demonstrations

100% Exact Symbolic SymPy & Rational Trigonometry Proofs of Nicomachus’s Arithmetical Geometry

In strict accordance with GEOMETOR’s Content / Code Separation, the historical Book Codex (Introductio Arithmetica (Ἀριθμητικὴ Εἰσαγωγή) — Quadrivium, Ratios, Gnomons & Cube Harmony) contains zero embedded Python scripts. All executable symbolic proofs, ratio trees, figurate gnomon dissections, semicircle mean constructions, and polyhedral cube models live in geometor.nicomachus and the runnable demos/*.py scripts documented below—verified over exact SymPy rationals \(\mathbb{Q}\) and quadratic radicals \(\mathbb{Q}(\sqrt{ac})\) with Model(..., auto_intersect=False) and zero floating-point approximations.

1. Book I, Chs. 13, 16, 17–23 & Book II, Chs. 1–2 — Ratio Taxonomy & the Equality Unfolding Tree

Codex Reference: Unit II — Inequality & the Equality Tree (Critical Notes) · Script: demos/equality_tree.py

Nicomachus I.23 & II.1–2 — Unfolding All Inequality from Equality (1, 1, 1) Direct rule N₁: (a, a+b, a+2b+c) · Anastrophe rule N₂ = N₁R · Inverse reduction N₁⁻¹ (gold dashed) N₁ N₁ (direct) N₂ (anastrophe) N₁ Book II.1–2 Inverse Subtraction N₁⁻¹ returns every ratio to (1, 1, 1) (1, 1, 1) · α, α, α Equality · 1:1 (1, 2, 4) · α, β, δ 1. Multiplex · 2:1 (1, 3, 9) · α, γ, ϑ Multiplex · 3:1 (4, 6, 9) · δ, ϛ, ϑ 2. Superparticularis · 3:2 (1, 4, 16) Multiplex · 4:1 (9, 12, 16) Epitritos · 4:3 (4, 10, 25) · δ, ι, κε 4. Mult. superpartic. · 5:2 (9, 15, 25) · ϑ, ιε, κε 3. Superpartiens · 5:3 (9, 24, 64) 5. Mult. superpartiens · 8:3
Book I, Ch. 23 & Book II, Chs. 1–2: Applying Nicomachus's three commands (a, b, c) -> (a, a+b, a+2b+c) directly and after sibling reversal (c, b, a) unfolds every rational inequality from primordial Equality (1, 1, 1).

  • Symbolic Verification: Starting from Equality (1, 1, 1), direct application of NICOMACHUS_MATRIX_N1 \((a, a+b, a+2b+c)\) generates the multiple spine (1, 2, 4), (1, 3, 9), (1, 4, 16). Reversing a multiple triple before applying the rule (NICOMACHUS_MATRIX_N2) births the superparticulars (4, 6, 9) (hemiolios \(3:2\)) and (9, 12, 16) (epitritos \(4:3\)), while direct vs. reversed children of superparticulars generate multiple superparticulars (4, 10, 25) and superpartients (9, 15, 25). Conversely, reducetoequality reduces every coprime geometric progression \((p^2, pq, q^2)\) back to (1, 1, 1).

from geometor.nicomachus import (
            classify_nicomachean_ratio,
            generate_equality_tree,
            reduce_to_equality,
            verify_equality_identities,
        )

        info = classify_nicomachean_ratio(9, 8)
        assert info["species_key"] == "superparticularis"
        tree = generate_equality_tree(max_depth=3)
        assert tree[0].terms == (1, 1, 1)
        steps = reduce_to_equality(9, 24, 64)
        assert steps[-1].after == (1, 1, 1)
        assert verify_equality_identities()["verified_tree_nodes"] == 16

2. Book II, Chs. 6–20 — Polygonal Alpha-Gnomons, Same/Other & Nicomachus’s Sum-of-Cubes Theorem

Codex Reference: Unit III — Figurate Gnomons & Cubes (Critical Notes) · Script: demos/figurate_gnomons.py

Nicomachus II.20 — Sum of Consecutive Cubes Equals the Squared Triangle 1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)² = 10² = 100 · Odd-block sums & L-gnomon square dissection 2³ = 8 3³ = 27 4³ = 64 0 T₁=1 T₂=3 T₃=6 T₄=10 Nicomachus's Consecutive Odd-Block Theorem (II.20) k = 1 : 1 = 1 = 1³ k = 2 : 3 + 5 = 8 = 2³ k = 3 : 7 + 9 + 11 = 27 = 3³ k = 4 : 13 + 15 + 17 + 19 = 64 = 4³ k = 5 : 21 + 23 + 25 + 27 + 29 = 125 = 5³ Why Shell k Has Exact Area k³: Tₖ² − Tₖ₋₁² = (Tₖ − Tₖ₋₁) · (Tₖ + Tₖ₋₁) = k · k² = k³ (By II.12, two consecutive triangles Tₖ₋₁ + Tₖ = k²!)
Book II, Ch. 20: Consecutive odd-number blocks {1}, {3, 5}, {7, 9, 11}, {13, 15, 17, 19} sum to cubes 1^3, 2^3, 3^3, 4^3 and tile the square of side T_4 = 10 as exact L-shaped gnomons.

