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Convex Pentagons for Edge-to-Edge Tiling, III

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Abstract

In previous papers, the cases in which there is a possibility that a convex pentagon generates an edge-to-edge tiling were sorted, and the remaining 42 cases in which there is uncertainty about whether a convex pentagon can generate an edge-to-edge tiling were shown. In this paper, the latter 42 cases are investigated using a computer. As a result, we find that convex pentagons that can generate edge-to-edge monohedral tiling of the plane can be sorted into eight types.

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Notes

  1. DDE-2 represents the sub-cases of the tentative 3-valent node \(\{D,D,E\}\). When convex pentagons are assembled around a tentative 3-valent node, there are eight possible ways to assemble the three pentagons. As a result, some sub-cases arise for edge fitting around the 3-valent node. For example, as shown in Fig. 2, \(\{D,D,E\}\) has two sub-cases, DDE-1 and DDE-2. Notice that the equations following the colons for each label in the figure represent the conditions on edge-lengths. The convex pentagon of this example can use only DDE-2 (the two arrangement way of DDE-2) from the edge conditions. Refer to Section 4 in [8] for details.

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Acknowledgments

The authors would like to give heartful thanks to Emeritus Professor Hiroshi Maehara, University of the Ryukyus, and Professor Shigeki Akiyama, University of Tsukuba, whose enormous support and insightful comments were invaluable. The authors would also like to thank Associate Professor Genta Ueno, The Institute of Statistical Mathematics, for a critical reading of the manuscript. The comments of the referees also helped to clarify the article.

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  1. The Interdisciplinary Institute of Science, Technology and Art, Suzukidaini-building 211, 2-5-28 Kitahara, Asaka-shi, Saitama, 351-0036, Japan

    Teruhisa Sugimoto

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  1. Teruhisa Sugimoto
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Correspondence to Teruhisa Sugimoto.

Appendix A.1

Appendix A.1

Let \(T_{x}\) be a set of convex pentagonal tiles that belongs to a type x for x = 1, 2, 4, 5, 6, 7, 8, 9. Figure 7 shows examples of the Venn diagram of \(T_{x}\). Figure 8 shows examples of the convex pentagonal tiles that are contained in each intersection of Fig. 7.

Fig. 7
figure 7

Venn diagram of \(T_{x}\)

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Fig. 8
figure 8

Convex pentagonal tiles that are contained in each intersection of Fig. 7. Representative tilings of type 2 by convex pentagonal tiles of a, i, j, m, and o are always non-edge-to-edge. Representative tilings of type 1 by convex pentagonal tiles of c, e, f, and p are always non-edge-to-edge

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Sugimoto, T. Convex Pentagons for Edge-to-Edge Tiling, III. Graphs and Combinatorics 32, 785–799 (2016). https://doi.org/10.1007/s00373-015-1599-1

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