Demonstrations

100% Exact Symbolic SymPy Models & Rational Trigonometry Proofs of Plato’s Constructions

In strict accordance with GEOMETOR’s Content / Code Separation, the historical Book Codex (Loci Mathematici et Geometrici — Plato on Geometry, Proportion & Noetic Ascent) contains zero embedded Python scripts. All executable constructions live in the geometor.plato Python package and runnable demos/*.py scripts documented below, where every coordinate, intersection, quadrance, spread, and harmonic interval is verified symbolically over \(\mathbb{Q}\) and closed-form quadratic extensions (\(\mathbb{Q}(\sqrt{2})\), \(\mathbb{Q}(\sqrt{3})\), \(\mathbb{Q}(\phi)\), and the Theodorus tower \(\mathbb{Q}(\sqrt{2}, \dots, \sqrt{17})\)) with Model(..., auto_intersect=False) and zero floating-point approximations.

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1. Meno 82b–85b — Doubling the Square by the Diagonal

Codex Reference: Unit I — Meno (Critical Notes) · Script: demos/meno_diagonal.py

Exact symbolic model of Socrates' three-stage inquiry with Meno's attendant: given 2-foot square ``ABCD`` (quadrance 4), false 4-foot square ``AEFG`` (quadrance 16), false 3-foot square ``AMNP`` (quadrance 9), and diagonal square ``BHKD`` (quadrance 8).

• Symbolic Verification: Given A00 and B20, has quadrance \(Q(A, B) = 4\). Doubling the side to E40 quadruples quadrance (\(Q(A, E) = 16\)); taking the 3-foot side M30 yields \(Q(A, M) = 9\) (leaving no rational integer solution to \(x^2 = 8\)). Drawing the diagonal BD (B20, D02) bisects ABCD into two triangles of area \(2\) and establishes the diagonal square BHKD (H22, K04 in the enclosing grid, or erected directly on BD) with exact quadrance \(Q(B, D) = 4 + 4 = 8 = 2 \cdot Q(A, B)\).

from geometor.model.invariants import quadrance
        from geometor.plato import construct_meno_diagonal_doubling

        model = construct_meno_diagonal_doubling()
        pt_a = model.get_element_by_ID("A")
        pt_b = model.get_element_by_ID("B")
        pt_d = model.get_element_by_ID("D")
        pt_e = model.get_element_by_ID("E")
        pt_m = model.get_element_by_ID("M")
        assert quadrance(pt_a, pt_b) == 4
        assert quadrance(pt_a, pt_e) == 16
        assert quadrance(pt_a, pt_m) == 9
        assert quadrance(pt_b, pt_d) == 8
        

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2. Theaetetus 147d–148b — Theodorus’s Spiral (\(\sqrt{2} \dots \sqrt{17}\)) & Dynameis

Codex Reference: Unit II — Theaetetus (Critical Notes) · Script: demos/theodorus_spiral.py

Exact compass-and-straightedge spiral from ``P1`` through ``P17`` around center ``O``. Square-number radii ``P1, P4, P9, P16`` are rational lengths (*mēkē*); all thirteen non-square radii are quadratic surds (*dynameis*).

• Symbolic Verification: At each step \(n \in \{1, \dots, 16\}\), a perpendicular line is erected at \(P_n\) on carrier \(O P_n\) and intersected with the unit circle centered at \(P_n\) to select the counterclockwise vertex \(P_{n+1}\) with unit step quadrance \(Q(P_n, P_{n+1}) = 1\) and exact radial quadrance \(Q(O, P_n) = n\) for all \(1 \le n \le 17\).

from geometor.model.invariants import quadrance
        from geometor.plato import classify_theaetetus_powers, construct_theodorus_spiral

        report = classify_theaetetus_powers(17)
        assert [r.area_n for r in report if "mekos" in r.line_kind] == [1, 4, 9, 16]

        spiral = construct_theodorus_spiral(max_n=17)
        origin = spiral.get_element_by_ID("O")
        for n in range(1, 18):
            assert quadrance(origin, spiral.get_element_by_ID(f"P{n}")) == n
        

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3. Republic VI 509d–511e — The Divided Line & Proportional Equality of the Middle Segments

Codex Reference: Unit III — Republic VI–VII (Critical Notes) · Script: demos/divided_line.py

Exact golden-section realization of Plato's Divided Line ``AE`` cut at ``C`` with sub-cuts ``B`` and ``D``, verifying ``|BC| = |CD|`` and the Euclid VI.13 semicircle mean proportional ``|BG| = |BC|``.

