TILING PROBLEM: CONVEX PENTAGONS FOR EDGE-TO-EDGE

MONOHEDRAL TILING AND CONVEX POLYGONS FOR

APERIODIC TILING

TERUHISA SUGIMOTO

Abstract. We show that convex pentagons that can generate edge-to-edge mono-hedral tilings of the plane can be classified into exactly eight types. Using these

results, it is also proved that no single convex polygon can be an aperiodic pro-

totile without matching conditions other than “edge-to-edge.”

1. Introduction

A collection of sets (the “tiles”) is a tiling of the plane if their union is the entireplane, but the interiors of different tiles are disjoint. If all tiles in a tiling are ofthe same size and shape, then the tiling is called monohedral, and the polygon inthe monohedral tiling is called the prototile of the monohedral tiling, or simply, thepolygonal tile [6, 16–19].

In the problem of classifying the types of convex pentagonal tiles, only the pentag-onal case remains unsettled. At present, known convex pentagonal tiles are classifiedinto essentially 15 different types (see type 1, type 2, ... , type 15 in [6, 11, 16, 18,20, 22]). However, it is not known whether this list of types is complete (perfect)or not1 [2, 6–8, 13, 16–22]. Note that the problem of classifying the types of convexpentagonal tiles and the problem of classifying the pentagonal tilings (tiling patterns)are quite different.

Tiling by convex polygons is called edge-to-edge if any two polygons are eitherdisjoint or share one vertex or one entire edge in common. In an edge-to-edge tiling,the vertices are called nodes. Thus, the vertices of several polygons meet at a node ofan edge-to-edge tiling, and the number of polygons meeting at the node is called thevalence of the node [16–20].

In tilings by the 15 types of convex pentagonal tiles, there are edge-to-edge andnon-edge-to-edge tilings. A convex pentagonal tile belonging only to type 3, type10, type 11, type 12, type 13, type 14, or type 15 cannot generate an edge-to-edgetiling. The other eight types can generate an edge-to-edge tiling. (In Table 1, therepresentative tilings of type 1 or type 2 are non-edge-to-edge as shown, but there

1In 1918, Reinhardt showed five types of convex pentagonal tiles in his dissertation [12]. In 1968,

Kershner published the article which showed eight types of convex pentagonal tiles which addedthree new types [9]. For several years, it was generally accepted as fact that Kershner’s list of types

of convex pentagonal tiles is complete [6]. However, after Gardner wrote an expository article onconvex polygonal tiles in 1975 [4], Richard James discovered one type and Marjorie Rice discoveredfour types [5, 6, 13]. In 1985, the 14th type was discovered by Rolf Stein [6, 15, 21]. In July, 2015,

Casey Mann et al. discovered the 15th type [11,22]. See Appendix A for details of type 15.

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exist such convex pentagons that can generate edge-to-edge tilings belonging to type1 or type 2 [16,18,20]). Our main result is the following theorem2 [17, 19].

Theorem 1. If a convex pentagon can generate an edge-to-edge monohedral tiling,then it belongs to one of eight types in Table 1.

Remark. These eight types are not necessarily “disjoint.” Some of them can containthe same convex pentagon.

A tiling of the plane is periodic if the tiling can be translated onto itself in twononparallel directions. More precisely, a tiling is periodic if it coincides with itstranslation by a nonzero vector. A set of prototiles is called aperiodic if congruentcopies of the prototiles admit infinitely many tilings of the plane, none of which areperiodic. Thus, an aperiodic set of prototiles admits only nonperiodic tilings. (Atiling that has no periodicity is called nonperiodic. On the other hand, a tiling byaperiodic sets of prototiles is called aperiodic. Note that, although an aperiodic tilingis a nonperiodic tiling, a nonperiodic tiling is not necessarily an aperiodic tiling.) Forexample, the Penrose tiling is a nonperiodic tiling, and it can also be considered anaperiodic tiling that is generated by the aperiodic set of prototiles with a matchingcondition [1,6,7,14]. (A matching condition is a condition concerning edge-matching.In some cases, it can be represented by assigning colors and orientations to severaledges of the prototiles.) For aperiodic tilings, there are the following two problems [7].

(i) Is there a single aperiodic prototile (with or without a matching condition), thatis, one that admits only aperiodic tilings by congruent copies?(ii) It is well known that there is a set of three convex polygons that are aperiodicwith no matching condition on the edges. Is there a set of prototiles with a size lessthan or equal to two that is aperiodic?

