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

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

doi:10.1107/S0108767308040592

Acta Crystallographica Section A
Acta Crystallographica
Section A FOUNDATIONS AND ADVANCES

IUCr IT WDC

home archive editors for authors for readers submit subscribe open access

view article

Figure 4
Magnified views of fragments of the P, D and G surfaces (see Fig. 3

), with three-dimensional Euclidean symmetry operations marked on the surfaces. The arcs AB, BC and CA correspond to the two-dimensional in-surface (non-Euclidean) reflections R₂, R₁ and R₃, respectively, for all three surfaces (see main text). Left: The P surface, with $[Im\overline{3}m]$ symmetry (including isometries that swap sides of the surface; these are coloured differently here for clarity): A, B and C are located at the 12d (site symmetry $[\overline{4}m2]$ ), 24h (mm2) and 8c ( $[\overline{3}m]$ ) sites, respectively; AB and BC lie in orthogonal mirror planes, CA is a twofold axis. Middle: The D surface ( $[Pn\overline{3}m]$ ): A, B and C are located at the 6d ( $[\overline{4}2m]$ ), 12f (222) and 4c ( $[\overline{3}m]$ ) sites, respectively; AB and BC are twofold axes, CA lies in a mirror plane. Right: The G surface ( $[Ia\overline{3}d]$ ): A and C are located at the 24d and 16a sites, with point-group symmetries $[\overline{4}]$ and $[\overline{3}]$ , respectively; B lies on a twofold axis that is normal to the surface.

FOUNDATIONS
ADVANCES

ISSN: 2053-2733

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

doi:10.1107/S0108767308040592

Follow Acta Cryst. A

Search term		doi		Advanced search
Author		volume	page