GEOMETOR

Exploring the Architecture of All That Is

GEOMETOR is an open-source research initiative, symbolic software engine, and digital scriptorium investigating geometry as humanity’s most concise language for documenting nature, proportion, and deductive truth.

When we watch a botanical meristem pack florets at the golden angle, examine crystal lattices, or trace projective collineations, one foundational question recurs: How do the simplest axiomatic relationships unfold into the full complexity of form and mind?

Across forty-six interconnected repositories, GEOMETOR recasts 2,400 years of classical treatises, projective and rational geometry, natural morphogenesis, epistemological philosophy, and machine reasoning into one consistent, exact symbolic grammar—uniting two complementary modes of work:

  • Scholarly Editions (*Book Codices*): Multi-language critical editions of public-domain works (Greek, Latin, Italian, German, French, and English) paired with calibrated .euclid-graphic SVGs and runnable geometor.model constructions.

  • Mathematical Catalogs: One-item-per-page symbolic catalogs documenting the theorems, golden constructions, triangle centers, rational trigonometry laws, and open-source proving engines of modern mathematics (etc, wildberger, divine, knott, codes) with full scholarly citation.

From Two Points in an Empty Field

Every GEOMETOR construction begins on an unmarked planar field with the minimum possible information: two given points, A and B. Their separation establishes the unit measure (\(1\)). Rather than approximating coordinates in floating-point arithmetic, we project continuous straight lines and circles from known points and select exact algebraic intersections.

Use the playback controls below the figure (or hover any highlighted symbol in the steps) to watch how two points unfold through the Vesica Piscis and Equidistant Poles into the Divine Proportion:

A B C C′ M P Q O H G (φ)
given A, B — Two starting points at \((0, 0)\) and \((1, 0)\) establish the planar field, unit segment / A B /, and infinite baseline [ A B ].
construct circle(A, B) → ( A B ), circle(B, A) → ( B A ) — Sweeping two symmetric unit circles forms the Vesica Piscis.
intersect ( A B ) ∩ ( B A ) → C, C′ — Their intersections select the Equidistant Poles, unfolding the equilateral triangle < A B C > (equilateral_triangle), rational spread \(s = \frac{3}{4}\) (\(\sqrt{3}\)), and the orthogonal axis [ C C′ ] bisecting / A B / at midpoint M.
construct Extending rays [ C′ P ] and [ C′ Q ] through A and B to the twin circles forms the enlarged side-\(2\) equilateral frame < C′ P Q >; its medians [ A Q ] and [ B P ] meet [ C C′ ] at circumcenter O, from which we sweep the outer circumcircle ( O C′ ).
∴

( O C′ ) ∩ [ A B ] → H, G — By George Odom’s midline theorem, / A B / and / B G / divide / A G / in Extreme and Mean Ratio:

\[\frac{|AG|}{|AB|} = \frac{|AB|}{|BG|} = \phi = \frac{1 + \sqrt{5}}{2}\]

From this single unfolding emerge the regular pentagon and icosahedron, projective harmonic ranges, rational spread polynomials, Fibonacci parastichies in living plants, and formal constructive proofs.

One Language of Form: Method & Principles

  • Exact Symbolic Algebra over Floating-Point Approximation: Dynamic geometry tools built on IEEE-754 floats can only report that two segments look numerically close to \(1.6180339\dots\) Powered by SymPy, geometor.model preserves every coordinate, polynomial root, quadrance, spread, and cross-ratio in exact algebraic form—proving mathematical identity rather than decimal coincidence.

  • Projection & Selection: Points hold information; lines [ A B ] and circles ( A B ) project infinite loci across the field; intersections ∩ select derived points carrying their complete ancestral dependency DAG back to the initial givens.

  • Earned Understanding (*Noetics of Geometry*): Grounded in Plato, Aristotle, Proclus, Thomas Aquinas (materia intelligibilis), Nicholas of Cusa, and Petrus Hoenen’s De Noetica Geometriae, geometric necessity is neither empirical guesswork nor an uninterpreted game of marks—it is apprehended directly within concrete construction.

  • Scholarly Editions & Mathematical Catalogs: Powered by photon-platform/publish, GEOMETOR pairs multi-track Book Codices of historical public-domain sources with structured one-item-per-page Mathematical Catalogs for modern research works.

The Five Pillars of Exploration

I. The Symbolic Core, Discovery & Software Catalog

  • model (GitHub · PyPI): Exact SymPy compass-and-straightedge construction engine with rational trigonometry invariants (quadrance, spread, cross_ratio), @proposition proof ledgers, and ancestral DAGs.

