DOI: 10.5281/zenodo.17477955
title: Synergetics Symmetrical Polynomial (Quaternion) Equations Chart

description: This is a digitized format related to my Prime Number Pattern Symmetry Charts. These Synergetics Symmetrical equations show the first 144 frequency of 6 symmetrical polyhedron's vector, tensor, tetrahedral volume, as well as triangular and square surface area tiles in whole rational units. These equations help us to understand how Universe can 'quantize' space into whole rational units of energy vectors, gravity tensors, tetrahedral volumes and triangular and square surface area tiles.

Fuller discovered that when he used the tetrahedron as the basic unit of volume he could equate other polyhedron’s volumes with the tetrahedron’s in whole rational units. Currently mankind uses the edge of the cube and its volume as the base unit of measurement and this sets up an irrational volumetric involvement domain.

Notice in the chart below how the various polyhedrons all hold whole rational units of volume when compared with the tetrahedron with the same edge length in contrast to the irrational ratios when the cube’s edge is used as ‘unity’.

The 1 and 2 frequency or 1 and 2 meter edge length for these polyhedrons ratios are listed in the chart below.

Synergetics section 223.03: “F = edge frequency, i.e., the number of outer layer edge modules.”

In section 1071.23: “Radial depth equals Frequency” as can be seen in the Sphere, RD and RT.

Since Synergetics uses closest packed same sized spheres to create the various polyhedrons it measures these polyhedrons different ‘sizes’ by way of calculating the total number of ‘intervals’ between the total number of spheres along the edge length of the polyhedron. The Frequency or ‘size’ of the polyhedron is measured by counting the total number of intervals or ‘vectorial interconnections’ between the spheres.

The number of vectors on the edge length is equal to the frequency. In this chart the edge length is equal to the ‘frequency’ which is equal to the total number of 1 unit length vectors which interconnect the 1 unit diameter spheres. If we give the unit vector an anthropic unit of measure of 1 meter it will enable easier mathematical understanding.

By multiplying all the irrational cubic volumes shown on the left by the second root of 72, or 8.48528, we derive whole rational tetrahedral volumes on the right. This ‘quantizes’ volumetric space in whole rational units based on the tetrahedron’s volume of 1.

Cubic meters with 1 meter edge:                                      Tetrahedral meters with 1 meter edge:

Tetrahedron                            0.11785…                           Tetrahedron                                   1 

Octahedron                             0.4714…                            Octahedron                                    4 

Cuboctahedron                       2.357…                              Cuboctahedron                             20 

Tetrakaidecahedron               11.313…                             Tetrakaidecahedron                      96 

Truncated tetrahedron            2.7105                                Truncated tetrahedron                  23 

Cube edge √2=1.414214…    2.828428                             Cube edge √2=1.414214…         24

The reason the cube is a rational volume in the 6 axis metric is that the edge is 1.414214 not ‘1.’  

 

With radius of 1 meter:     

Rhombic Dodecahedron = RD  5.65685                          Rhombic Dodecahedron               48

Sphere with radius 1 meter =  4.188789                          Sphere of radius 1                         35.54

With radius of .99948333 meter:

Rhombic Triacontahedron = RT  4.714045                     Rhombic Triacontahedron             40

 

Cubic meters with 2 meter edge:                                     Tetrahedral meters with 2 meters edge: 

Tetrahedron                                0.9428…                        Tetrahedron                                      8

Octahedron                                3.7712…                         Octahedron                                     32

Cuboctahedron                          18.856…                         Cuboctahedron                              160 

Tetrakaidecahedron                   90.508…                         Tetrakaidecahedron                       768 

Truncated tetrahedron               21.684                             Truncated tetrahedron                   184 

Cube edge √8=2.828428…       22.62742                         Cube edge √8=2.82428…              192 

 

With radius of 2 meter:     

Rhombic Dodecahedron             45.2548                           Rhombic dodecahedron                384

Sphere with 2 meter radius        33.51031                         Sphere 2 meter radius             284.344

With radius of 1.9989666 meter:

Rhombic triacontahedron           37.71236                        Rhombic triacontahedron                320

What’s even more incredible is that not only are the volumes now all calculatable as whole rational units of the tetrahedron’s volume but the total number of vectors, tensors and surface area tiling of 6 symmetrical polyhedrons are also calculatable in whole rational units in ratio to their edge length or 'Frequency'.

Notice that the equations below which calculate the total number of vectors, tensors and volumes all start with the same coefficient number. This symmetrical polynomial association further identifies the isotropic vector matrix’s polyhedral field domain as the underlying basis for the symmetrical holographic EMR field which lies at the core of our holographic Universe.

Synergetics 203.05: As Werner Heisenberg says, “if nature leads us to mathematical forms of great simplicity and beauty… to forms that no one has previously encountered, we cannot help thinking that they are ‘true’ and that they reveal a genuine feature of nature.” [ “Physics and Beyond” by Werner Heisenberg. Harper & Row. 1970 p.68.]

