West Chester University

Department of Mathematics

West Chester PA, 19383

Phone: (610) 430-4411

E-mail: vnitica@wcupa.edu

*Department of Mathematics, Anderson Hall 331a, West Chester University, West Chester,
PA 19383*

The goal of this note is to take a new look at some of the most amazing objects discovered
in recreational mathematics. These objects, having the curious property of making
larger copies of themselves, were introduced in 1962 by Solomon W. Golomb [2], and soon afterwards were popularized by Martin Gardner [3] in Scientific American. In Golomb's terminology a plane figure (or tile) is defined
to be replicating of order * k* (or rep-

**Question 1 ** Which polygonal rep-* 4* tiles are rep-

An equivalent formulation of the question is to find for which rep-* 4* tiles

**Question 2** (Finite Tiling Problem) Given a finite plane region * T*, and a finite set ∑ of tiles, find if

Let us only note here that methods do exist, coloring arguments among others, which
can be used to prove *non-existence* of tilings. We refer the reader to [1],[4], and [5] for other techniques, some of them stronger then coloring arguments. An open question
about non-existence of certain tilings related to reptiles will be mentioned during
our exposition. The answer to Question 1 will be positive most of the time. The main
technique used in the proof of our results is mathematical induction. Our proofs are
not necessarily the shortest, or the most elegant, possible. We encourage the reader
to try to find his own proofs before reading ours, and hope that this exercise will
get him hooked in this fascinating field.

As shown in Fig. 1, there are three trapezoids that are rep-* 4* tiles. We will denote them by

**Proposition 3** All three trapezoids * T1, T2, T3*, are rep-

*Proof*. For all three cases the proof is done by induction. We check first our hypothesis
for ** k = 2, 3**, and then show that rep-

**Figure 2:**

**Figure 3:**

*Proof for T2.* The cases k = 2, 3 are dealt with Fig. 4.

**Figure 4:**

**Figure 5:**

To show that rep-** k^{2}** implies rep-(

**Figure 6:**

**Figure 7:**

**Figure 8:**

To show that rep-** k^{2}** implies rep-

**Figure 9:**

There are three hexagons that can be dissected into four replicas (see **Fig. 9**). The first one is made out of three squares, the second one out of four squares,
and the third one out of five squares. We will denote them by * H1*,

**Proposition 4: ** All three hexagons * H1, H2, H3*, are rep-

Rep-tiles Revisited *Proof*. The proof is again done by induction, but here the length of the induction step
is different for each type of tile.

**Figure 10:**

*Proof for **H1**.* We show cases ** k =2, 3** in

**Figure 11:**

If ** k ≅ 1** (mod

**Figure 12:**

**Figure 13:**

If ** k ≅ 2** (mod

**Figure 14:**

**Figure 15:**

**Figure 16:**

*Proof for H2.* Observe that the proof is immediate if

**Figure 17:**

**Figure 18:**

**Figure 19:**

*Proof for H3.* We show cases

**Figure 20:**

There is a pentagon that can be dissected into four replicas (see **Fig. 20**). This figure is known as the "sphinx". It consists of five equilateral triangles.
We will denote it by * P*.

**Proposition 5** The pentagon * P* is rep-

**Figure 21:**

*Proof*. We show cases ** k=2,3** in

**Figure 22:**

If ** k ≅ 0** (mod

**Figure 23:**

If ** k ≅ 1** (mod

**Figure 24:**

The last two convex polygonal rep-**4** tiles we consider are obtained by changing slightly two of the tiles investigated
so far. Note that * H1* is being made out of three-fourths of a square. One can actually use here three-fourths
of any rectangle (see

**Figure 25:**

**Figure 26:**

A more interesting modification is to take the second hexagon * H2* and to skew it as in

**Figure 27:**

There are two stellated polygons in the list, both shown in **Fig. 27**. We will denote them by * S1* and

**Proposition 6**

is rep-**S1**for any*k*^{2};**k ≥ 1**is rep-**S2**for any k even;*k*^{2}is not rep-**S2**for any*k*^{2}odd.**k**

- J. H. Conway and J. C. Lagarias: Tiling with polyominoes and combinatorial group theory J. Combin. Theory Ser. A 53 1990 183-208
- S. W. Golomb: Replicating figures in the plane Mathematical Gazette 48 1964 403-412
- M. Gardner: On "rep-tiles", polygons that can make larger and smaller copies of themselves Scientific American 208 1963 154-157
- I. Pak: Ribbon tile invariants Trans. AMS 352 2000 5525-5561
- W. P. Thurston: Conway's tiling groups Amer. Math. Monthly 97 1990 757-773

**Rep-tiles Revisited**

This document was generated using the **LaTeX**2`HTML` translator Version 99.2beta8 (1.46)

Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.

Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.

The command line arguments were: **latex2html** `reptiles.tex`

The translation was initiated by Viorel Nitica on 2002-03-25