Johannes Kepler

Polyhedral Cosmography, Close-Packed Crystallography, Constructible Polygons, and Celestial Harmony

Geometria ante rerum ortum Menti divinae coaeterna, Deus ipse (quid enim in Deo, quod non sit Ipse Deus?) exemplaria Deo creandi mundi suppeditavit, et cum imagine Dei transivit in hominem.

Geometry, coeternal with the Divine Mind before the origin of things—being God Himself (for what is in God that is not God Himself?)—supplied God with the exemplars for creating the world, and passed over into humanity along with the image of God.

—Johannes Kepler, Harmonices Mundi Liber III, Cap. I (Linz, 1619)

Overview

GEOMETOR Kepler (geometor-kepler) unites a 4-Track Bilingual Digital Scriptorium Book Codex (English, Latin, Summary, Notes) with 100% exact symbolic SymPy / ``geometor.model`` constructions and proofs across the three foundational geometric treatises of Johannes Kepler (1571–1630):

  1. *Mysterium Cosmographicum* (1596 / 1621): Kepler’s July 19, 1595 Graz discovery nesting the five regular Platonic solids between the six heliocentric planetary orbs of Nicolaus Copernicus, proving polar-duality insphere/circumsphere equality ($\text{Cube} = \text{Octahedron}$ with $Q_{\text{in}}/Q_{\text{out}} = \frac{1}{3}$; $\text{Dodecahedron} = \text{Icosahedron}$ with $Q_{\text{in}}/Q_{\text{out}} = \frac{5 + 2\sqrt{5}}{15}$) and celebrating the Two Great Treasures of Geometry—the Pythagorean Theorem and the Divine Section $\phi = \frac{1 + \sqrt{5}}{2}$.

  2. *Strena seu de Nive Sexangula* (*On the Six-Cornered Snowflake*, 1611): Dedicated to Johannes Matthäus Wackher von Wackenfels, founding mathematical crystallography and natural morphology: discovery of the space-filling Rhombic Dodecahedron ($\sqrt{2}:1$) and golden Rhombic Triacontahedron ($\phi:1$), the Kepler Sphere-Packing Conjecture ($\frac{\pi}{3\sqrt{2}} \approx 74.048\%$), and the self-propagating Fibonacci recurrence $1, 1, 2, 3, 5, 8, 13, 21\dots \to \phi$ in pentamerous blossoms.

  3. *Harmonices Mundi Libri V* (*The Harmony of the World*, 1619 — Books I, II & V) & the 1597 Kepler Right Triangle: Euclidean degrees of geometric knowability vs. the irreducible cubic non-constructibility of the regular heptagon ($x^3 + x^2 - 2x - 1 = 0$); complete classification of the 11 uniform Archimedean planar tilings, pentagonal/decagonal star tessellation (Figure Aa), the 13 Archimedean convex solids, and the Small & Great Stellated Dodecahedra ($\{5/2, 5\}, \{5/2, 3\}$); the Kepler Right Triangle $(1, \sqrt{\phi}, \phi)$ over $\mathbb{Q}(\sqrt{\phi})$; planetary aphelion/perihelion consonances; and Kepler’s Harmonic Third Law $(T_1/T_2)^2 = (a_1/a_2)^3$.

Architecture of the Repository

  • Digital Scriptorium Book Codex (Cosmos & Harmony Codex): Five multi-track units (:layout: book) pairing original GEOMETOR English translations with verbatim public-domain Neo-Latin primary texts (Frisch 1858–1868), executive geometric summaries, scholarly commentary, 9 interactive .euclid-graphic SVG plates, and linked apparatus registries (Concordance, Bibliography, Persons, Glossary).

  • Exact Symbolic Demonstrations (Symbolic Demonstrations): Pure-symbolic Python constructions (Model(…, auto_intersect=False)) free of floating-point coordinates, verifying every polyhedral ratio, sphere-packing density, planar tiling, heptagon minimal polynomial, Kepler triangle quadrance, and planetary power law.

  • Python API Reference (Modules): Documented symbolic modules in geometor.kepler (mysterium_spheres, strena_packing, harmonices_geometry, cli, sphinx_ext).

  • Project Context & Log (About GEOMETOR Kepler, log): Provenance, editorial standards, and development history.