Hi again,

Here is a secret discovery I made once. I hope you find it as interesting as I did. It is an algorithm that creates what I call “fundamental geometries.” These are the primary geometries that maximize the distances between vertices, in a constrained manner. You won’t believe it until you see it. You will need to do some programming, though.

MULTIDIMENSIONAL GEOMETRY ALGORITHM

Initialize an array, M, of random unit vectors (to the desired dimensionality). These will eventually become the number of vertices in the geometry.

  1. Calculate a random unit vector, V (to the desired dimensionality).
  2. Perform a scalar product calculation on each entry of M against V, and save the index position into M where the largest scalar product was found
  3. Add a fractional portion of V to M[index], and normalize M[index]

Repeat steps 1, 2 and 3… thousands of times.

RESULT:

At the end of this process you will see that the array M will contain the vertices of a fundamental geometry to the desired dimensionality. For instance, four vertices in R2 will form a square, but in R3 they form a pyramid. Eight vertices in R3 forms a cube, and in R4 also forms a cube but has internal mirror images at the expected probability. You will have to experiment with this in order to understand what it is I am saying.

Thanks again!