  • explorer (GitHub): Interactive SVG workbench, symbolic CLI, scrubbable construction timeline, and publication sheet exporter.

  • divine (GitHub · PyPI): Automated golden-section & harmonic-range scanner paired with a Catalog of Cut-the-Knot golden ratio constructions (Odom, Hofstetter, Bogomolny).

  • codes (GitHub): Catalog of open-source geometric computing and theorem-proving engines (GeoCoq, JGEX / Area Method, GeoGebra Discovery, SymPy Geometry, CGAL).

  • trees: Monotone planar tree drawing algorithms grounded in Stern–Brocot trees and Farey rational sequences.

II. The Unfolding Canon (The Elementrium & Constructive Geometry)

  • euclid (GitHub): Sir Thomas L. Heath’s 13-Book Elements (465 propositions, 1,416 folios, 831 calibrated SVGs) plus Euclid’s Data (Δεδομένα, 94 propositions of geometric analysis in Greek & English).

  • elements (GitHub): Topological codification of Euclid into the 601-node G-Index (G.1–G.601) in a 3-Track Book Codex (Pseudocode, Executable Python, and Heath Prose).

  • byrne (GitHub): Oliver Byrne’s chromatic First Six Books of the Elements of Euclid (1847) analyzed as visual geometric pseudocode with a Rougeux/G-Index concordance.

  • archimedes (GitHub): Works of Archimedes & The Mechanical Method (Heath 1897/1912)—circle measurement, quadrature of the parabola, spirals, sphere & cylinder, and the Arbelos/Salinon lemmas.

  • apollonius (GitHub): Apollonius of Perga’s Treatise on Conic Sections (Heath 1896, Books I–VII) on cone symptomata, conjugate diameters, poles/polars, normals, and analytical loci.

  • heron (GitHub): Heron of Alexandria’s Metrica & Dioptra (Schöne 1903 Greek + English)—Heron’s Area Formula, iterative root extraction, Platonic/Archimedean volumes, and land-division problems.

  • menelaus (GitHub): Menelaus of Alexandria’s Sphaerica (Books I–III)—spherical congruence without parallels, Menelaus’s planar & spherical transversal theorems, and cross-ratio invariance.

  • ptolemy (GitHub): Claudius Ptolemy’s Almagest (Book I) & Planisphaerium (Heiberg 1898/1907)—single-circle Golden Pentagon–Decagon construction, Ptolemy’s Cyclic Quadrilateral Theorem, base-60 chords, and stereographic projection.

  • pappus (GitHub): Pappus of Alexandria’s Synagoge (Hultsch Greek + English: Three Means, Arbelos Chain, Honeycomb Isoperimetry, Domain of Analysis, Hexagon Theorem) alongside 6 \(S_3\) Steiner lines and Pascal’s 60-line Hexagrammum Mysticum.

  • ceva (GitHub): Giovanni Ceva’s De lineis rectis se invicem secantibus statica constructio (1678 Latin + English)—Archimedean barycentric-lever concurrency, Menelaus duality, \((2n+1)\)-gons, and 3D tetrahedra.

  • feuerbach (GitHub): Leonhard Euler (E325, 1765 Latin) & Karl Wilhelm Feuerbach (Eigenschaften, 1822 German)—the Euler Line, center quadrances, Nine-Point Circle, and Feuerbach’s Tangency Theorem (\(X_{11}\)).

  • staudt (GitHub): Karl Georg Christian von Staudt’s Geometrie der Lage (1847) & Beiträge (1856–60)—metric-free projective geometry, harmonic nets, the Algebra of Throws (Würfe), and the projective regular 17-gon.

  • petersen (GitHub): Julius Petersen’s Methods and Theories for Geometrical Constructions (1879)—intersection of two loci, parallel translation, rotation/spiral similarity, and circular inversion.

  • casey (GitHub): John Casey’s A Sequel to the First Six Books of Euclid (1888)—coaxial circles, inversion, Casey’s Generalized Ptolemy Bitangent Theorem, and Brocard–Neuberg triangle geometry.

  • lemoine (GitHub): Émile Lemoine’s Géométrographie (1902 French + English)—operational Simplicity (\(S\)) and Exactness (\(E\)) of compass-and-straightedge constructions plus Symmedian (\(X_6\)) geometry.

