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arXiv · 2608.01109

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

George M. Georgiou
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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 $(\lambda,\lambda,\beta)$ satisfying $\lambda\sqrt2\leq\beta<\lambda\sqrt3$; and eight in all remaining cases.

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George M. Georgiou. 2026-08-02. Congruent Triangular Faces, Reflections Allowed: Universal Realization and the Minimum Face Count in Problem B22. https://arxiv.org/abs/2608.01109

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