Every Convex Polyhedron with Directly Congruent Scalene Triangular Faces Is a Tetrahedron

Abstract

Let T be an oriented scalene Euclidean triangle. We prove that every convex polyhedron whose outward-oriented facets are directly congruent to T is a disphenoid tetrahedron. Consequently, such a polyhedron exists if and only if T is acute; in particular, right and obtuse scalene triangles admit no orientation-preserving realization. Together with Georgiou's reflection-allowed minimum-face-count theorem and the fact that direct and reflected congruence are indistinguishable for isosceles triangles, this completes the orientation-preserving existence and minimum-face-count classification for triangles in Problem B22. The key local rule is the all-r neighbouring structure, or rrr-structure: the three side labels advance cyclically around every vertex, so the vertex degree is divisible by three and the incident face angles sum to an integral multiple of π. Convexity and Euler's formula then force four vertices. With facets understood as polygonal cells, the same argument excludes directly congruent, distinct-sided n-gonal facets for n ≥ 4 and yields a curvature formula for closed oriented polyhedral surfaces.