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Figure 3
Relationship between the F-type 3D AKN tiling with edge length of a and that with edge length of τ2a. (a) Rhombic triacontahedron OD [{\scr C}] with edge length of a, (b) acute rhombohedron and (c) obtuse rhombohedron units in the former tiling. The black and grey balls correspond to `even' and `odd' vertices, respectively. The dashed line represents the threefold axis of each rhombohedron. The arrangement of [{\scr C}]'s on (d) the fivefold plane spanned by (1, 0, 0, 0, 0, 0) and (0, 1, 1, 1, 1, 1), (e) the threefold plane spanned by [(1,0,\bar{1},0,\bar{1},0)] and [(0,1,0,\bar{1},0,1)] in the F-type 6D superstructure that generates the F-type 3D AKN tiling with edge length of a. The arrangement of [\tau^{-2}{\scr C}]'s, which generates the F-type AKN tiling inflated by τ2, in (f), (g), is obtained by the similarity transformation of order 2 from that of [{\scr C}]'s in (d), (e). The black and grey line segments in (d)–(g) represent the 2D sections of [{\scr C}]'s or [\tau^{-2}{\scr C}]'s that generate `even' and `odd' vertices, respectively. The thick black line shows the 2D section of the 6D unit cell.

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ISSN: 2053-2733
Volume 80| Part 6| November 2024| Pages 422-438
https://doi.org/10.1107/S2053273324008568
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