Congruent Triangular Faces, Reflections Allowed: Universal Realization and the Minimum Face Count in Problem B22

Abstract

Problem B22 in Unsolved Problems in Geometry asks which triangles occur as the common face of a convex polyhedron, how many copies are needed, and how they may be arranged. We settle the existence and minimum-face-count questions for triangles in the version that allows reflected copies; we do not classify all attainable face counts, nor the possible arrangements. Every nondegenerate Euclidean triangle occurs: we exhibit an explicit convex polyhedron, combinatorially an octahedron, all eight of whose faces are congruent to a prescribed triangle. We then determine the minimum number of faces for \emph{every} triangle. It is four for an acute triangle; six for a right or obtuse isosceles triangle with side lengths $(λ,λ,β)$ satisfying $λ\sqrt2\leqβ<λ\sqrt3$; and eight in all remaining cases.