Semi-Regular Solids, Stomachion & Sand-Reckoner — Analytical Summary

Scope: Pappus V.19 · Heiberg Vol. II, pp. 242–291, 416–424, 446–455 · Heath (1897), pp. xxxvi, 221–232, 319–326

Polyhedral Symmetry, Combinatorial Dissection & Cosmic Exponential Scale

Unit 7 brings together the three treatises in which Archimedes transcends pure planar and quadric exhaustion to pioneer uniform polyhedral classification, enumerative combinatorics, Diophantine Pell arithmetic, and heliocentric cosmic scale:

  1. Thirteen Semi-Regular Polyhedra: Exhausts all vertex-transitive convex polyhedra with regular faces of 2 or 3 kinds (Pappus V.19), splitting cleanly into the octahedral/cubic family over (Solids I–VI), the icosahedral/dodecahedral golden family over (Solids VIII–XII, directly modeled in geometor/pacioli and geometor/divine), and the two chiral snub cubics (Solids VII & XIII).

  2. Stomachion & Cattle Problem: Exact integer/rational dissection of into 14 commensurable polygons (Stomachion), alongside the monumental Pell equation (Cattle Problem).

  3. The Sand-Reckoner (*Arenarius*): Unites experimental parallax instrument design (correcting for pupil width!), Aristarchus's heliocentric cosmos, and a two-tier myriad-octad number tower reaching with the explicit index law .

Scale Hierarchy of The Sand-Reckoner

Magnitude / Tier

Archimedean Naming

Modern Power of 10

Myriad myriads (base of Orders)

End of First Order

Sand filling Traditional Cosmos

units of 7th Order

Sand filling Aristarchus’s Star-Sphere

myriads of 8th Order

End of First Period (-th Order)

Unit of 2nd Period, 1st Order

Crown of the System (-th Period)

Myriad-myriad of -th Order, -th Period