Tautangles

Tautangles are a theoretical concept in plane geometry that concerns special configurations of angles along a polygonal chain. They arise from imposing a local tension-like constraint between consecutive angles, so that a single global pattern emerges from simple pairwise relations. The term is used mainly in recreational or exploratory contexts rather than as a standard textbook topic.

Formal definition: Consider a polygonal chain with vertices v0, v1, ..., vn in the plane, and let e_i denote the exterior angle at vertex v_i, measured from the incoming to the outgoing edge. A tautangle configuration with tautness constant C is one in which e_i + e_{i+1} = C for every interior vertex i, for some fixed constant C. If the chain closes into a polygon, the usual sum of exterior angles around a simple polygon equals 360 degrees, which imposes a compatibility condition between C, n, and the chain’s geometry. The construction is invariant under similarity transforms, meaning only angle relations matter, not edge lengths.

Properties and examples: Solutions exist for various lengths and constants. When C = 180 degrees, adjacent exterior angles sum to a straight angle, producing straight or near-straight local behavior in certain realizations. For repeating patterns where the exterior angles alternate between two values a and C − a, the chain exhibits a periodic tautangle structure. The study emphasizes how local angle restrictions influence global shape.

Relation and usage: Tautangles intersect ideas from angle chasing, polygonal chains, and geometric constraint problems. They are discussed mainly as a conceptual tool for examining how local relationships govern global configurations, with occasional applications in puzzle design and educational explorations of geometric reasoning. See also angle, exterior angle, polygonal chain, geometric constraint.