Inner and Outer Beauty - Beauty in Chemistry - page 4

Fig. 3 M.C. Escher's Study of Regular Division of the Plane with Human Figures (1944). As in
Escher's other tessellation diagrams, translational and point symmetry operations are used to
completely fill the plane with repeated, ordered objects. Copyright 2011 The M.C. Escher
Company - Holland. All rights reserved. www.mcescher.com
This illustrates a type of symmetry only seen in crystals and other extended
arrays. That is, the symmetry operation combines both elements of point symmetry
(as seen in molecules) and translation (which generates arrays). Here you can see
that the repeat of this operation yields a vertical translation of one unit. The two-
and three-dimensional space groups are realizations of the more general topic of
group theory, which has been one of the tremendous scientific achievements in the
last two centuries in the field of pure mathematics.
1 Symmetrical Extended Arrays
Cubic Space Division (1952) [ 7 ] (Fig. 4 ) anticipates a current chemical interest in
open arrays as storage materials.
While this general topic has an ancient lineage, it is an area of intense current
research. What is now described as the Hofmann clathrate was first reported in 1897
[ 8 ]. However, the structure was not known until 50 years later when reported by
Powell and coworkers [ 9 ]. Single crystal X-ray diffraction showed the structure in
Fig. 5 , in which a two-dimensional array of nickel cyanide encapsulates trapped
benzene molecules.
A general review of the principles and structures of metal coordination chemistry
arrays was published in 1964 by Bailar, one of the founders of modern inorganic
chemistry [ 10 ]. The chapter “Coordination polymers” included both inorganic and
organic bridging ligands. The extension of this chemistry into something more like
Escher's vision in Fig. 4 was described by Hoskins and Robson [ 11 ], stating in their