II. The Unfolding Canon (The Elementrium)

Classical Synthesis, Triangle Centers, Projective Geometry & Rational Trigonometry

Pillar II recasts foundational mathematical treatises and modern synthetic canons into interactive Book Codices where historical prose, formal pseudocode, exact SymPy proofs, and vector diagrams illuminate one another.

euclid — Sir Thomas L. Heath’s Thirteen Books of Euclid’s Elements

A complete three-volume critical digital edition of Heath’s Euclid’s Elements spanning 1,416 archival folios, 130 definitions, 5 postulates, 5 common notions, 465 propositions, and 831 recalibrated interactive SVGs. Presented as a 2-Track Book Codex (Proposition and Heath Commentary) with bidirectional :g: hover highlighting and :just: citation tooltips.

elements — Elements 2.0 & The 601-Node G-Index

Codifies Euclid’s logical dependency DAG into the sequential G-Index (G.1 through G.601), ensuring no definition, postulate, common notion, or proposition is invoked before its prerequisites are established. Each proposition is published as a 4-Track Book Codex:

  1. Formal Pseudocode with interactive step-by-step SVG playback (|< < ▶ > All);

  2. Executable Python built in geometor.model;

  3. Heath Classical Prose synchronized to the same figure;

  4. QHD Studio Broadcast with harmonic acoustic cues.

fundamentals — The Fred W. Seelig Constructive Archive

Preserves and executes the mathematical notebooks of Fred W. Seelig across 55 foundational topics (F.1–F.55), 353 manuscript folios, and 59 analytical spreadsheets in a 3-Track Codex—bridging classical Euclidean constructions, harmonic duality, Conway triangle notation, circle-square perimeter/area proportions, and pyramid geometry.

altshiller-court — Nathan Altshiller-Court’s College Geometry

Complete scholarly Book Codex of Nathan Altshiller-Court’s classic College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed., 1952)—spanning 334 folios and 699 numbered articles across 10 chapters. Bridges Book VI of Euclid to nineteenth- and twentieth-century synthetic mastery: the four-stage constructive method (Analysis, Construction, Proof, Discussion), geometric loci, similitude and homothety, the Nine-Point Circle and Feuerbach’s Theorem, the Simson Line, Menelaus and Ceva, Harmonic Division and Apollonian Circles, Inversion, and Lemoine/Symmedian/Brocard geometry.

etc — Clark Kimberling’s Encyclopedia of Triangle Centers

Transforms Clark Kimberling’s monumental Encyclopedia of Triangle Centers (53,144 centers parsed; 4,000+ centers compiled across 62 collections with 16,000+ interactive SVGs) into an inspectable laboratory. Every compiled center \(X_n\) is evaluated symbolically and rendered across four canonical reference triangles—Acute (5, 6, 7), Egyptian Right (3, 4, 5), Kepler Golden Right (1, √φ, φ), and Kimberling Scalene (6, 9, 13)—accompanied by Euler/Nagel/Brocard central lines, circle families, and Conway notation foundations.

pappus — Projective Geometry & Hexagrammum Mysticum

Investigates projective incidence, cross-ratio invariance, and conic hexagons through exact SymPy symbolic proofs:

  • Pappus’s Hexagon Theorem (Book VII, Prop. 139): Full symmetric group \(S_3\) analysis of all 6 hexagon permutations, proving their 6 Pappus lines concur 3-by-3 at the 2 Steiner Star Points \(S_+\) and \(S_-\).

  • Pascal’s *Hexagrammum Mysticum*: Symbolic verification of the complete configuration on a conic—60 Pascal lines, 20 Steiner points, 60 Kirkman points, and 15 Plücker lines.

  • Harmonic Conjugates & Golden Conics: Unmarked straightedge construction of the harmonic fourth via the complete quadrangle, plus golden projectivities and conic hexagons over \(\mathbb{Q}(\phi)\).

wildberger — Divine Proportions & Rational Trigonometry

Constructive symbolic companion to Norman J. Wildberger’s Divine Proportions: Rational Trigonometry to Universal Geometry (2005). Provides exact SymPy derivations, geometor.model constructions, and interactive SVGs for Quadrance and Spread, the five fundamental laws (Pythagoras, Triple Quad Formula, Cross Law, Spread Law, Triple Spread Formula), Spread Polynomials \(S_n(s)\) (including pentagonal golden spreads over \(\mathbb{Q}(\sqrt{5})\)), Chromogeometry (blue Euclidean, red Lorentzian/relativistic, and green bilinear forms), and rational conic geometry.