Synergetics coordinates

Synergetics coordinates is Clifford Nelson's attempt to describe, from another mathematical point of view, Buckminster Fuller's '60 degree coordinate system' for understanding nature. Synergetics is the word Fuller used to label his approach to mathematics.[1]

Geometric definition

A system of synergetics coordinates uses only one type of simplex (triangle, tetrahedron, pentachoron, ..., n-simplex) as space units[disambiguation needed], and in fact uses a regular[disambiguation needed] simplex, rather like Cartesian coordinates use hypercubes (square, cube, tesseract, ..., n-cube.)

Synergetics coordinates in two dimensions

The n Synergetics coordinates axes are perpendicular to the n defining geometric objects that define a regular simplex; 2 end points for line segments, 3 lines for triangles, 4 planes for tetrahedrons etc.. The angles between the directions of the coordinate axes are Arc Cosine (-1/(n-1)). The coordinates can be positive or negative or zero and so can their sum. The sum of the n coordinates is the edge length of the regular simplex defined by moving the n geometric objects in increments of the height of the n-1 dimensional regular simplex that has an edge length of one. If the sum of the n coordinates is negative the triangle (n = 3) or tetrahedron (n = 4) is upside down and inside out.

Algebraic examples

Regular triangular coordinates are in a grid of equilateral triangles and are of the form $(x,y,z)$ such that $x,y,z$ are equal to or greater than 0.

Regular tetrahedral coordinates are in a Euclidean 3-space 'grid' of equilateral tetrahedra and are of the form $(w,x,y,z)$ such that $w,x,y,z$ are equal to or greater than 0.