A Tangle is a smooth closed curve formed by the union of arcs of congruent circles, known as links. We use the Gauss-Bonnet theorem to prove an interesting result about the number of concave v. convex links in a Tangle. This in turn allows us to show that all “triangular Tangles” (planar Tangles formed from sixths of circles) may be constructed by a sequence of fundamental operations starting from a circle.

Based on the paper “Properties of regular Tangles” by Bowen, Pruitt, and T. (https://doi.org/10.48550/arXiv.2405.20793)