## Systema: Geodesates, Nodes and Links

“To every action there is an equal and opposite reaction.” — Isaac Newton

A predominant issue arising from my work is the discovery of the difference between a node and a link.  A node type represents a state type while a link type represents a transaction between state types.  However I am finding there are a limited number of node types (self-ordered states) and link types (self-ordered state actions).

In the diagram below, each polyhedron is a first frequency geodesate and has a unique polytrope/polytype combination.  A polytrope is the number of edges per polyhedron vertex.  A polytype is the number of polyhedron vertexes.  This is not the final version.  I am still working to purify my geodesate concept.

What I am revealing here is that each of the seven Node Types on the Left has only one Link Type on the right.  In the same way that an association is composed of a source node type and target node type, an association is composed of a source link type and target link type.

Here is an example of a homogenous Entity to Entity association:

Here is an example of a hetergeneous Entity to Positity association:

Having considered this it is now possible to conclude that there are a unique set of nodes each with a unique link which can be used to build homogeneous or heterogeneous associations.  In otherwords, each node type can perform only one action type.  It is the reaction type of the target node type that makes the action reaction combination unique in the system.

Let’s look at some examples of node type and link type associations:

1. To identify a positity, positifies an identity.
2. To objectify a projectity, projectifies an objectity.
3. To chronify a chronity, chronifies a chronity.
4. To projectify a quantity, quantifies the projectity.
5. To qualify an identity, identifies a quality.

Fourty-nine possible type combinations exist.  I think there are even more types which I will explore with Archimedean Solids and higher frequency Geodesates in later posts.

## Systema: Geodesates as Singularities

“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:

1. 3 edges per 4 vertices – 6 edges  – Tetrahedron
2. 4 edges per 6 vertices – 12 edges  – Octahedron
3. 3 edges per 12 vertices – 18 edges – Truncated Tetrahedron
4. 5 edges per 12 vertices  – 30 edges – Icosahedron
5. 3 edges per 20 vertices  – 30 edges – Dodecahedron
6. 3 edges per 24 vertices – 36 edges – Truncated Cube
7. 4 edges per 30 vertices – 60 edges – Icosadodecahedron
8. 3 edges per 60 vertices – 90 edges – Truncated Icosahedron
9. 3 edges per 60 vertices – 90 edges – Truncated Dodecahedron
10. 5 edges per 60 vertices – 150 edges – Snub Dodecahedron
11. 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.