Surfaces of infinite genus

Within a topological context, most surfaces are hyperbolic, yet our knowledge of this geometric class is less developed than for the other two classes. This book is entirely concerned with these hyperbolic surface geometries. In particular, we focus on the most exotic topological species within the hyperbolic realm, namely surfaces of infinite genus. 

Three-periodic hyperbolic surfaces of infinite genus carve space into two intertwined sub-volumes, both resembling three-dimensional arrays of interconnected tubes. They are simple candidates for the interfaces in bicontinuous structures, consisting of two continuous subvolumes [4, 5]. As such they have attracted great interest as models for microstructured complex fluid interfaces, biological membranes, and structures of condensed atomic and molecular systems, to be explored in subsequent Chapters. 

Many more examples of interpenetration in inorganic chemistry lead to a recognition of the ubiquity of hyperbolic surfaces of infinite genus -exemplified by three-periodic minimal surfaces - that demands consideration. In the giant structure of Cu4Cd3 the Cu atoms are separated from the Cd atoms by a surface that resembles a minimal surface. In diamond, cubic ice and cristobalite, all the atoms are located on one side of the surface and the space on the other side is empty. If ice is subjected to very high pressure, the same structure appears on both sides of a minimal surface (double ice or ice IX), with almost double the density of ordinary ice (Fig. 2.8). Similarly, diamond is expected to transform to a double-diamond structure with metallic properties at sufficiently high pressure. 

This result suggests a remarkably simple picture of these frameworks as dense packings of flexible discs (each containing a single Si02 group) on periodic hyperbolic surfaces close to three-periodic miiumal surfaces of infinite genus. 

The polyhedron corresponding to the Neovius surface has the same arrangement of points as that for the infinite semi-regular polyhedral surface 6.43 discussed above, but the spaces between the points are differently filled with polygons so that each of the 48 points per cubic cell has the configuration of 8.4.8.6 and this leads to a surface of genus of 9. This surface has two kinds of flat points and is thus not regular (Mackay Terrones 1991). 12 tubes in the [110] directions connect cavities. 

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