New Polyhedral Families from the UMCR: The Three-Ring Prismatoid AP3L, the Crystallon, and The Genesis Solids

What does a cosmological parameter look like in three dimensions?

The UMCR introduced kβ = 1.327, a dimensionless number measured from 1,701 supernovae that governs cosmic expansion. The preceding paper showed that kβ selects geometrically distinguished members from three classical icosahedral families, the first three Santillana Solids. This paper pushes further, into lower symmetries and entirely new territory.

Two things emerge. First, the Crystallon R(kβ): the fourth Santillana Solid, a compact six-faced rhombohedron whose face-diagonal ratio equals kβ exactly. Its face angle of 74.00° falls within 0.25° of the ankerite–dolomite mineral series, a coincidence between a cosmological fit and carbonate crystal geometry that was not engineered.

Second, the AP3L family: a previously uncataloged infinite family of convex polyhedra built from three coaxial vertex rings, with a clean universal topology V = 3n, E = 7n, F = 4n+2. Within it live six new named solids: the Genesis Solids.

Three are equal-edge members, one per convex regime of the family. The Trikahedron (n=3) is the only AP3L solid whose faces are exclusively triangular, 14 of them, all equal-edge. The Chrysohedron (n=4) carries a double golden identity: its outer and lateral edges equal √2 exactly, while its medial ring edge equals φ exactly, both emerging by algebraic necessity. The Homohedron (n=5) is the extremal case: all three edge classes collapse into one, making it the only AP3L member with a single edge length across all 35 edges.

The remaining three enter through a different door. The Trinhedron has exactly two edge classes with ratio φ, forced by the golden identity φ = 2cos(π/5). The Cosmohedron encodes kβ as the circumradius of its medial ring. The Santillanahedron, the polar dual of the Trinhedron, is a trivalent solid with 22 vertices whose coordinates are all algebraic in φ, a property the geometry produces on its own, without being asked.

All topology verified via Euler's formula. All metric claims derived analytically and confirmed numerically. Novelty confirmed against OEIS, Grünbaum, Johnson, Goldberg–Coxeter, and crystallographic databases.