wildberger

Rational Trigonometry, Universal Geometry & Chromogeometry in Exact SymPy Algebra

geometor.wildberger synthesizes the algebraic principles of Norman J. Wildberger’s Divine Proportions: Rational Trigonometry to Universal Geometry (Wildberger2005) and Chromogeometry (Wildberger2008) into a 28-Principle Executable Codex (W.01 through W.28).

Where classical trigonometry introduces square-root distances and transcendental angles (\(\sin, \cos, \arccos\)) at the very first step, Rational Trigonometry grounds geometry in two purely quadratic invariants—quadrance \(Q(A_1, A_2)\) and spread \(s(l_1, l_2)\)—unlocking exact polynomial laws over \(\mathbb{Q}\), quadratic extensions such as \(\mathbb{Q}(\sqrt{5})\), arbitrary bilinear metrics, and finite fields \(\mathbb{F}_p\) (\(p \ne 2\)).

Architecture of the Codex

Every principle (W.01 through W.28) is compiled as a 2-Track Symbolic Dossier (zero scanned book reproductions, 100% original synthesis and executable proofs):

  1. Track 1 — Synthesis (index.rst): Formal .. enunciation::, interactive Explorer-styled SVG construction (w-NN.svg) with two-way hover highlighting on every :g: label, constructive .. signature:: block, mathematical synthesis, key algebraic identities, live SymPy verification table, Graphviz dependency DAG, and cross-repository canon links to geometor/euclid, geometor/elements, geometor/fundamentals, geometor/etc, geometor/pacioli, and geometor/pappus.

  2. Track 2 — Code (code.rst): Standalone runnable Python script (model.py) building the exact geometor.model.Model with zero floating-point approximations and rendering w-NN.svg.

The Four Thematic Parts

Part

Principles

Core Algebraic Laws & Discoveries

I. Foundations & Universal Affine Geometry

W.01 – W.05

Arbitrary fields \(\mathbb{F}\) (\(\operatorname{char}\mathbb{F}\ne 2\)), line proportion \(\langle a : b : c \rangle\), quadrance, null line isotropy, spread, cross, twist, altitudes, rational reflections \(\Sigma_l(m)\), and projective cross-ratio bridges.

II. The Five Laws & Polynomial Trigonometry

W.06 – W.12

Pythagoras’ Law (\(Q_1 + Q_2 = Q_3\)), Triple Quad Formula & Archimedes' function / quadrea \(\mathcal{A}\), Spread Law, Cross Law, Triple Spread Formula, spread polynomial Chebyshev/logistic hierarchy \(S_n(s)\), and Quadruple Quad / Spread formulas.

III. Triangle Centers, Polygons, Circles & Conics

W.13 – W.21

Isosceles/Equilateral/Barycentric laws, Stewart/Menelaus/Ceva, circumcenter/orthocenter/incenter, Euler line & Nine-Point Circle over \(\mathbb{F}_p\), cyclic quadrilaterals & quadratic Ptolemy’s Theorem, Kepler’s Golden Triangle & \(\mathbb{F}_{19}\) pentagrams, Quadrolas & Grammolas, and calculus-free Taylor Conic tangents.

IV. Chromogeometry, Surveying, Physics & 3D Space

W.22 – W.28

Finite-field spreads in \(\mathbb{F}_p\), Blue/Red/Green Chromogeometry (\(N_B = \tfrac{1}{2}(C_R + C_G)\)), exact triangle/Regiomontanus/Snellius–Pothenot/Hansen solvers, projectile parabolas & Snell’s Spread Law, rational Platonic dihedral spreads (\(8/9, 1, 8/9, 4/9, 4/5\)), rational polar curves \([s, Q]\), and the cubic Ellipson.