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.

Disclosure

“oblem B22 are also left open here: the classification of all attainable face counts, as opposed to the minimum; the description of the possible arrangements; and the analogous problem for congruent n-gons. Acknowledgments The author used OpenAI’s ChatGPT 5.6 Sol to assist with the initial discovery of the result and drafting of the proof, and Anthropic’s Claude Opus 5 for subsequent review and refinement. The author independently verified the final arguments and assumes responsibility for”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:18:30.695768Z

Structural counts