Tesselations

Multiplicity Wyckoff letter Site symmetry Coordinates of equivalent positions 

From the forgoing, it is seen that in order to specify a two-dimensional crystal structure, all that needs to be specified are (i), the lattice parameters, (ii), the plane group symbol, (iii), a list of atom types together with the Wyckoff letter and the atomic coordinates. For complex crystals, it simplifies the amount of information to be recorded enormously. 

A non-periodic (aperiodic) tesselation or tiling is one that cannot be created by the repeated 

---

The speed of the method comes from two sources. First, all of the macroscopic cells of the same size have exactly the same internal structure, as they are simply formed of tessellated copies of the original cell, thus each has exactly the same multipole expansion. We need compute a new multipole expansion only once for each level of macroscopic agglomeration. Second, the structure of the periodic copies is fixed we can precompute a single transfer... 

The PCM algorithm is as follows. First, the cavity siuface is determined from the van der Waals radii of the atoms. That fraction of each atom s van der Waals sphere which contributes to the cavity is then divided into a nmnber of small surface elements of calculable surface area. The simplest way to to this is to define a local polar coordinate frame at tlie centre of each atom s van der Waals sphere and to use fixed increments of AO and A(p to give rectangular surface elements (Figure 11.22). The surface can also be divided using tessellation methods [Paschual-Ahuir d al. 1987]. An initial value of the point charge for each surface element is then calculated from the electric field gradient due to the solute alone ... 

The hexahedral tessellation presents more of a problem because of its nonorthogonality. Bayes [bayes87b] suggests two update schemes, both based on situating the sites on a cubic lattice (1) update according to... 

After exhaustive simulations, Bayes has found only three rules on the cubic lattice ( = (4555), = (5766), = (5655)) and two rules on the hexahedral tessellation ( = (3333) and = (4633)) tliat can arguably be called worthy of the name ([bayes87a], [bayesQO] and [bayesQl]). We follow Bayes nomenclature in calling these rules, respectively, Life-4555, Life-5766, Life-5655), Life-3333 and Life-4633. 

While Svozil certainly does a credible job in questioning how a local fermion field theory on a tessellated space might still be made able to circumvent the species doubling problem and thereby be converted (at least formally) into a CA-like theory, one could also argue that Svozil does not take his argument as far as it could be... 

Iaing75a] Laing, R. Artificial Molecular Machines A Rapproachment Between Kinematic and Tessellation Automata, in Proceedings of the International Symposium on Uniformly Structured Automata and Logic, Tokyo (1975). 

Tessel, RE., and Woods. J.H. Fenfluramine and N-ethyl amphetamine Comparison of reinforcing and rate-decreasing actions in the rhesus monkeys. Psychopharmacologia 43 239-244, 1975. 

Tessel. RE. Woods, J.H. Counsell, R.E. and Lu, M. Structure-activity relationships between meta-substituted N-ethylamphetamines and locomotor activity in mice. J Pharmacol Exp Ther 192(2) 310-318, 1975. 

Antagonism of several characteristic effects of amphetamine and cocaine by the alpha adrenergic receptor antagonist prazosin is a most recent example of noradrenergic mechanisms in the actions of psychomotor stimulants (Tessel and Barrett 1986). We investigated whether or not prazosin may attenuate the disruptive effects of amphetamine on social and aggressive behavior in mice and squirrel monkeys (Miczek, unpublished observations). Pretreatment with prazosin (0.4 mg/kg) attenuated the disruption of attack... 

A minor problem with 2D peak capacity is that it is based on a square or rectangular tessellation (Guiochon et al., 1982) instead of a cubic closest packed... 

The way of the best choice to model PS s structure on both molecular and supramolecular levels begins with allocation of primary building units (PBUs), which without gaps and overlaps would fill a 3D space occupied by a PS. An universal method for allocation of such PBUs in both ordered and randomly arranged PSs, formed of packings of convex particles (or pores), is based on the construction of the assembles of Voronoi polyhedra (V-polyhedra) and Delaunay simplexes (or D-poly-hedra), which form Voronoi-Delaunay tessellation [100],... 

Figure 9.27 Example of Voronoi (V, dotted line) and Delaunay (D, solid line) tessellations for a 2D case.

The geometry of V-polyhedra and V-tessilations was elaborated by Voronoi and Delaunay [143,144], They have shown that all PBU/Ps have a convex shape, each facet is common for two neighboring PBU/Ps, each edge is formed by no less than d PBU/Ps (d is the dimension of V-tessellation), and no less than d + 1 PBU/Ps intersect at each vertex. We write no less but an exact coincidence usually takes place. 

It is significant that both V- and D-tessellations are unequivocally interrelated because they form geometrically dual graphs. For a brief expression of a relation between these two lattices let us designate a total number of facets in V- and D-lattices as/yE and E, and the total number of edges as BV1 and liV) z, respectively. For dual graphs the relation is [100,146,147]... 

Concluding this section we note that VD-tesselations may have numerous different applications. For example, each lion, which lives in Savanna assumes himself as a host of the whole territory that is closer to his den than to the den of his neighbor [100], Thus, the boarders of the territory of each lion may be estimated by V-polyhedra. By analogy, each archaeologist attempts to identify the parts of region under the influence of different neolithic clans by VD-tesselation, the urban planner locates public schools in a city, a physiologist examines a capillary supply to muscle tissue, etc. [141],... 

B. Boots, A. Okabe, and K. Sugihara, Spatial Tessellation. Concepts and Applications ofVoronoi Diagrams, Wiley, New York, 1992. 

Figure 7 (Left) Typical configuration of particles before particle i is removed. (Right) After particle i is removed, tessellations must be reconstructed within the superpolyhedron (bold, dashed line). The volume and surface area of the cavity that once held the removed particle can then be determined. (Adapted from Ref. 71).

---