As Einstein explained, “One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the representation of reality.” [Einstein A. The Meaning of Relativity. Methuen. 1956: 169-170.]

Einstein’s quote helps us to understand what Fuller is talking about in section 964.31 where he says “In the quantum and wave phenomenon, we deal with individual packages. We do not have continuous surfaces. In synergetics we find the familiar pattern of second powering (F^2 or c^2) displaying a congruence with the points, or separate little energy packages, of the shell arrays. Little energy actions, little separate stars: this is what we mean by quantum. Synergetics provides geometrical conceptuality in respect of energy quanta”.

The ‘shell arrays’ are the outer layer of the omnidirectionally expanding polyhedron’s tensor field as can be seen in the equations below where a+bi becomes a+bF^2 or 2+10F^2 thereby replacing and ‘quantizing’ the imaginary infinities created by the complex numbers and the Hamiltonian Quaternions as I explain in my papers, "Understanding Nature's Coordinate System and the 6 Axis Metric" and Analyzing Hamiltonian Quaternions in the Light of Synergetics". Available here: 

https://zenodo.org/records/17479104

Athur M. Young told me that Synergetics geometry represents the underlying geometry of space-time because Fuller introduced the concept of 'Frequency' to accommodate the interassociative principles related to how space/size and time correlate in association with all of physical Universe's intertransformative requirements. Arthur let me know that Fuller had not finished developing the equations for the total number of vectors, tensors and tetrahedral volumes for the various polyhedrons so he encouraged me to physically count all the vectors and tensors themselves. He told me I should draw up a chart listing all the totals and that I should try and discover the equations for these totals. I completed this task with these original hand drawn charts in 2010 which are available here: 

 https://zenodo.org/records/17478990

Arthur studied math, physics and Relativity Theory for 5 years under Oscar Veblen and he told me that the energy vectors and gravity/curvature tensors from synergetics represented the 'T' tensors and 'R' tensors from Relativity theory.

By setting up a symmetrical accounting system for the total number of energy vectors and gravity tensors, (or 'T' tensors and 'R' tensors), within the underlying structural geometry of space-time, based on basic algebraic equations related to the total number of vectors and tensors needed to 'structure' the various polyhedrons 'In Space and Time', Universe can therein use similar types of basic algebraic equations to transform the vectors, tensors and volumes form any one polyhedron into any other.

For example, every odd frequency tetrahedron’s total tetravolume can transform itself into every cube’s total number of closest packed spheres or gravity tensors by adding one and dividing by 2. These cube’s spherical gravity tensors can then all transform themselves into the total number of energy vectors in the octahedron by subtracting F+1.

Example 1: The three frequency tetrahedron’s 27 tetravolume adds one = 28 and divides by 2 = 14 spheres in 1 frequency cube.

14 - (F+1) or (1+1) = 12 vectors in 1 frequency octahedron.

Example 2: The five frequency tetrahedron's 125 tetra volume adds 1 = 126 and divides by 2 = 63 spheres in two frequency cube.

63 - (F+1) or (2+1) = 60 vectors in the 2 frequency octahedron.

This process continues ad infinitum.

Once again, this basic mathematical and geometrical system related to Synergetics geometry and the isotropic vector-tensor matrix can help human minds to better understand the Universal transformation ‘process’ phasings between vectors, tensors and volumes at the ‘Quantum States of Matter’.

This is called “Polyhedral Symmetry Phasing” and it is one of the Paradoxes that continue to confound Earthian minds.

By using these same types of basic polynomial transformation equations we can see that there is an underlying mathematical correlation between Fuller's transformational polyhedral models and these equations. For example, when the 1 Frequency cuboctahedron with 20 tetravolume transforms itself into the 1 Frequency octahedron with 4 tetravolume, within the jitterbug transformation model, it simply subtracts 16F^3 from the equation 20F^3 to become the total volume in the octahedron which is 4F^3. See Color Chart 4 from Synergetics for more information on the jitterbug transformation model. Available here: https://rwgrayprojects.com/synergetics/plates/figs/plate04.html

Notice that the total number of tetrahedral volumes, energy vectors and gravity tensors, as well as triangular and square surface area tiles, are all ‘quantized’ as discrete, finite individual aspects of the larger polyhedral system based on the frequency, ‘F’ or edge length of the polyhedron.

This is directly related to what Fuller is talking about in Color Chart 9 and section 982.61A where he shows how the 'Unit Vector' sets the 'Dimensional volumes' within his cosmic hierarchy of omni-concentric polyhedrons in whole rational units compared to the tetrahedron. This rationally 'phasing' inter-transformative complex of energy vectors and gravity tensors is what led him to believe his system of geometry "... may come to be identified as the Unified Field, which, as an operationally transformable complex, is only conceptualizable in its equilibrious state".

