Exact colinearity of centroids of iterated midpoint hexagons

TL;DR

This paper proves that for a specific midpoint iteration on hexagons, the centroids of the resulting hexagons from the second iteration onward always lie on a fixed line, revealing a unique geometric property.

Contribution

It uncovers a novel algebraic and geometric property of hexagons under midpoint iteration, showing exact colinearity of centroids after the first iteration.

Findings

Centroids of iterated hexagons from the second iteration onward are colinear.

This colinearity is a unique algebraic feature specific to hexagons.

The property does not generalize to other polygons.

Abstract

We study the iteration that replaces a planar hexagon by the hexagon formed by joining the midpoints of consecutive edges. While this iteration quickly drives any polygon toward a point and their shapes asymptotically regularize, we show a stronger and unexpected rigidity holds for hexagons: from the second iterate onward, the centroids of the filled hexagons all lie exactly on a fixed line. This exact colinearity reflects a special algebraic feature of the hexagonal case and does not hold generally for any other polygons.