DOI: 10.5281/zenodo.17478989
title: Synergetics Polynomial Equations and Prime Number Pattern Symmetry Charts. (Hand Drawn Charts 2010)

description: In Synergetics Buckminster Fuller uses closest packed unit radius spheres to create different sizes or 'frequencies' of various polyhedrons. This is my hand drawn chart from 2010 which shows the tetrahedron's first 144 frequency or 'sizes' along with the fractionation process which 'reveals' the 6 internally 'phasing' polyhedrons. If we fractionate spheres off of the tetrahedron's vertices every second frequency we reveal the octahedron. When we fractionate spheres off of the octahedron's vertices every second frequency we reveal the vector equilibrium or cuboctahedron. When we fractionate spheres off of the octahedron's vertices every third frequency we reveal the tetrakaidecahedron and when we fractionate spheres off the tetrakaidecahedron's vertices every frequency we reveal the cube created with FCC closest packing.

The charts were drawn on 2 foot by 8 foot paper so if you want to print them out you may need to take them to a print shop.

This chart lists the symmetrical trinomial and quadrinomial equations related to Synergetics geometry. Fuller discovered equations like 10F^2+2 for the total number of spheres in the Vector Equilibrium's (cuboctahedron) external shell layer but he never discovered the trinomial and quadrinomial equations related to his geometry.

This equation and the others he discovered hold identical algebraic expressions to the complex numbers a+bi where F^2 takes the place of the imaginary component 'i' and transforms the equation a+bi into the binomial equation 2+2F^2 for the tetetrahedron, 2+4F^2 for the octahedron,  2+10F^2 for the icosahedron and cuboctahedron. It was Arthur Young, Oscar Veblen's student for 5 years at Princeton, who alerted me to this identity. He told me Fuller hadn't discovered the equations for the total number of spheres which accumulate in the continuing different sizes nor for the total number of 'kissing points' or 'vectors' and he recommended I do so.

After I studied Synergetics I drew up this chart and formalized the equations. As we can see, these new equations hold identical algebraic expressions with the Hamiltonian Quaternions. Where Q=a+bi+cj+dk, we see how these new equations from synergetics which I discovered enable us to take the i, j and k out of the equations and replace them with F, F^2 and F^3 to become a+bF+cF^2+dF^3 for the tensor sphere equations and bF+cF^2+dF^3 for the vector equations where a,b,c and d are real number variables related to each individual polyhedrons individual vector and tensor totals in ratio to their 'edge length' frequency.

The second and third charts show how the total number of vectors, tensors and tetrahedral volumes can all be reduced to their individual prime number patterns and indig patterns. indig means integrated digit. When we take any number and continue to add the numbers until they are only one digit long, this is an indig. ie. 259=2+5+9=16 and 1+6=7 so the indig of 259 is '7'.

Fuller never discovered that the prime numbers are always used in a patterned way within the developing sizes or frequencies of his polyhedrons nor did he discover that the equations from his geometry hold identical algebraic expressions with the complex numbers and the Quaternions but I bet he would have thought it Amazing!

noel_coughlin01@yahoo.com