Geometric polyhedral complex

Definition 2.38. A geometric polyhedral complex P in R is a collection of convex polytopes in R such that... 

Most of the terminology, such as skeleton, subcomplex, join, carries over from the simplicial situation. One new property worth observing is that a direct product of two geometric polyhedral complexes is again a geometric polyhedral complex, whereas the same is not true for the geometric simplicial complexes. 

In general, borates are structurally complex, since the boron atoms can be in 3 and/or 4 coordination and oligomer, ring, and chain polymers are all found (Christ and Clark, 1977 Wells, 1975). We shall not attempt to describe fully the complexity of these structures but will concentrate on the fundamental polyhedral units. The molecular geometric and electronic structures of these materials can be studied using many of the site-specific spectroscopies previously discussed. The bulk properties of the materials also change, of course, depending upon the molecular structure. 

Polyhedral foams have a complex structure composed of liquid films (formed between the faces of adjacent polyhedral bubbles) and Plateau border (PB) channels (formed at the edges). Although the dimensions of the films and Plateau border channels vary almost randomly in a typical foam, all foams are known to obey a set of geometrical rules formulated by the Belgian scientist Plateau ... 

Consider a polyhedral figure with /i-vertices embedded into a (p-l)-dimensional space. If the distances between all the vertices are the same the polyhedron is called a simplex. A tetrahedron is a simplex with four vertices defined in 3-dimensional spaces. The three p-type GTO example of the present CQS4 chapter generates a simplex in a 2-dimensional space. A complex corresponds to a similar geometrical object with dilferent vertex distances. 

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