The cube

(21)    The cube is a positive and a negative tetrahedron together. The mid points of the edges of the tetrahedrons are at the centers of the faces of a cube. The cube in the following graphic has an edge length of because + = . The cube's edges are not shown, just the face diagonals.

(22)    Three Cartesian coordinates refer to the perpendicular movement of the three unique non parallel planes that define the six faces of a cube. Their intersection after they move defines a point in three dimensional space.

(23)    Four Synergetics coordinates represent the independent perpendicular movements of the four planes of a regular tetrahedron. Their intersections after the planes move define a point in four dimensional space, represented by a regular tetrahedron in three dimensional space. Whether you move the planes toward the center (a negative direction) of the reference tetrahedron, or away from the center (positive), you always get another regular tetrahedron. The edge length of the tetrahedron defined by the intersections of the planes, in terms of unit vectors or Euclid distance, is the sum of the coordinates if the coordinates are in terms of the height of a tetrahedron of edge length one. If the sum is negative, the tetrahedron is upside down and inside out. If the sum is zero it represents a "fix", location, or three dimensional point, and the zero edge length tetrahedron corresponds to three Cartesian coordinates.

(24)    The function StoP transforms Synergetics coordinates which add up to zero, to perpendicular 90 degree coordination. All movements of the four planes are kept with reference to a volume zero tetrahedron at {0,0,0,0}. The four planes of the reference tetrahedron coincide with the four hexagonal cross section planes of the cuboctahedron. The edge length of the resulting tetrahedron found by adding the four coordinates is both the StraightLineDistance (Euclidian distance) and the (unit) VectorDistance from vertex to vertex.

(25)    The VectorDistance function is the distance going by way of adjacent neighbors similar to the "Manhattan distance" with Cartesian coordinates. The synergetics coordinates add up to zero. There are six adjacent neighbors in two dimensions, twelve neighbors in three dimensions, d(d+1) neighbors in d dimensions with d+1 synergetics coordinates. When the coordinates are integers they are the positions of the centers of unit diameter closest packed circles in two dimensions and closest packed spheres in three dimensions. Notice the difference between VectorDistance and StraightLineDistance. VectorDistance gives the results I need for complex coordinates (that is why the Abs function is not used).

(26)    The edge length of a tetrahedron is just the sum of the four synergetics coordinates, like the tetrahedron {1,2,3,4} as shown below.

(27)    Four synergetics coordinates that add up to zero are points in three dimensional space.

(28)    If you drop the last coordinate of {1,1,1,-3} and find the vertices of what is left {1,1,1}, the three points are on a plane and form the base of a regular tetrahedron whose top point is {1,1,1,-3}. {1,1,1,-3} is the endpoint of a line perpendicular to the midpoint of the triangle {1,1,1}, and it's equidistant from the three vertices of {1,1,1}.

(29)    Therefore an equiangular triangle in the plane is a way of representing a point in three dimensional space. If the triangle has an edge length of zero the point is in the plane. If the triangle is positive the point is above the  plane. If the triangle is negative the point is below the plane.

(30)    Five synergetics coordinates that add up to zero are points in four dimensional space. Just as the first three synergetics coordinates of a point in three dimensional space can represent the point as a triangle in the plane, the first four synergetics coordinates of a point in four dimensional space can represent the point as a tetrahedron in three dimensional space. It is implied that to transform the point represented by the tetrahedron in synergetics coordinates, to find the perpendicular fourth coordinate, find the midpoint of the tetrahedron and make a line perpendicular to all three mutually perpendicular XYZ Cartesian coordinate axes until the end point of the line is the same distance from all four vertices of the tetrahedron as they are from each other. The four perpendicular Cartesian coordinates are usually called XYZW.

Created by Wolfram Mathematica 6.0  (17 March 2008)