The Periodic Table of the Polyhedral Universe deals with all possible 3-D polyhedral envelopes which comply with the celebrated Euler’s formula of V-E+F=2(1-g). It is constructed on the basis of the polyhedral primary parameters, considered to be the Valav, Zaav.
and-g (average valency, average sum of angles in a vertex and genus of the manifold, respectively).
When taken as coordinates of a Cartesian 3-D space, these primary parameters provide for an environment in which every conceivable 3-D polyhedron, whether topologically spherical, toroidal or hyperbolical (sponge-like), has a unique point representation. These point representations form into a discrete set, possessed with a tightly knit ordered structure of a highly periodic nature. Every property, shared by a group of Polyhedra and expressible in terms of the a.m. parameters, discloses characteristic location pattern and everv discernible pattern of polvhedral location Doints represents a distinct shared property.
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The Table’ delves into generation of many new polyhedral entities, expanding considerably the familiar scope of ‘proper Polyhedra’, adding to the list ‘Dihedral, ‘polydygonal’ Polytoroidal and ‘Floral’ Polyhedra, and the queer Univertex Polyhedra, with curved faces, the sum of angles of which is Ea ay.= – 27.
All location patterns, when perceived and comprehended simultaneously, contribute to insights into the nature of polyhedra
and their structural-topological-energetical behavior and on the basis of their point location analysis, facilitate accountability and predictability.
The insights gained from the ‘Periodic Table’ led the way to defining the objectives of morphological research into the domain of “Uniform Sponge Polyhedra’ and their genetic surfaces and subsequently also into the domain of uniform networks in the 3-D space.
Of special interest are the portrayal of the structural-energetical properties and the mode of their development through the entire polyhedral universe, as represented in the Periodic Table, with two important insights to consider and appreciate:
1.The Structural Plate-Lattice duality of ‘finite’ (spherical) ‘toroidal’ and ‘hyperbolic’ dually related polyhedra.
2.The phenomenon of the ‘Grand Divides’, representing location patterns of all potentially stable polyhedral plate and lattice structures, in all g-domains.
The evolution of the ‘Periodic Table’s’ – meta structure of all known and conceivable polyhedra (mostly periodic and regular), was envisioned in 1969, to be developed in the following years, and was published at the Technion, 1.1.T, in a research report entitled” ‘The Hierarchical Structure of the Uniform-Regular Polyhedra’ (Burt, 1975). It promoted the basic observation that all polyhedral envelopes, whether finite (spherical), developable-toroidal or hyperbolic, form into a continuous ordered spectrum, defined by the same set of parameters.
A crucial comprehensive development evolved with the realization that the ‘Periodic Table’, to be all-inclusive, must be conceived and founded on topological perspectives and insights rather than on geometrical-symmetrical ones, thus leading to the final selection of the serving primary parameters of Val av: Edav. And g.
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