Description (摘要) This paper constructs a new normalized compact constant γ by synthesizing the classical intrinsic invariants of the regular tetrahedron, realizing a unique "fingerprint" representation of the simplest convex polyhedron in 3D Euclidean space. Three independent and orthogonal information sources—geometric extremum (the ratio of circumradius to inradius), group-theoretic intrinsic property (the order of the vertex stabilizer of the Td symmetry group), and metric compactness (the volume ratio of the regular tetrahedron to its circumscribed sphere)—are integrated by geometric mean for unbiased aggregation, which is a natural and necessary method for synthesizing such dimensionless multiplicative intrinsic quantities. The core compact constant is derived as γ=343/3π≈0.902, and a rigorous theorem is proved: a convex polyhedron is a regular tetrahedron if and only if its vertex number V=4 and the circumradius-inradius ratio R/r=3. This study demonstrates a generalizable methodology for constructing comprehensive feature identifiers for highly symmetric geometric objects, reflects the mathematical idea of "simplicity is essence" with strict derivation and extremely concise form, and provides a new paradigm for characterizing regular polyhedra by combining extremum principles and discrete invariants. The construction idea can be directly extended to other Platonic solids, as verified by the parallel derivation of the characteristic constant for the regular hexahedron (cube).
Jian Wen (Tue,) studied this question.