Counting intersections of perpendicular cylinder curves on regular polygon surfaces

This preprint counts intersections and complementary regions formed by horizontal and vertical closed curves on a regular polygon with 4g sides identified in opposite pairs. A direct midpoint argument shows which pairs of outer cylinders contribute zero, two, or four crossings. When N distinct closed curves are chosen in every cylinder, the arrangement has (2g² − 1)N² crossings and (2g² − 1)N² + 2 − 2g complementary regions, each an open disk. The octagon and dodecagon illustrate how crossings on identified sides are counted. The manuscript relates the calculation to published cylinder decompositions and the known octagon tiling. Independent human review is pending.