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Figure 15
(a) A unit-distance graph, the corresponding directed proto-graph, vertex subsets and the 2D and 3D modes. Modes (1), (2) and (3) cannot be applied to produce unit-distance graphs that are geometrically distinct from the original unit-distance graph in (a) without forcing unequal edge lengths and are therefore invalid. Mode (4) involves moving vertices 1 and 2 in 3D as shown in (b) to produce a planar unit-distance graph that is geometrically identical to the original unit-distance graph in (a). Mode (4) may be applied to every second repeat unit as shown in (c) to produce a non-planar unit-distance graph that is geometrically distinct from the original unit-distance graph in (a). Mode (4) may also be applied by moving vertices 1 and 2 to symmetrically non-equivalent positions in each repeat unit as shown in (d) to produce an additional geometrically distinct non-planar unit-distance graph, and therefore mode (4) is valid. Blue 3D sheets show the geometry and orientation of each unit-distance graph. Legend as in Figs. 10[link] and 12[link].

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ISSN: 2053-2733
Volume 80| Part 3| May 2024| Pages 258-281
https://doi.org/10.1107/S2053273324002432
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