Smallest known rep-tile which is a 3-torus
A k-rep-tile in 3D is a polyhedron P which can be dissected into k congruent parts, each of which is similar to P. In 1997 C. Goodman-Strauss asked at an Oberwolfach meeting whether there is a rep-tile which is connected, but not simply connected. In fact, such examples are easily constructed in 3D. G. van Ophuysen found this one at the same meeting. It is a 24-rep-tile. That is, only 24 copies of it can be assembled yielding a scaled copy of it. This seems to be the smallest example so far.
This non-convex polytope P is made from 12 rectangular boxes of side length 1, a, b. Two copies of P can be assembled into a rectangular box of edge length 3, 2a, 4b. With a=3 1/3 , b=9 1/3 /2, this box is similar to the smaller ones. In other words: with these values of a and b we obtain 1:a:b = 2a:4b:3. Twelve of such larger boxes can be assembled into a larger copy of P, namely, 2a P.
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Submitted: Fri Nov 28 18:18:49 CET 2008.
Revised: Tue Dec 23 11:05:56 CET 2008.
Accepted: Mon Feb 1 10:17:41 CET 2010.
Univ. BielefeldIwan Suschko