Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples

Ramsden, S.J.; Robins, V.; Hyde, S.T.

doi:10.1107/S0108767308040592

Figure 2
This table of images presents the various stages in our map from the hyperbolic plane (top) to Euclidean space (bottom). (a) Delaney–Dress symbol enumeration with $[\star{\tt 2223}]$ symmetry gives a subgroup tiling, QC164, containing five distinct flags. $[\star{\tt 2223}]$ domains are bounded by orange edges; a single domain is shaded. (b) The h-net, hqc167, is a maximally symmetric version of the net defined by the edges and vertices of the tiling QC164. (c) We geometrize the subgroup tiling to form a U-tiling, UQC183, that respects the local and translational symmetries of the TPMS (in this paper the P, D and G surfaces). The single $[\star{\tt 2223}]$ (shaded) domain from (a) adopts the geometry of a pair of $[\star{\tt 246}]$ triangles. (d) The U-tiling is projected onto a genus-3 torus (tritorus), giving an O-tiling. (e) The edges of the embedded surface reticulation define an o-net. (f) A hyperbolic translational cell built from the $[\star{\tt 2223}]$ domains of (c) is highlighted by the orange polygon. (g) This polygon projects onto the D surface to give an E-tiling, EDC183, whose rhombohedral unit cell is shown. (h) The projected tile boundaries on the D surface form the triply periodic e-net, edc183, with space group P4₂ 32. A single unit cell is shown, with quadrilateral cycles highlighted in purple for clarity. (i) The s-net, sqc7388, is the highest-symmetry embedding of the e-net topology, with space group $[Pm\overline{3}n]$ .

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ISSN: 2053-2733

Volume 65| Part 2| January 2009| Pages 81-108

doi:10.1107/S0108767308040592

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