Sibley, Thomas Q.

Wagon, Stan

English

College of Saint Benedict and Saint John’s University: DigitalCommons@CSB/SJU

College of Saint Benedict and Saint John's University DigitalCommons@CSB/SJU Mathematics Faculty Publications Mathematics 3-2000 Rhombic Penrose tilings can be 3-colored Thomas Q. Sibley College of Saint Benedict/Saint John's University, tsibley@csbsju.edu Stan Wagon Follow this and additional works at: https://digitalcommons.csbsju.edu/math_pubs  Part of the Geometry and Topology Commons Recommended Citation Sibley TQ, Wagon S. 2000. Rhombic Penrose tilings can be 3-colored. The American Mathematical Monthly 107(3): 251-253. The American Mathematical Monthly © 2000 Mathematical Association of America NOTES Edited by Jimmie D. Lawson and William Adkins Rhombic Penrose Tilings Can Be 3-Colored Tom Sibley and Stan Wagon The four-color theorem tells us that any tiling in the plane can be 4-colored, where adjacent tiles are to get different colors. Anyone who tries to color a Penrose tiling will wonder whether they are all 3-colorable; this question was first proposed by John H. Conway [2, p. 27]. We show, by a very simple argument, that any tiling by Penrose rhombs is 3-colorable; our method yields an algorithm, the results of which are illustrated in Figure 1. We were led to this result by a Mathematica implementation of a 4-coloring algorithm for plane maps based on Kempe chains; see [5] and [7]. It had no difficulty in 3-coloring rhombus tilings. For background on Penrose tilings see [2], [4], and [8]. Note that there is no simple characterization of the 3-colorable planar maps, and the problem of recognizing 3-colorable planar maps is NP-complete [3]. Figure 1. A 3-coloring of a Penrose rhomb tiling with 228 tiles. In fact, our main result is much more general, and has nothing to do with Penrose geometry. Call a map with polygonal countries tidy if whenever two countries meet, they do so either in a single point, or in a complete edge (or several complete edges) of each polygon of the pair. March 2000] NOTES 251 This content downloaded from 152.65.133.188 on Mon, 24 Nov 2014 16:15:16 PMAll use subject to JSTOR Terms and ConditionsTheorem. Any tidy plane map whose countries are parallelograms is 3-colorable. Proof: The Erdos-de Bruijn theorem [1] states that an infinite map is 3-colorable whenever every finite submap is. Given a tidy finite map it suffices to show that it has a country having at most two neighbors. Recalling the Meisters two-ear theorem for polygons [6, p. 16], let us use the term elbow for such a country. Given an elbow, remove it, color the remainder by induction, and then replace the elbow and color it with the free color, which must exist since it has at most two neighbors. Thus we need to show only that every finite and tidy collection of parallelograms has an elbow. We do that in a separate lemma. The Three-Elbow Lemma. Any tidy, finite collection of parallelograms in the plane contains an elbow; if the collection has at least three elements then it must contain at least three elbows. Proof: The basic idea is that if there were no elbows then the boundary could never turn adequately in the right direction so as to close up. A checkerboard is a good example: the squares at the corners-where the boundary turns-are elbows. We now make this precise. Suppose, to get a contradiction, that we have a map as hypothesized with no elbows. We may work with a submap that is connected in the sense that there is a path from any country to any other that does not pass through any vertices of the map; such a submap cannot be disconnected by the removal of a single point. Let P be the simple, closed polygon that forms the exterior of the submap and let n be the number of vertices on P (Figure 2). The interior angles of P sum to (n - 2)Yr. But at each vertex there is an edge coming in and an edge going out. The lack of elbows means that these edges belong to different countries. The parallelogram belonging to the edge coming in contributes two angles to the polygon's interior angles, and they sum to IT. It follows that the sum of all the interior angles of P is at least nm, a contradiction. Since an elbow can subtract strictly less than iT from this interior angle sum, there are at least three elbows. This succinct proof-a simplification f our original proof-is due to Michael Schweitzer (Berlin). z~~~~~~~~I 4 Figure 2. The thick line is part of the boundary of a tidy tiling by parallelograms: if there is no elbow, then each edge contributes X to the interior-angle sum. 252 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 107 This content downloaded from 152.65.133.188 on Mon, 24 Nov 2014 16:15:16 PMAll use subject to JSTOR Terms and ConditionsThe maps shown in Figure 3 ((a) is due to Rick Mabry) show that our results do not hold for convex quadrilaterals in general, or for untidy maps with parallelo- grams. Both maps require four colors. And these maps are minimal with respect to the number of countries used (thanks to David Castro for a proof of this for case (a)). (a) (b) Figure 3. (a) A collection of 7 convex quadrilaterals that requires four colors. (b) An untidy rectangular tiling that needs four colors. It is not at all clear whether these ideas yield anything for the Penrose kites and darts. Figure 4 shows a kite-and-dart configuration that is elbowless (there are many such); but it is easily 3-colored, and it seems as if larger kite-and-dart ilings can be 3-colored. Figure 4. A collection of 15 Penrose kites and darts, each of which has degree 3 or 4. REFERENCES 1. N. G. de Bruijn and P. Erdos, A colour problem for infinite graphs and a problem in the theory of relations, Indag. Math. 13 (1951) 369-373. 2. M. Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman, New York, 1989. 3. M. R. Garey and D. S. Johnson, Computers and Intractibility: A Guide to the Theory of NP-Complete- ness, W. H. Freeman, New York, 1979. 4. B. Griinbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1986. 5. J. P. Hutchinson and S. Wagon, Kempe revisited, Amer. Math. Monthly 105 (1998) 170-174. 6. J. O'Rourke, Computational Geometry in C, Cambridge University Press, New York, 1993. 7. S. Wagon, An April Fool's hoax, Mathematica in Education and Research 7:1 (1998) 46-52. 8. S. Wagon, Mathematica in Action, 2nd ed., Springer/TELOS, New York, 1999. St. John's University, Collegeville, MN 56321 tsibley@csbsju.edu Macalester College, St. Paul, MN 55105 wagon@macalester.edu March 2000] NOTES 253 This content downloaded from 152.65.133.188 on Mon, 24 Nov 2014 16:15:16 PMAll use subject to JSTOR Terms and Conditions

Rhombic Penrose tilings can be 3-colored

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Rhombic Penrose tilings can be 3-colored

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