On the Penrose and Taylor–Socolar hexagonal tilings

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

doi:10.1107/S2053273317003576

Figure 17
$[v,v^{{\prime}}]$ are the ends of an edge e of a triangle T⁺ from the nesting determined by $[{\cal T}^{{+}}({\bf r})]$ . At its midpoint z we see the edge $[u,u^{{\prime}}]$ of a triangle T from $[{\cal T}({\bf q})]$ . The black triangles all come from the triangulation of $[{\cal T}({\bf q})]$ , the largest ones being of level 3. The edge e is maximal, in the sense that it is not part of an edge of some larger triangle from $[{\cal T}^{{+}}({\bf r})]$ . Thus, at its ends, the stripes of the large hexagons at v and $[v^{{\prime}}]$ are in the directions of the other sides of T⁺. The inner hexagons along $[vv^{{\prime}}]$ are centered at double hexagon centers and their stripes are all oriented in the same direction, namely perpendicular to $[vv^{{\prime}}]$ . At the left we have separated out the outer hexagons that overlay the small hexagons along $[vv^{{\prime}}]$ . We see their matching arrows and how their stripes align to form the edge $[vv^{{\prime}}]$ (in green). The colors (red/blue) of the short diameters of these large hexagons are determined by (or determine, whichever way one wants to put it) the color rule that we see in Fig. 10

, though note that the stripe of the large hexagons is perpendicular to that of the small ones, so the right/left crossing rule is opposite! The fact that the stripe orientation changes at the end dictates that the edge $[vv^{{\prime}}]$ is an interior edge of a larger triangle. The shift indicated by the orientation of the arrows matches this.

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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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