I've explored axis-collections/systems that could be used to identify/measure locations of points in 1, 2, and 3 D spaces (i.e., line, plane, solid). Cartesian coordinates are a "boundary system" in that they are formed from the minimum number of values that can represent a point in a 1, 2, 3 (or higher) dimensional space - 1 number for a point on a line, 2 numbers for a point on a plane, 3 numbers for a point in a solid, 4 numbers for an event in Einstein's space/time, etc.
However, Cartesian coordinates are not the only choice.
I've developed a shorthand to identify a given dimensional system. The first number indicates the number of measures being taken. The second number indicates the dimensional space in which the measure is being taken.
Cartesian examples: 1M1, 2M2, 3M3.
Quasi-Cartesian examples (take opposite of each axis in "pure" cartesian system): 2M1, 4M2, 6(0)M3.
Non-Cartesian examples: 3M2, 4M3, 6(1)M3, 6(2)M3 (there are three different ways to use 6 axes to subdivide and measure 3D space), 10M3, 15M3, and on.
The tetrahedron incorporates 4M3 (center to vertices, center to middles-of-faces), 3M3 and 6(0)M3 (center to centers of edges).
The octahedron incorporates 3M3, 6(0)M3 (center to vertices), 4M3 (center to centers of faces), 6(1)M3 (center to centers of edges).
The cube incorporates 3M3, 6(0)M3 (center to centers of faces), 4M3 (center to 6(1)M3 (center to centers of edges).
The icosahedron and dodecahedron incorporate 10M3 and 15M3 (which I will not itemize here).
Note that cube and octahedron both incorporate 6(1)M3 in the same manner - center to centers of edges. That makes them a "dual" - a pair of Platonic solids that nest together.
The simplest pair is tetrahedron with tetrahedron (the intersection points form the vertices for the octahedron). The most complex pair is dodecahedron with icosahedron.
The intersection points of nested cube and octahedron form the vertices of the shape Buckminster Fuller called the Vector Equilibrium (VE). It is quite close in shape to the icosahedron.
The VE has 8 triangular sides (center 1/4 of octahedron's sides) and 4 square sides (center 1/2 of cube's sides).
If one adds a diagonal to one of the VE's 6 square sides, changing its shape to two triangles, then there is natural patterning of a diagonal across each of the other square faces, forming a figure with 20 triangular sides (8+6*2).
(The choice of starting diagonal determines which of two icosahedrons is formed from a VE.)
It seems to me that natural systems must tend towards forming structure that relates to various dimension systems.Crystals, for instance, might have structures based on one or more Platonic solids. Living forms are more complex than crystals. They sometimes exhibit the Golden Mean in their structures (e.g., sunflower seed pattern), which requires (mathematically) the number 5. In the Platonic solids, only the icosahedron and dodecahedron use 5 directly in their structure - 12 pentagonal (5-sided) faces; 20 (5*4) triangular faces. This leads me to wonder if plants (e.g.) use the 10M3 or 15M3 vector systems in forming their structure.
I welcome feedback.
Sigurd Andersen : sigurd@sover.net