Regular tesselation

Table 1.2 Relation between average ring size, n, connectivity, z, and Euler-Poincare cheuracteristic per vertex, xl i a range of networks that are regular tessellations of surfaces. The nature of the network is set by the value of cages, planar networks and three-dimensional frameworks are characterised by positive, zero and negative respectively.

You can create semi-regular tessellations by combining different types of shapes, like rectangles and triangles or octagons and squares. Think of some combinations that will work. 

Figure 3.15 The regular tesselations (a) equilateral triangles (b) squares (c) regular hexagons...

The geometrical characteristics of the reciprocal lattice rows parallel to c, each taken as a whole, are termed "row features". In the Trigonal model all mica polytypes have the same row features, described by the regular tessellation 3,6 (Takeda and Donnay 1965 see the section Tessellation of the hp lattice ), and the nine R, were classified into three types (Fig. 16) ... 

Assuming the mica two-dimensional lattice in the (001) plane is hp [ray = 0], the lattice can be described through a regular tessellation 3,6, i.e. an assemblage of equal... 

The relation between cp and cp is derived taking into account that a node belonging to one set is related to the two nearest nodes of the other set by two reflection lines that intersect at the origin. For the regular tessellation, only one set of six nodes with the same r exists, each node being 60° apart in this case cp = cp = 0° (mod 60°). 

There are only three regular tesselations of the plane triangles, squares, and hexagons. 

Tessellation of hexagons Figure 2 The three regular tessellations... 

Dre84] A. W. M. Dress, Regular polytopes and equivariant tessellations from a combinatorial point of view, in Algebraic Topology, SLN1172, Gottingen, 1984,56-72. 

Figure 7. A tessellation of the hyperbolic plane H2 by regular heptagons. The 7.62 tessellation (thicker lines) is obtained by truncation, a hexagon replacing each three-fold vertex. Polygons are connected as they are locally in Vanderbilt and Tersoff s surface.

Fig. 14 A picture of the previous regularly crosslinked polymer which emphasizes the designated cyclic monomer, while maintaining the divisions of the tessellated plane. This model produces the canonical systemic name...

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...

The limitation on the types of n-fold rotational axes can be easily visualized by considering the analogous task of completely tiling a two-dimensional plane with polygon tiles, a process called tessellation. Congment regular polygons (equilateral and... 

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