The Maximum Number of Intersection Points of Overlapping Regular Polygons

This note studies how many intersection points can appear when several regular polygons are overlaid in the plane. Because a regular polygon is convex, two regular k-gons cross in at most 2k points; the proof of this bound is short and is given in full. For n copies of the same k-gon the maximum grows to kn² − kn, and closed forms are derived for mixtures of polygons of different sizes. The general rule from the original project is restated here with its index error corrected. The note also settles a question left open in the original work: the same arrangement cuts the plane into exactly kn² − kn + 1 bounded regions, a result that follows from Euler's formula for planar graphs. Every formula was verified by an independent Python program over all tested combinations of polygon size and count; the verification code is available in the linked repository. The work began as a research project at Halil Inalcık Bilim ve Sanat Merkezi (BILSEM) and received an award at MOSTRATEC (Mostra Internacional de Ciência e Tecnologia). This is a revised and extended version prepared in 2026.