*“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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The following derivation of the law of equivalence, which has not been published before, has two advantages. Although it makes use of the principle of special relativity, it does not presume the formal machinery of the theory but uses only four modified laws (I hope it is not just sci-fi):

- Distance-space and Time-space are both three dimensional coordinate systems
- Momentum which is distance-space velocity and “Chromentum” (for lack of a better word) which is time-space velocity are conserved
- Radiations exert their own pressure; that is, the light-event is a complex of radiation moving in a fixed distance-space direction and time-space direction with momentum and chromentum.
- Light and events are aberrant.

We now consider the following system.

Let the body B rest freely in space and time with respect to the system K0. Two complexes of radiation S, S’ each of energy E/2 durate in the positive and negative x0 direction respectively and are eventually absorbed by B. With this absorption the energy of B increases by E. The body stays at rest in space and time with respect to K0 by reasons of symmetry.

Where* **c *is the velocity of light and *e*** **is the velocity of event.

Now we consider this same process with respect to the system K, which moves with respect to K0 with the constant distance-space velocity *vd *and time-space velocity *vt* in the negative Z0 direction. With respect to K the description of the process is as follows:

The body B moves in the positive *z* direction with distance velocity vd and time velocity v*t. *The two complexes of radiation now have directions with respect to K which make an angle *a* with the *x* axis. The law of light and event aberration states that in the first approximation *a = ( c / e ) / vd * vt , *where c is the velocity of light and *e *is the velocity of event. From the consideration with respect to K0 we know that the distance-space velocity and time-space velocity of B remains unchanged by the absorption of S and S’.

Now we apply the law of conservation of momentum and chromentum with respect to the *z* direction to our system in the coordinate frame K.

*Before the absorption*let*M*be the mass of*B*;*M * vd * vt*is then the expression of the momentum and chromentum of B (according to this new mechanics). Each of the complexes has the energy*E / 2*and hence, by a new conclusion of Maxwell’s theory, it has the momentum and chromentum*E / 2 * (c / e**)*. Rigorously speaking this is the momentum and chromentum of*S*with respect to*K*. However when*vd*and*vt*are small with respect to*c*and*e*, the momentum and chromentum with respect to*K*is the same except for a quantity of second order of magnitude (*( vd ^ 2 * vt ^ 2) / ( c ^ 2 / e ^ 2 )*compared to 1). The z-component of this momentum and chromentum is*E / 2 * ( c / e ) sin a*or with sufficient accuracy (except for quantities of higher order of magnitude)*E / 2 * ( c / e ) * a*or*E / (2 ***(vd * vt ) / (c ^ 2 / e ^ 2 ) ) S*and*S’*together therefore have a momentum and chromentum*E * ( ( vd * vt ) / ( c ^ 2 / e ^ 2 ) )*in the*z*direction. The total momentum and chromentum of the system before absorption is therefore*After the absorption*let*M’*be the mass of B. We anticipate here the possibility that the mass increased with the absorption of energy E (this is necessary so that the final result of our consideration be consistent). The momentum and chromentum of the system after absorption is then

We now assume the law of conservation of momentum and chromentum and apply it with respect to the *z* direction. This gives the equation

or

This equation expresses the law of the equivalence of energy and mass and the nature of events as part of the complex of radiation. Since energy according to the usual definition leaves an additive constant free, we may choose the latter that

As a consequence of this equation there is no dark matter and the acceleration of matter in the universe is due to a transition from a distance singularity to a time singularity. Distance-space is observing exponential expansion.

Time-space is finite. Which leads to this:

Which bring us to my post on Singularity, Pluralarity and Lorentz Transformation