ABSTRACT
One of the things that happens when subdivisions are created between vertices is that these subdivisions create concentric circles of interior points. If S > 1, then S is either even or odd and therefore can be written as either S = 2 k or S = 2 k +1. This formal way of talking about even and odd is done to highlight k , the number of concentric circles and levels from Level 1 to Level k . Consider k = 1 (and S = 2 or 3). There is a single internal circle in this instance. But S = 4 or 5 produces k = 2 levels of concentric circles. The concentric circles come from the symmetry about the midpoint of any vertex frame line. Points that have the same number of subdivisions from an end are equidistant from the circle's center.
Take S = 4. The first (subdivision) point and the third point (3 = 4 − 1) are at the same distance from the center of the circle on Level 1, and the second point (which is also the midpoint) is closer to the center than points 1 and 3.
Take S = 5. The first (subdivision) point and the fourth point (4 = 5 − 1) are at the same distance from the center of the circle on Level 1, and the second and third points (3 = 5 – 2) are at the same distance from the center of the circle but are closer to the center than points 1 and 4 on Level 2.