For Problem (i), the affirmative solution (a convex hexagon with a matching condi-tion) was proved in recent years [14]. The aperiodic tiling by a set of three convexpolygons (one hexagon and two pentagons) with no matching condition on the edgesin Problem (ii) is based on the Penrose tiling [6]. Note that the aperiodic tiling isedge-to-edge.

Using Theorem 1, it can be proved that every convex polygon that generates anedge-to-edge tiling can generate a periodic tiling. Hence, we have the following theo-rem3 [17].

Theorem 2. Without matching conditions other than “edge-to-edge,” no single con-vex polygon can be an aperiodic prototile.

2. Outline of the Proof of Theorem 1

Our proof is based on Bagina’s Proposition (see Proposition 2.1 in [2], Section 1in [16] or, Subsection 2.2 in [18]), which implies the following:

2 We know that the same result was obtained by Bagina in 2011 after this result was obtained in2012 [3, 17].

3 Any triangle and any convex quadrilateral can generate a periodic edge-to-edge tiling. Allconvex hexagons that can generate a monohedral tiling are categorized into three types and can

generate a periodic edge-to-edge tiling. There is no convex polygon with seven or more edges thatcan generate a monohedral tiling [4, 6, 7, 10,12].

TILING PROBLEM 3

Table 1. Eight types of convex pentagonal tiles that can generateedge-to-edge tilings.

4 TERUHISA SUGIMOTO

In any edge-to-edge tiling of the plane by convex pentagonal tiles, there is a pentagonaround which at least three nodes have valence 3; in other words, there exists a tilewith at least three 3-valent nodes.

Now, let G be a convex pentagonal tile candidate that can generate an edge-to-edgetiling. Then, by Bagina’s Proposition, it has at least three vertices that will become3-valent nodes in the tiling. We choose (any) two of them as V1 and V2 and let v1and v2 be the conditions on the angles at V1 and V2, respectively. (For example, ifV1 = A, then v1 could be A + B + D = 360◦, 2A + B = 360◦, etc.) By substitutingall possible angle conditions in v1 and v2 and considering the lengths of the matchingedges, we can produce 465 patterns of pentagons as the candidates G. Examiningthese pentagons one by one, we sort them into (i) geometrically impossible cases, (ii)the cases that cannot generate an edge-to-edge tiling, (iii) the eight types in Table 1,and (iv) uncertain cases. In this first-stage sorting, there remain 34 uncertain cases(for details, see [16]). These remaining 34 cases are further refined into 42 cases byimposing extra edge conditions [18]. To check these 42 cases, we further prepare thefollowing lemma and corollary [18,19].

Lemma 1. If the density of k-valent nodes for k ≥ 5 in an edge-to-edge monohedraltiling by a convex pentagon is greater than zero, then there exists a tile with four ormore 3-valent nodes.

Corollary 1. If a convex pentagon with exactly three vertices that can simultaneouslybelong to tentative 3-valent nodes is a convex pentagonal tile that can generate an edge-to-edge tiling, then an edge-to-edge monohedral tiling by the tile is a tiling of the planewith only 3- and 4-valent nodes.

Note that the density of k-valent nodes is the ratio of the number of k-valent nodes tothe number of pentagons in an edge-to-edge monohedral tiling by a convex pentagon(see Subsection 2.1 of [18] for the exact definition of the density).

Applying Bagina’s Proposition, Lemma 1, Corollary 1, etc., we could conclude thatevery convex pentagon in the remaining 42 cases either belongs to the eight types inTable 1 or cannot generate an edge-to-edge tiling. A computer was used to performsuch an investigation.

Acknowledgments. The author would like to thank Emeritus Professor HiroshiMaehara, University of the Ryukyus, for his helpful comments and suggestions, Pro-fessor Shigeki Akiyama, University of Tsukuba, for his mathematical helpful com-ments and guidance, and Emeritus Professor Masaharu Tanemura, The Institute ofStatistical Mathematics, for a critical reading of the manuscript.

Appendix A.

In July, 2015, Casey Mann, Jennifer McLoud-Mann, and David Von Derau dis-covered the convex pentagonal tile of type 15 (see Figure 1) by a computerizedsearch [11,22].

TILING PROBLEM 5

Figure. 1. Convex pentagonal tiles of type 15.