Synergetics color chart 9 is available here:http://www.rwgrayprojects.com/synergetics/plates/figs/plate09.html

These equations help us to understand how Universe uses purely algebraic polynomial equations as the underlying representation of physical reality by structuring exact numbers of energy vectors and gravity tensors to enclose all volumetric and areal space as polyhedral continuums in whole rational discrete units, and this shows us “that a finite system of finite energy can be completely described by a finite set of numbers”, as Einstein hypothesized.

With Synergetics we discover a purely algebraic theory for the representation of reality as we replace the “space-time continuum” with frequency modulation, as I explain in my various papers. In my papers I show how we can unite the four fundamental forces with Planck's constant 'h', the fine structure constant 'a', the Gravitational constant 'G' and the masses and Mev/c^2 of the electrons, quarks, protons and neutrons in whole rational units of energy vectors, gravity tensors and tetrahedral volumes using only the basic mathematics related to these algebraically expressed polynomial equations.

All equations originally from Synergetics are marked with an asterisk (*). All others are from my work.

Tetrahedron:                                                                 Tetrakaidecahedron:

Tensors: (F^3+6F^2+11F+6) / 6                                 Tensors: (96F^3+90F^2+36F+6) / 6

Vectors: F^3+3F^2+2F                                                 Vectors: 96F^3+42F^2+6F

(*) Volume: F^3                                                         (*) Volume: 96F^3

 

Octahedron:                                                                 Truncated Tetrahedron:

Tensors: (4F^3+12F^2+14F+6) / 6                             Tensors: (23F^3+42F^2+25F+6) /6

Vectors: 4F^3+6F^2+2F                                               Vectors: 23F^3+21F^2+4F                    

(*) Volume 4F^3                                                             Volume: 23F^3

 

Vector Equilibrium: (Cuboctahedron)                           Cube:

Tensors: (20F^3+30F^2+22F+6) / 6                           Tensors: (24F^3+36F^2+18F+6) / 6

Vectors: 20F^3+12F^2+4F                                           Vectors: 24F^3+12F^2

(*) Volume: 20F^3                                                          Volume: 24F^3

 

The total number of spheres which accumulate on the surface area of these polyhedrons =

(*) 2F^2 + 2 = Spheres/Points on the surface area of the Tetrahedron’s external layer.

(*) 4F^2+2 = Spheres/Points on the surface area of the Octahedron’s external layer.

(*) 6F^2+2= Spheres/Points on the surface area of the Orthogonal Cube’s external layer.

(*) 10F^2 +2 = Spheres/Points on the surface area of the VE & Icosahedron’s external layer.

12F^2+1 = Spheres/Points added to each IVM Cube to create next Frequency IVM Cube.

The surface areas of these polyhedrons are also quantized in whole rational second powering accumulations of triangular and square surface tiles which therein equals the total surface area.

(*) Tetrahedron:                    4F^2 = total number of triangular tiles = triangular surface area.

(*) Octahedron:                    8F^2 = total number of triangular tiles = triangular surface area.

Truncated Tetrahedron:    28F^2 = total number of triangular tiles = triangular surface area.

Cube :                                  12F^2 = total square area.

(*) Cuboctahedron:             8F^2 = total number of triangular tiles = triangular surface area.

                                            + 6F^2 = total number of square tiles = square surface area.

Tetrakaidecahedron:          48F^2 = total number of triangular tiles = triangular surface area.

                                           + 6F^2 = total number of square tiles = square surface area.

(*) To convert triangular area to square area multiply by the second root of 3/16 = .4330127018922

(*) To convert square area to triangular area multiply by the second root of 16/3 = 2.309401076759

 

This leads to whole rational volumetric accounting for the symmetrical polyhedrons when measured in ratio to the unit edge length tetrahedron instead of the unit edge length cube.

These Synergetics volume ratios listed above enable children to more easily and quickly ascertain the cubic volume associated within the x,y,z cubical mensuration system as follows:

In this association the edge length is the Frequency when measuring the tetrahedron, octahedron, truncated tetrahedron, tetrakaidecahedron or cuboctahedron = ‘F’. 

When measuring the sphere, the rhombic triacontahedron or the rhombic dodecahedron the insphere radius is the Frequency or ‘F.’

 

Multiply the edge length, or insphere radius, of the polyhedrons listed above to the third power.

Next multiply this number by the synergetics constant variable for each polyhedron's volume.

Next multiply by the Synergetics constant .11785, or the ‘inverse square’ root of 72.

This example gives your academia approved cubic volume:

Octahedron with 2 inch edge length. F=2: Multiply 2^3 = 8.

Next multiply by constant 4 because the volume is 4F^3= 32.

Next multiply by .11785 = 3.7712 cubic inches of volume.

With a 1 inch edge length octahedron we obtain .4714 cubic inches of volume.