On the Penrose and Taylor–Socolar hexagonal tilings

Lee, J.-Y.; Moody, R.V.

doi:10.1107/S2053273317003576

Figure 10
In the center of the figure we see the large pair of triangles making a diamond shape between the two extreme vertices v and $[v^{{\prime}}]$ , which are assumed to be vertices arising from centers of the large hexagons of the double hexagon tiling. The diamond is made up of an opposite pair of fully internally nested triangles. The triangles are all in their stretched form, but the line thicknesses indicate the nesting relationships. On the left side we see the corresponding arrangement of double hexagons that surround the small hexagons between v and $[v^{{\prime}}]$ . The arrows must match, but their common orientation in the horizontal direction is irrelevant here. There is a color change as we cross each triangle edge at right angles. The key point is what happens at the ends of the diamond as the color line crosses edges (indicated by the thickest lines) which are not at right angles to it. The main edge at v is shown by the heavy black line. The rules for coloring hexagons show that the color stripe is fully red here, see Fig. 4

. The main edge at v has to be matched with its partner at $[v^{{\prime}}]$ , where the color strip changes to fully blue. Notice the correct color change at the arrow. The scenario shown on the right side of the figure, where the main edge at $[v^{{\prime}}]$ is shown in purple, cannot occur because of the color change violation at the brown arrow.

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ISSN: 2053-2733

Volume 73| Part 3| May 2017| Pages 246-256

https://doi.org/10.1107/S2053273317003576

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