References

[1] S. Akiyama, A note on aperiodic Ammann tiles, Discrete & Computational Geometry, 48 (2012)

221–232.[2] O. Bagina, Tiling the plane with congruent equilateral convex pentagons, Journal of Combina-

torial Theory, Series A 105 (2004) 221–232.[3] —, Tiling the plane with convex pentagons (in Russian), Bulletin of Kemerovo State Univ, 4 no.

48 (2011) 63–73. Available online: http://bulletin.kemsu.ru/Content/documents/Bulletin_

Kemsu_11_4.pdf (accessed on 7 November 2015).[4] M. Gardner, On tessellating the plane with convex polygon tiles, Scientific American, 233 no.

1 (1975) 112–117.

[5] —, A random assortment of puzzles, Scientific American, 233 no. 6 (1975) 116–119.[6] B. Grunbaum, G.C. Shephard, Tilings and Patterns. W. H. Freeman and Company, New

York, 1987. pp.15–35 (Chapter 1), pp.113–157 (Chapter 3), pp.471–487, pp.492–497, pp.517–518

(Chapter 9), pp.531–549, and pp.580–582 (Chapter 10).[7] T.C. Hallard, J.F. Kennet, K.G. Richard, Unsolved Problems in Geometry. Springer-Verlag,

New York, 1991. pp.79–80, pp.95–96 (C14), and pp.101–103 (C18).

[8] M.D. Hirschhorn, D.C. Hunt, Equilateral convex pentagons which tile the plane, Journal ofCombinatorial Theory, Series A 39 (1985) 1–18.

[9] R.B. Kershner, On paving the plane, American Mathematical Monthly, 75 (1968) 839–844.

[10] M.S. Klamkin, A. Liu, Note on a result of Niven on impossible tessellations, American Mathe-matical Monthly, 87 (1980) 651–653.

[11] C. Mann, J. McLoud-Mann, D. Von Derau, Convex pentagons that admit i-block transitivetilings (2015), http://arxiv.org/abs/1510.01186 (accessed on 23 October 2015).

[12] K. Reinhardt, Uber die Zerlegung der Ebene in Polygone, Inaugural-Dissertation, Univ.

Frankfurt a.M., R. Noske, Boran and Leipzig, 1918. Available online: http://gdz.sub.

unigoettingen.de/dms/load/img/?PPN=PPN316479497&IDDOC=57097 (accessed on 7 November

2015).

[13] D. Schattschneider, Tiling the plane with congruent pentagons, Mathematics Magazine, 51(1978) 29–44.

[14] J.E.S. Socolar, J.M. Taylor, An aperiodic hexagonal tile, Journal of Combinatorial Theory, 18(2011) 2207–2231.

[15] R. Stein, A new pentagon tiler, Mathematics magazine, 58 (1985) 308.

[16] T. Sugimoto, Convex pentagons for edge-to-edge tiling, I, Forma, 27 (2012) 93–103. Avail-able online: http://www.scipress.org/journals/forma/abstract/2701/27010093.html (ac-

cessed on 8 August 2015).

[17] —, Convex pentagons for edge-to-edge tiling and convex polygons for aperiodic tiling (in Japan-ese), Ouyou suugaku goudou kenkyuu syuukai houkokusyuu (Proceeding of Applied Mathematics

Joint Workshop) (2012) 126–131.

[18] —, Convex pentagons for edge-to-edge tiling, II, Graphs and Combinatorics, 31 (2015)281–298, doi: 10.1007/s00373-013-1385-x. Available online: http://dx.doi.org/10.1007/

s00373-013-1385-x (accessed on 8 August 2015).

6 TERUHISA SUGIMOTO

[19] —, Convex pentagons for edge-to-edge tiling, III, Graphs and Combinatorics (ac-

cepted) doi:10.1007/s00373-015-1599-1. Available online: http://dx.doi.org/10.1007/

s00373-015-1599-1 (accessed on 8 August 2015).

[20] T. Sugimoto, T. Ogawa, Properties of tilings by convex pentagons, Forma, 21 (2006) 113–

128. Available online: http://www.scipress.org/journals/forma/abstract/2102/21020113.

html (accessed on 8 August 2015).

[21] D. Wells, Curious and Interesting Geometry, The Penguin Dictionary of (Penguin science).

Penguin Books, London, 1991. pp.177–179.[22] Wikipedia contributors, Pentagon tiling, Wikipedia, The Free Encyclopedia, https://en.

wikipedia.org/wiki/Pentagon_tiling (accessed on 23 October 2015).

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