The Vector Equilibrium

(34)    There are twenty points in four dimensional space represented by five Synergetics coordinates which add up to zero that have a VectorDistance of one and also have integer coordinates.

(35)    The last coordinate can be dropped from each one and the VertexesOf the remaining four coordinates produce twenty tetrahedrons. Twelve with an edge length of zero and eight of edge length one.

(36)    The graphic above is why I think that I have the same coordinate system that R. Buckminster Fuller had. The figure above is a cuboctahedron which Bucky called the Vector Equilibrium. He wrote that the XYZ Cartesian coordinate axes should go through the square faces of the VE, but I don't know the advantage to that. The XY Cartesian plane is coincident with one of the hexagonal planes of the VE in this system. Bucky also said that the radius of the closest packed spheres should be one, but it is in this system, to make the diameter equal to one.

(37)    The function StoP (short for Synergetics to perpendicular coordinates) has an inverse PtoS.

(38)    PtoS and StoP were derived by computing the height of an equilateral triangle of edge length 3 and the altitude of a regular tetrahedron of edge length 4. The formulas were plugged in by hand at first. The first vertices of {1,1,1} and {1,1,1,1} are {1,1,-2} and {1,1,1,-3} respectively. I noticed that their StraightLineDistance is the square root of the quantity (the sum of the squares divided by two). Append[Table[1,{k}],-k] gives the first vertex of the simplex Table[1,{k+1}], and their StraightLineDistances are the square roots of the triangular numbers as shown below.

(39)    All these vertices are made to point as "up" as possible. That explains the logic behind the definitions of StoP and PtoS.

(40)    Synergetics coordinates simplexes with an edge length of zero can be added and subtracted from each other. That is just vector addition and subtraction. They can be multiplied and divided by each other according to special rules. Then I call them B numbers or "Bucky numbers" after R. Buckminster Fuller. Floating point numerical approximations to fractional powers can be computed because each B number corresponds to a matrix and the MatrixPower function can be used. Unity for numbers is B[1,1,1,1,-4]. The last coordinate of a B number equals minus the sum of the other coordinates. The first four coordinates of a number can be shown as a tetrahedron. Here is a graphic of the B number times B[1,-1,0,0,0] for x from 0 to 2 color coded as Hue[].

(42)    Eric W. Weisstein's MathWorld entry about Synergetics Coordinates, albeit with the first coordinate oriented at 6 o'clock instead of 2 o'clock like mine, is on line at:
Weisstein, Eric W. "Synergetics Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SynergeticsCoordinates.html

(43)     Both volumes of the books Synergetics were on line the last time I looked at:
http://www.rwgrayprojects.com/synergetics/synergetics.html

Created by Wolfram Mathematica 6.0  (17 March 2008)