:39
  • $\begingroup$ One probably crucial difference between triangles and tets could be: although tets do not have an obvious partition into arbitrarily large number of congruent pieces (all triangles allow such partition for n any perfect square), there is in 3D (unlike on plane), more 'room' for non-convex solids to curve and wind around one another so one might just have some non-trivial congruent partitions of the tet for n values like perhaps 16 or 20. $\endgroup$
    – Nandakumar R
    Commented May 20 at 17:42
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    $\begingroup$ I asked this question on Math StackExchange in the regular case three years ago here, along with some investigations into possible strategies for subdivision. $\endgroup$
    – RavenclawPrefect
    Commented May 27 at 0:16

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This MSE question exhibits two non-regular tetrahedra which can be decomposed into 8 smaller copies congruent to themselves; this yields $8^n$ for any $n$.

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  • $\begingroup$ This certainly is a partial answer - at least, I don't see a way to cut a regular tet into n pieces where n is any power of 8. Remark: It is not clear to me if allowing non-convex pieces would help at all. thanks for the pointer to your question. $\endgroup$
    – Nandakumar R
    Commented May 29 at 4:51
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A family of so-called Hadwiger-Hill simplices in $\mathbb{R}^d$ (https://en.wikipedia.org/wiki/Hill_tetrahedron) can be cut into $m^d$ congruent simplices, all similar to the original simplex; that is, they are $m^d$-reptile. See [2, Figure 1] for an illustration.

Maehara [1] described a construction of $d$ simplices that can be obtained from the right-angled Hadwiger-Hill simplex by repeatedly cutting in half. It can be shown that all these simplices are $(2m)^d$-reptile [3, Introduction]. As a consequence, the right-angled Hadwiger-Hill simplex can be cut into $2^km^d$ congruent pieces, for arbitrary values of $k,m$.

References:

[1] H. Maehara, Some replicating simplices other than Hill-simplices, Beitr. Algebra Geom. 55 (2) (2014), 429–432. https://doi.org/10.1007/s13366-013-0152-8

[2] J. Matoušek and Z. Safernová, On the Nonexistence of k-reptile Tetrahedra. Discrete Comput Geom 46, 599–609 (2011). https://doi.org/10.1007/s00454-011-9334-z

[3] J. Kynčl and Z. Patáková, On the nonexistence of k-reptile simplices in R^3 and R^4 (with Z. Patakova), The Electronic Journal of Combinatorics 24 (2017), Issue 3, P3.1, 44 pp. https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p1

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