Why Do All Triangles Form a Triangle?

Abstract

We provide a geometric explanation, based on symmetry, for why the moduli space of all triangles up to similarity is itself a triangle. Symmetries occur because the lengths of the sides define triples in ℝ3 so are acted on by the symmetric group 핊3, which is isomorphic to the symmetry group 픻3 of an equilateral triangle. The moduli space for triangles is a fundamental domain for the action of 픻3 on an equilateral triangle in ℝ3 determined by all triangles with unit perimeter and is chosen from a subdivision into six congruent triangles. Isosceles and equilateral triangles occupy special locations determined by their symmetries. The sides of a right triangle lie on one of three double cones in ℝ3, and those of unit perimeter lie on a segment of a hyperbola in the moduli space.