Space generated by a network will have a density based on the number of links each node has to others. In the polar space derivation, this density is the number of steps in the rotations that create the space. A discrete rotation can have any number of steps, so if a perfect circle has infinite steps, a triangle is a “3-circle”, a square a “4-circle”, a pentagon a “5-circle” and so on (Figure 2.6). These N-circles approximate an ideal circle as N increases. It might seem that more rotation steps are better but wargamers and online games don’t use octagons because they don’t “fill” a flat surface, as side-by-side octagons leave gaps. Squares fill the board but only give four interaction directions so wargamers prefer hexagons as they both fill the board and give six interaction directions.

If the quantum network emulating space is dense, each point will have many connection directions but a large N-circle can’t fill a Euclidean space perfectly. Does this exclude it from emulating our space? For example, not all paths in such a space would be reversible, so following a route taken in reverse may not return to the exact same node, though it would be a true vicinity. In essence, a discrete space based on polar coordinates will have “holes” in it, so billiard ball point particles could pass right through each other!

This might seem to disqualify a space based on discrete rotations but entities in our world are described by quantum probability clouds not billiard balls. When quantum entities “collide” they overlap over an area, so a space with a few holes in it doesn’t matter. That quantum entities exist as quantum probability clouds avoids the problems of N-circle space. Since a polar space must have a finite number of directions for any quantum event, quantum realism predicts that direction, like length, is quantized, so there will be a minimum Planck angle.