• Symbolic Verification: Cutting baseline AE in ratio \(1 : r\) at C and each main section in the same ratio \(1 : r\) at B and D forces \(|AB| : |BC| : |CD| : |DE| = 1 : r : r : r^2\) for every \(r \neq 1\) (verified symbolically over sp.Symbol('r', positive=True)) and specifically over \(\mathbb{Q}(\phi)\) (\(r = \phi = \frac{1 + \sqrt{5}}{2}\)), where the semicircle on AK meets the perpendicular at B in G with \(Q(B, G) = Q(B, C) = Q(C, D) = \phi^2\).

from geometor.model.invariants import quadrance
        from geometor.plato import construct_divided_line, verify_divided_line_identity

        ident = verify_divided_line_identity()
        assert ident["middle_segments_equal"] is True

        line_model = construct_divided_line()
        pt_b = line_model.get_element_by_ID("B")
        pt_c = line_model.get_element_by_ID("C")
        pt_d = line_model.get_element_by_ID("D")
        pt_g = line_model.get_element_by_ID("G")
        assert quadrance(pt_b, pt_c) == quadrance(pt_c, pt_d)
        assert quadrance(pt_b, pt_g) == quadrance(pt_b, pt_c)
        

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4. Timaeus 31b–36d — Three Means in a Semicircle & the 34-Pitch World-Soul Scale

Codex Reference: Unit IV — Timaeus: World-Soul (Critical Notes) · Script: demos/timaeus_world_soul.py

Pappus's semicircle construction of arithmetic mean ``|OG| = 3``, geometric mean ``|JG| = 2√2``, and harmonic mean ``|GH| = 8/3`` on segments ``|AJ| = 2`` and ``|JB| = 4``, paired with the equatorial Circle of the Same and ecliptic Circle of the Different.

• Symbolic Verification: Starting from the seven integers 1, 2, 3, 4, 8, 9, 27, inserting harmonic means \(\frac{2ab}{a+b}\) and arithmetic means \(\frac{a+b}{2}\), and filling every \(4:3\) fourth with two \(9:8\) whole tones leaves the exact Pythagorean leimma \(\frac{256}{243}\), generating 34 exact rational pitches (\(24\) tones and \(9\) leimmata: \((9/8)^{24} \cdot (256/243)^9 = 27\)), cleared of fractions by Crantor's least integer multiplier 384 (384 .. 10368).

import sympy as sp
        from geometor.plato import CRANTOR_INTEGER_MULTIPLIER, EPOGDOOS, LEIMMA, build_timaeus_scale

        pitches = build_timaeus_scale()
        assert len(pitches) == 34
        assert pitches[0].crantor_integer == CRANTOR_INTEGER_MULTIPLIER == 384
        assert pitches[-1].crantor_integer == 10368
        assert sp.simplify(EPOGDOOS**24 * LEIMMA**9 - 27) == 0
        

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5. Timaeus 53c–56c — Elementary Right Triangles & Rational Trigonometry Spreads

Codex Reference: Unit V — Timaeus: Polyhedra (Critical Notes) · Script: demos/timaeus_triangles.py

Euclid I.1 two-circle construction of equilateral face ``ABC`` subdivided by its three medians/altitudes at centroid ``G`` into six ``1 : √3 : 2`` scalene right half-triangles (quadrances ``1, 3, 4``; spreads ``1/4, 3/4, 1``).

• Symbolic Verification: Both of Timaeus’s elementary planar building blocks satisfy polynomial Rational Trigonometry identities without transcendental angles:

  • Scalene Half-Equilateral (\(\triangle GM_1C\)): quadrances \((1, 3, 4)\), spreads \(\left(\frac{1}{4}, \frac{3}{4}, 1\right)\) (“the greater side triple in power of the lesser”, 54b5).

  • Isosceles Right Triangle (cube face ABCD bisected by diagonals at center O): quadrances \((1, 1, 2)\) around O, spreads \(\left(\frac{1}{2}, \frac{1}{2}, 1\right)\).

import sympy as sp
        from geometor.model import quadrance, vertex_spread
        from geometor.plato import construct_timaeus_cube_face, construct_timaeus_triangles

        tri = construct_timaeus_triangles()
        pt_g = tri.get_element_by_ID("G")
        pt_m1 = tri.get_element_by_ID("M1")
        pt_c = tri.get_element_by_ID("C")
        assert (quadrance(pt_g, pt_m1), quadrance(pt_m1, pt_c), quadrance(pt_g, pt_c)) == (1, 3, 4)
        assert vertex_spread(pt_g, pt_c, pt_m1) == sp.Rational(1, 4)
        assert vertex_spread(pt_m1, pt_g, pt_c) == sp.Rational(3, 4)
        assert vertex_spread(pt_g, pt_m1, pt_c) == 1

        cube = construct_timaeus_cube_face()
        assert cube is not None
        

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6. Parmenides, Philebus & Laws VII — Compass, Straightedge & Dimensional Surds

Codex Reference: Unit VI — Foundations (Critical Notes)

Unit circle centered at ``O`` cut along optical sightline ``AB`` (Parmenides 137e), generating length ``1`` (``OC``), plane diagonal ``√2`` (``OD``), and space-diagonal power ``√3`` (``OE``) by compass and straightedge (Philebus 51c; Laws VII 819e).

• CLI Quickstart: Run all symbolic invariant checks, inspect the 34-pitch Timaeus table, or recompile all SVGs and 4-track RST units via geometor-plato:

geometor-plato verify
        geometor-plato scale
        geometor-plato concordance
        geometor-plato figures
        geometor-plato compile