## 3-Torus Rep-Tile

Electronic Geometry Model No. 2010.02.001#### Authors

Dirk Frettloeh and Iwan Suschko#### Description

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.

Model produced with: JavaView v3.95.001

**Keywords** | | rep-tile |

**MSC-2000 Classification** | | 52B10 |

#### References

- G. van Ophuysen:
*Tagungsbericht 20 1997* (1997), .

#### Files

#### Submission information

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.

#### Authors' Addresses

Dirk Frettloeh
Univ. Bielefeld

Fakultaet Mathematik

Univ. Bielefeld

33501 Bielefeld

dirk.frettloeh@udo.edu

www.math.uni-bielefeld.de/baake/frettloe

Iwan Suschko
Univ. Bielefeld

Fakultaet Mathematik

Univ. Bielefeld

33501 Bielefeld