A cosmological parameter determined from supernova data, kβ = 1.327, turns out to select geometrically distinguished members of three classical polyhedral families. This is not a metaphor or an analogy. The number that governs the recursive expansion of spacetime in the UMCR is the same number that uniquely defines the circumradius ratio of a dodecahedron-icosahedron compound, the face-diagonal ratio of a rhombic-30 isohedron, and the pyramid-height ratio of an icosahedral kleetope. These three solids are named the Santillanahedron, the Recursohedron, and the Observahedron. Their metric and topological properties are derived analytically and verified numerically. No existing polyhedral catalogue names any member of these families by a physically measured cosmological parameter. This paper establishes their formal definition, proves their geometric properties, and deposits their .off files as reproducible records. The geometry was not designed to fit the physics, it was found there.
Jean Santillana (2026) studied this question.