In the derivation of polar space, the rotation steps creating it are finite. A discrete rotation can have any number of steps, so if a perfect circle has infinite steps, a triangle is three rotations, a square four, a pentagon five, and so on (Figure 2.6). These N-rotations approximate a perfect circle as N increases. More rotation steps seem better but wargamers use hexagons not octagons because a board of octagon tiles has gaps in it. In simple terms, they don’t fill the surface completely. Squares and hexagons do, but the latter give more interaction directions, so are preferred.
In general, a rotational space could be incomplete, and if it has gaps in it, not all paths will be reversible. Taking a route in reverse might not return to the same point, though it would be a true vicinity. Even so, a polar space with holes in it would allow billiard ball particles to pass right through each other! Would this incompleteness exclude a discrete polar space from describing our space?
It might seem so, but in our world, entities are quantum probability clouds not billiard balls. When these clouds collide, they overlap an area so a space with a few holes in it doesn’t matter. If matter entities are quantum probability clouds, a polar space with holes in it still works.
If our space is polar, the circle of neighbors around a point will be a finite number N, so it will have that many transfer connections. If each connection is a direction, this predicts that a point in space has a finite number of planar directions, so direction, like length, is quantized, as a minimum Planck angle [Note 1]. Experiments with high frequency light may be able to confirm that planar quantum events have a minimum angular effect.
[1] If a point has N neighbors in a circle around it, the minimum Planck event angle is 360°/N.