THE MATHEMATICS OF STRUCTURES

Michael Jacob

Inorganic Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm Sweden

Structure and composition specify the chemical and physical properties of a material. For the development of modern solid state chemistry and physics, the understanding of the structure is of vital importance for constructing new materials with desirable properties, and for the understanding of the structure, the description is the foundation.

Classically, solid state material is described in terms of atomic packings, interconnected coordination polyhedra, bond networks, minimal surfaces or rod packings, all in order to reduce the information down to a level where it can be interpreted simpler. Now a more powerful and dynamical tool for structure description is being developed - the mathematics of structures - which will be presented here. These mathematics are based on the exponential scale mathematics, in which several equations can be combined on the exponential level to yield a new îhybridisedî function. With this method, geometries that earlier were not possible to describe with analytical functions in 3D real space can be constructed. Crystal structures can be described with single mathematical equations in terms of all the above mentioned packings, with the distinction that these descriptions are dynamical, i.e. are not restricted to only one type of illustration [2,3,6]. By changing the value for which the function is displayed, the isosurface value, the description can be scanned from, for example, an atomic packing over a packing of interconnected polyhedra to a nodal surface description, resulting in a more thorough and deeper understanding of the structure. The descriptionís dynamical property gives an indication of the forces and channel spaces present in the structure.

Apart from crystal structures a vast number of geometries can be described with these mathematics, as for instance polyhedra [1], molecules and clusters [6], and more specific the DNA double helix with its ten base pairs per pitch and bridging bonds [4]. Also, the mathematics can be used for creating finite periodic structures - structures that only exist within natural boundaries and which may be modulated in their repetition spacing [5,7].

1 S. Andersson, M. Jacob, S. Lidin, On the shapes of crystals, Z. Kristallogr. 210, 3-4 (1995).
2 S. Andersson, M. Jacob, S. Lidin, The exponential Scale and Crystal Structures, Z. Kristallogr. 210 (1995) 826-831.
3 S. Andersson, M. Jacob, On the structure of mathematics and crystals, Z. Kristallogr. 212 (1997) 334-346.
4 M. Jacob, Saddle, Tower and Helicoidal surfaces, J.Phys. II France 7 (1997) 1035-1044.
5 M. Jacob, S. Andersson, Finite Periodicity and Crystal Structures, Z. Kristallogr. 212 (1997) 486-492.
6 S. Andersson, M. Jacob, The exponential Scale, Supplement No. 13 of Zeitschrift für Kristallographie, R. Oldenbourg Verlag, München, 1997.
7 M. Jacob, S. Andersson, Finite periodicity, chemical systems and Ninham forces, Colloids and Surfaces 129-130 (1997) 227-237.