How do A.H. Stone’s paper flexagons change when they are flexed?

A flexagon, a geometric construction often associated with the art of paper folding, is a polygon made from a strip of paper designed to change its appearance through manipulation. When a flexagon is flexed along its folds, the structure undergoes a transformation where one of its visible faces disappears and a previously hidden face takes its place.12

This phenomenon occurs because the figure is folded in a specific way that allows it to reveal various arrangements of its faces as it is flexed. While Arthur H. Stone is credited with the discovery of these shapes in 1939, the fundamental mechanism remains the same: the model's geometry allows for the continuous cycling of different faces through the act of flexing.12