“No one untrained in geometry may enter my house” — Plato
Over the past year I have been working with associative and relational databases attempting to find out more about how to develop a better database architecture. This has taken me into many realms including network theory, chaos theory, state transition theory, geometry, logic, chemistry, biochemistry and physics. Recently, I began to put these things together and I think I have had a valuable insight. I call this insight “Geodesate Singularities”.
Geodesate Singularities regard networks as transitions between geodesates which are a group of convex polyhedrons. Convex polyhedron networks have vertexes as nodes and edges as links.
What is of primary importance to this concept is the vertex enumeration (number of vertexes) and the polytope (number of edges per vertice) in these convex polyhedrons as geodesates are regarded as the most stable states.
First frequency Geodesates are a subset of the Platonic Solids and the Archimedean Solids:
3 edges per 4 vertices – 6 edges – Tetrahedron 
4 edges per 6 vertices – 12 edges – Octahedron 
3 edges per 12 vertices – 18 edges – Truncated Tetrahedron
5 edges per 12 vertices – 30 edges – Icosahedron 
3 edges per 20 vertices – 30 edges – Dodecahedron
3 edges per 24 vertices – 36 edges – Truncated Cube
4 edges per 30 vertices – 60 edges – Icosadodecahedron
3 edges per 60 vertices – 90 edges – Truncated Icosahedron
3 edges per 60 vertices – 90 edges – Truncated Dodecahedron
5 edges per 60 vertices – 150 edges – Snub Dodecahedron
3 edges per 120 vertices – 180 edges – Great Rhombicosidodecahedron
Higher frequency Geodesates are triagulations of the above solids. I recommend downloading the Mathematica Player and the Mathematica Demonstrations Project Geodesate Demonstration to view the polygons for each frequency.
Again, what is important in the Geodesates are the number of vertexes (nodes) and edges (links).
My hypothesis is when the growth of a network achieves the vertex enumeration and polytope of a geodesate at the first frequency or higher, a singularity state exists in the network order and results in a state transition of the network when exceeded.
Increasing a Geodesate’s frequency involves dividing the faces of the chosen polygon into sub-triangles:
The first frequecy subdivision is termed as 1V, second as 2V, third as 3V and fourth as 4V.
1V Icosahedron Geodesate
12 Vertexes – 12 5 Edge Polytopes
2V Icosahedron Geodesate
42 Vertexes – 12 5 Edge Polytopes – 30 6 Edge Polytopes
3V Icosahedron Geodesate
92 Vertexes – 12 5 Edge Polytopes – 80 6 Edge Polytopes
4V Icosahedron Geodesate
162 Vertexes – 12 5 Edge Polytopes – 150 6 Edge Polytopes
I think geodesate singularites have implications for Telic, Organic, Chemic, Physic, Static and Gegonic networks. This has implications for Ray Kurzweil’s Singularities, Malcolm Gladwell’s Tipping Points, Stuart Kauffman’s Self-Organization and Howard Rheingold’s Cooperation Theory.
Convex polyhedrons and geodesates could create and limit new organizational structures for enterprise goals, personnel, products, measures, spaces and schedules.
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