 | Acta Crystallographica Section A Acta Crystallographica Section A FOUNDATIONS AND ADVANCES |
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![[Figure 12]](uv5024fig12mag.jpg)
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Figure 12
(a) A planar unit-distance graph, the corresponding vertex and edge subsets, and the 2D and 3D modes of geometric modification. Mode (1) involves moving vertex 2 in 2D (in the plane of the unit-distance graph) as shown in (b) and requires that the 1–2 and 2–4 edges are of unequal length (red dashed lines), and thus mode (1) is invalid. Mode (9) involves moving vertex 2 in 3D (out of the plane of the unit-distance graph) as shown in (c) and results in a geometrically distinct unit-distance graph while retaining equal edge lengths, and thus mode (9) is valid. Mode (11) involves moving vertices 1 and 2 in 3D as shown in (d). Here, equal edge lengths are retained but a geometrically distinct unit-distance graph is not produced; instead, the unit-distance graph is rotated in 3D, and thus mode (11) is a rotational mode. Mode (13) involves moving vertices 2 and 3 in 3D as shown in (e), and results in a geometrically distinct unit-distance graph while retaining equal edge lengths, and thus mode (13) is valid. (f) Four geometrically distinct graphs that correspond to mode (9) and the movement of vertex 2 to positions `a', `b', `c' and `d'. Each valid mode corresponds to an infinite number of geometrically distinct unit-distance graphs. Dashed black lines and arrows show the movement of a vertex labelled n to the position n′. Valid, invalid and rotational modes are shown in green, red and purple, respectively. |
![[Figure 12]](uv5024fig12.jpg)
| FOUNDATIONS ADVANCES |
ISSN: 2053-2733
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