  • Symbolic Verification: In construct_nicomachus_cubes_model(order=4), a geometor.model.Model(auto_intersect=False) constructs the \(T_n \times T_n\) square bordered at triangular coordinates \(T_k = \frac{k(k+1)}{2}\). Each \(k\)-th L-gnomon has exact symbolic area \(T_k^2 - T_{k-1}^2 = k^3\) and equals the sum of the \(k\)-element odd block {13, 15, 17, 19} = 64 = 43, while sieve_of_eratosthenes and euclid_nicomachus_perfect_numbers verify Book I, Chs. 13 & 16.

from geometor.nicomachus import (
            construct_nicomachus_cubes_model,
            euclid_nicomachus_perfect_numbers,
            sieve_of_eratosthenes,
            verify_figurate_identities,
        )

        sieve = sieve_of_eratosthenes(35)
        assert sieve["odd_primes"][:6] == (3, 5, 7, 11, 13, 17)
        assert [row["perfect_number"] for row in euclid_nicomachus_perfect_numbers(4)] == [6, 28, 496, 8128]
        cubes = construct_nicomachus_cubes_model(order=4)
        assert cubes.total_area == cubes.triangular_side**2 == 100
        assert verify_figurate_identities()["nicomachus_cubes_verified"] is True

3. Book II, Chs. 21–29 — Semicircle Means, Ten Proportions & the Cube’s Teleiotatē Analogia

Codex Reference: Unit IV — Ten Means & Cube Harmony (Critical Notes) · Script: demos/ten_means_and_cube.py

Nicomachus II.21–28 — The Three Classical Archetypes & Synoptic Decad of Ten Means Left: Semicircle on PQ = 6, QR = 12 proving A ≥ G ≥ H & A · H = G² · Right: Hoche p. 144 Ten Means Semicircle Construction (a = 6, c = 12) P(0) Q(6) O(9) R(18) S F G = 6√2 A = 9 H = 8 A = 9 > G = √72 ≈ 8.49 > H = 8 · A·H = 72 = G² Synoptic Decad Table (Book II.28.11, Hoche p. 144) 1. Arithmetic α, β, γ (1, 2, 3) (z−m):(m−x) = 1 2. Geometric α, β, δ (1, 2, 4) (z−m):(m−x) = z:m 3. Harmonic γ, δ, ϛ (3, 4, 6) (z−m):(m−x) = z:x 4. Subcontr. Harmonic γ, ε, ϛ (3, 5, 6) (m−x):(z−m) = z:x 5. Subcontr. Geom. I β, δ, ε (2, 4, 5) (m−x):(z−m) = m:x 6. Subcontr. Geom. II α, δ, ϛ (1, 4, 6) (m−x):(z−m) = z:m 7. Neo-Pythagorean I ϛ, η, ϑ (6, 8, 9) (z−x):(m−x) = z:x 8. Neo-Pythagorean II ϛ, ζ, ϑ (6, 7, 9) (z−x):(z−m) = z:x 9. Neo-Pythagorean III δ, ϛ, ζ (4, 6, 7) (z−x):(m−x) = m:x 10. Additive / Fibonacci γ, ε, η (3, 5, 8) (z−x):(z−m) = m:x
Book II, Chs. 21–28 & Pappus Synagoge III: Over diameter PR = a + c, radius OS = A, perpendicular ordinate QS = G, and radial projection SF = H satisfy H <= G <= A with exact right-triangle identity G^2 = A * H.

  • Symbolic Verification: In construct_three_classical_means_model(6, 12), exact circle-line and line-line intersections over \(\mathbb{Q}(\sqrt{2})\) establish radius OS = A = 9 (\(Q(O, S) = 81\)), ordinate QS = G = 6√2 (\(Q(Q, S) = 72\)), and radial projection SF = H = 8 (\(Q(S, F) = 64\)). In construct_cube_proportion_model(), the Cube’s \(6\) faces, \(8\) vertices, and \(12\) edges embed \(6 : 8 :: 9 : 12\) with projective harmonic cross-ratio -1 and every Pythagorean consonance (\(2:1, 3:2, 4:3, 9:8\)).

from geometor.model.invariants import quadrance
        from geometor.nicomachus import (
            compute_all_ten_means,
            construct_cube_proportion_model,
            construct_three_classical_means_model,
            verify_cube_proportion_identities,
            verify_ten_means_identities,
        )

        m = construct_three_classical_means_model(6, 12)
        assert quadrance(m.get_element_by_ID("O"), m.get_element_by_ID("S")) == 81  # A^2 = 9^2
        assert quadrance(m.get_element_by_ID("Q"), m.get_element_by_ID("S")) == 72  # G^2 = 6 * 12
        assert quadrance(m.get_element_by_ID("S"), m.get_element_by_ID("F")) == 64  # H^2 = 8^2

        assert len(compute_all_ten_means(4, 12)) == 10
        cube = construct_cube_proportion_model()
        assert cube.cube_counts == {"faces": 6, "vertices": 8, "edges": 12}
        assert cube.projective_cross_ratio == -1
        assert all(verify_ten_means_identities().values())
        assert all(verify_cube_proportion_identities().values())