Abstract and physical 3-D space is populated with inter-relating and inter-connected entities, generating assemblages with a network characteristics and hyperbolic ‘force fields’ in between, aptly described as ‘sponge surface configurations’. These networks and sponge surfaces can represent the structure of almost any plurality that may exist; from a reality of any battle field, cultural, economical or political, to transportation and communication systems, to social patterns, cosmological arrays and patterns of perception, knowledge and thought.
All networks come in dual pairs. Each network is uniquely determined by, and is a reciprocal of its dual (complementary) Companion.
A periodic network is formed by extended repetition of a locally symmetrical association of vertex figures. The resulting confiauration of vertices and axes-edaes could be described as a polyhedral network, conforming with the following relation: =, with E: V&Val.av. standing for the number of Edges, Vertices and average Valency value in a vertex, respectively.
Every dual pair of networks is associated with a continuous hyperbolical sponge surface which subdivides the space between the two, into two complementary subspaces.
This trinity of the dual pair and the associated-reciprocal sponge surface is the most conspicuous, all pervading geometric-topological phenomenon of our 3-D space, associated with its order and organization and more than anything else determines the way we perceive and comprehend its structure.
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By defining as ‘morphic’ those processes which display a movement toward greater 3-dimensional spatial order, symmetry or form (Wythe-1969) and morphology as the logical preoccupation with, and manipulation of those processes, than the research into the nature of networks and the associated sponge surfaces may be classified as the essence of morphology.
“The theory of these nets does not appear to be known, and in fact no attempt to derive them systematically seems to have been made until comparatively recently ” (A.F.Wells – Structure of Inorganic Chemistry – 1962). In spite of the efforts of Wells, the present condition of the field remains basically unchanged.
Uniform networks in 3-d space may be primarily classified into two main categories: 1. The Trinity Networks (constituting the trinity of the dual pair of networks and the reciprocal surface, subdividing the space between the two). 2. The Entangled networks, mostly Translation Networks which could be perceived as 3-dimensional representations of 4-D networks. The most appropriate secondary classifications of the Uniform networks is according to their valency (coordination number or the number of edges joined in a vertex) and density (in terms of a/a², when a is the edge length).
Since the range of valences of the Uniform Trinity Networks seems to be confined to:
Val. = 3+12, it is a fair assumption that their total number is finite, exhaustible, and in the range of few hundreds only! The number of the Uniform Entangled – Translation nets, although phenomenologically exhaustible, seems to be reaching to infinity.
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