允许反射的全等三角形面：问题B22的通用实现与最小面数
Congruent Triangular Faces, Reflections Allowed: Universal Realization and the Minimum Face Count in Problem B22
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中文总结 AI 辅助
针对允许反射的凸多面体公共三角形面的B22问题，构造了所有非退化三角形的通用八面体实现，并确定了各类三角形的最小面数。
中文摘要 AI 辅助
《几何未解决问题》中的B22问题询问：哪些三角形可作为凸多面体的公共面、需要多少个该类三角形，以及它们如何排列。我们针对允许反射副本的情形解决了三角形的存在性与最小面数问题，但未对所有可达面数或可能排列进行分类。所有非退化欧氏三角形均可实现：我们构造了一个组合上为八面体的凸多面体，其8个面均与给定三角形全等。随后我们确定了**每个**三角形的最小面数：锐角三角形为4；边长为(λ,λ,β)且满足λ√2≤β<λ√3的直角或钝角等腰三角形为6；其余所有情形均为8。
英文摘要
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.
