Semi-Regular Solids, Stomachion & Sand-Reckoner — Philological & Mathematical Notes

1. Archimedes → Pappus → Pacioli & Leonardo → Kepler

In Synagoge V.19 (pappus:synagoge:v), Pappus lists the thirteen solids in increasing order of complexity within the tetrahedral/cubic and icosahedral families (8, 14, 14, 14, 26, 26, 38, 32, 32, 32, 62, 62, 92 faces) and computes their vertex and edge numbers by face-incidence counting (). During the Renaissance, Piero della Francesca and Luca Pacioli (geometor:pacioli, De Divina Proportione, Venice 1509, with woodcuts after Leonardo da Vinci’s corpora vacua skeletal drawings) rediscovered the truncated and elevated (abscisi / elevati) members directly from the five Platonic solids of Euclid Book XIII, before Johannes Kepler (Harmonices Mundi, Linz 1619) restored all thirteen—including the enantiomorphic snub cube and snub dodecahedron—by systematic angular-defect analysis ( at every convex vertex, Eucl. XI. 21).

2. GEOMETOR Cross-Corpus Concordance

• Pacioli (`geometor/pacioli`) & Divine (`geometor/divine`) (geometor:pacioli): Solids VIII–XII (Icosidodecahedron, Truncated Icosahedron, Truncated Dodecahedron, Rhombicosidodecahedron, Truncated Icosidodecahedron) have exact vertex coordinates belonging purely to the golden quadratic field (), uniting Archimedes’ lost polyhedral treatise directly with Euclid XIII.16–17 and Pacioli’s Divine Proportion.

• Pappus (`geometor/pappus`) (geometor:pappus, pappus:synagoge:v): Bridges Pappus Book V (isoperimetric and polyhedral comparison) with Archimedes’ solid geometry.

• Biographical & Astronomical Witness: The Sand-Reckoner preserves two unique historical testimonies found nowhere else in antiquity: (1) the name and astronomical work of Archimedes’ father Phidias, and (2) the verbatim heliocentric hypotheses of Aristarchus of Samos—cited eighteen centuries later in Copernicus’s manuscript of De revolutionibus!