Demonstrations

100% Exact Symbolic SymPy Models, Möbius Inversion & Rational Trigonometry Proofs

In strict accordance with GEOMETOR’s Content / Code Separation, the historical Book Codex (De Docta Ignorantia et Scripta Mathematica — Coincidence of Opposites & Exact Geometry) embeds zero Python scripts. All executable geometric constructions live in the geometor.cusa package and the standalone demos/*.py scripts documented below, verified over \(\mathbb{Q}\), \(\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{6}, \sqrt{7}, \sqrt{15}, \sqrt{21})\), and \(\mathbb{Q}(n)\) using Model(..., auto_intersect=False) with zero floating-point approximations.

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1. De Docta Ignorantia I.13 — Tangent Circle Pencil & Conformal Möbius Inversion

Codex Reference: Unit III — Coincidence of Figures (Critical Notes) · Script: demos/coincidentia_circle_line.py

Tangent circle pencil C_1, C_2, C_4, C_8 above straight horizon AB (y = 0) with exact radical sagittae 1, 2 - sqrt(3), 4 - sqrt(15), 8 - 3*sqrt(7) along vertical secant x = 1.

Bounded conformal inversion in unit circle Omega at pole S = (0, -1), mapping C_R to finite rational circles C'_R centered at (0, -R/(2R+1)) converging smoothly to C'_inf = inverted line AB.

• Exact Symbolic Verification: Rationalizing the sagitta \(h(x, R) = R - \sqrt{R^2 - x^2} = \frac{x^2}{R + \sqrt{R^2 - x^2}}\) proves \(\lim_{R \to \infty} h(x, R) = 0\), \(\lim_{R \to \infty} R\,h(x, R) = \frac{x^2}{2}\), and curvature \(\kappa(R) = \frac{1}{R} \to 0\). Under unit-circle inversion at S \(= (0, -1)\), every circle \(C_R\) maps to the exact rational circle centered at \(\left(0, -\frac{R}{2R+1}\right)\) of radius \(\frac{R}{2R+1}\), whose \(R \to \infty\) limit at Minf \(= \left(0, -\frac{1}{2}\right)\) is identically the inverted straight line AB.

import sympy as sp
        from geometor.cusa import (
            construct_tangent_circle_pencil,
            verify_mobius_inversion_coincidence,
            verify_sagitta_and_curvature_limits,
        )

        model = construct_tangent_circle_pencil()
        assert not any(pt.has(sp.Float) for pt in model.points)

        limits = verify_sagitta_and_curvature_limits()
        assert limits["identity_holds"] and limits["sagitta_limit"] == 0

        mobius = verify_mobius_inversion_coincidence()
        assert mobius["limit_center_y"] == sp.Rational(-1, 2)
        assert mobius["limit_radius"] == sp.Rational(1, 2)
        

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2. De Docta Ignorantia I.14–15 — Flattening Triangles in Rational Trigonometry

Codex Reference: Unit III — Coincidence of Figures (Critical Notes) · Script: demos/coincidentia_triangle_line.py

Nested isosceles triangles B A_n C over fixed base BC = 2 with altitudes 1/n: pure rational apex spreads s_A = 4n^2/(n^2+1)^2 -> 0 and Triple Quad collinearity residual 16/n^2 -> 0.

• Exact Rational Trigonometry Proof: For B \(= (-1, 0)\), C \(= (1, 0)\), and apex \(A_n = (0, 1/n)\) (A1, A2, A4, A8 descending toward M), Wildberger's invariants belong purely to \(\mathbb{Q}(n)\): \(Q(A_n, B) = 1 + \frac{1}{n^2}\), \(s_B = s_C = \frac{1}{n^2+1} \to 0\), \(s_{A_n} = \frac{4n^2}{(n^2+1)^2} \to 0\), the Triple Spread Formula vanishes identically, and the Triple Quad Formula residual equals \(\frac{16}{n^2} \to 0\).

import sympy as sp
        from geometor.cusa import (
            compute_flattening_triangle_metrics,
            verify_triangle_line_coincidence,
        )

        m4 = compute_flattening_triangle_metrics(4)
        assert m4.spread_apex == sp.Rational(64, 289)
        assert m4.triple_quad_residual == sp.Integer(1)

        proof = verify_triangle_line_coincidence()
        assert proof["triple_spread_identity"] is True
        assert proof["limit_triple_quad"] == 0
        

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3. Polygon Exhaustion (I.3) & Scripta Mathematica (1445–1459)

Codex Reference: Unit I & Unit V · Script: demos/polygon_exhaustion_and_complementa.py

De Geometricis Transmutationibus (1445, §§9–11): quarter-side ray AE = sqrt(7)/4 extended in 5:4 sesquiquartal ratio to AH = 5*sqrt(7)/16, implying pi ~= 24*sqrt(21)/35 ~= 3.14233762.

Aurea Propositio in Mathematicis (1459, §§1–2) over Q(sqrt(3)) at theta = pi/6: balanced rectifying segment IKL satisfying |AI| + |AK| + |AL| = 3 and fifth-order Maclaurin accuracy theta^5/180.

• Exact Symbolic Verification: Verifies (a) monotone radical apothems \(r_n = \cos(\pi/n)\) across \(n \in \{3, 4, 6, 8, 12\}\), (b) Cusa’s 1445 sesquiquartal equilateral-triangle rectification \(\pi_{\text{Cusa},1445} = \frac{24\sqrt{21}}{35}\), and (c) Cusa’s 1458–1459 three-ray law \(L_{\text{Cusa}}(\theta) = \frac{6\sin\theta}{2 + \cos\theta}\) with exact Maclaurin series error \(\theta - \frac{3\sin\theta}{2+\cos\theta} = \frac{\theta^5}{180} + \frac{\theta^7}{1512} + \frac{\theta^9}{25920} + O(\theta^{11})\).

import sympy as sp
        from geometor.cusa import (
            verify_cusa_snellius_approximation,
            verify_transmutatio_1445,
        )

        tr = verify_transmutatio_1445()
        assert sp.simplify(tr["pi_cusa_1445"] - 24 * sp.sqrt(21) / 35) == 0

        au = verify_cusa_snellius_approximation()
        assert au["tangent_ratio_equal"] is True
        assert au["coeff_theta_5"] == sp.Rational(1, 180)