  • klein (GitHub): Felix Klein’s Erlangen Program (1872) & Famous Problems of Elementary Geometry (1895)—transformation groups, quadratic-radical constructibility, Gauss’s 17-gon, and the transcendence of \(e\) and \(\pi\).

  • altshiller-court (GitHub): Nathan Altshiller-Court’s College Geometry—one-item-per-page proposition codex with calibrated and hero SVGs across 10 chapters.

  • fundamentals (GitHub): The Fred W. Seelig Archive—55 foundational topics (F.1–F.55) and 353 manuscript folios uniting triangle synthesis, harmonic duality, Conway notation, and sacred proportions.

  • etc (GitHub): Catalog of Clark Kimberling’s Encyclopedia of Triangle Centers (53,144 centers parsed; 4,000+ centers compiled across 62 collections with 16,000+ interactive SVGs).

  • wildberger (GitHub): Catalog of Norman J. Wildberger’s Divine Proportions—exact SymPy proofs for Quadrance, Spread, pentagonal/golden Spread Polynomials, Triple Quad/Spread/Cross Laws, and Chromogeometry.

III. Philosophical, Renaissance, Harmonic & Axiomatic Scriptorium

  • plato (GitHub): Geometric & Mathematical Dialogues (Burnet Greek + Jowett English)—Meno diagonal square-doubling, Republic Divided Line & Quadrivium, and Timaeus World-Soul ratios & elemental triangles/solids.

  • aristotle (GitHub): Loci Geometrici et Noetici—critical selections from Posterior Analytics, Physics, Metaphysics IX.9, De Anima, and De Caelo in Bekker Greek, English, and geometric commentary.

  • nicomachus (GitHub): Nicomachus of Gerasa’s Introduction to Arithmetic (Hoche 1866 Greek + D’Ooge 1926 English)—the Quadrivium, Three-Term Unfolding from Equality \((1,1,1)\), polygonal gnomons, 10 means, and \(6:8::9:12\) Cube Harmony.

  • proclus (GitHub): Commentary on the First Book of Euclid’s Elements (c. 470 AD)—4-Track Codex (English, Friedlein Greek 1873, Summary, Notes) with a 164-locus concordance on phantasia and probole.

  • fibonacci (GitHub): Leonardo Pisano (Fibonacci)’s Liber Abaci (1202) & Practica Geometriae (1220)—Boncompagni Latin + English on the rabbit recurrence \(F_n\), continued fractions, and golden pentagon/decagon measurement.

  • aquinas (GitHub): Opera de Scientia Mathematica et Materia Intelligibili—4-Track Codex across 37 sections (Super Boetium De Trinitate qq. 5–6, In Post. Anal., In De Anima, In Phys., In Metaph.) with a 78-locus Hoenen concordance.

  • cusa (GitHub): Nicholas of Cusa’s De Docta Ignorantia (On Learned Ignorance, 1440 Latin + English)—the coincidentia oppositorum of infinite line, triangle, circle, and sphere.

  • pacioli (GitHub): Fra Luca Pacioli’s De Divina Proportione (1509, Leonardo da Vinci’s 60 polyhedra woodcuts) & geometric Distinctio of the Summa de Arithmetica (1494).

  • durer (GitHub): Albrecht Dürer’s Underweysung der Messung (Nuremberg, 1525 German + English)—spirals, projective conics, single-compass pentagon, unfolded polyhedral nets, letterforms, and perspective.

  • kepler (GitHub): Johannes Kepler’s Mysterium Cosmographicum (1596), Strena (1611), & Harmonices Mundi (1619)—nested Platonic solids, 6-cornered snowflake & rhombic dodecahedron, Scibilia polygons, Kepler Triangle \((1,\sqrt{\phi},\phi)\), and Harmonic Third Law.

  • hilbert (GitHub): David Hilbert’s Grundlagen der Geometrie / Foundations of Geometry (1899/1902)—German text, Townsend English translation, and metamathematical independence models.

  • poincare (GitHub): Henri Poincaré’s Science and Hypothesis (1902/1905 French + English)—conventionalism, transformation groups, and the conformal Poincaré disk model of hyperbolic geometry.

  • bertrand-russell (GitHub): Bertrand Russell’s An Essay on the Foundations of Geometry (1897) and The Principles of Mathematics Part VI Space (1903) tracing projective vs. metrical foundations.

  • hoenen (GitHub): Facing Latin–English edition of Petrus Hoenen, S.J.’s De Noetica Geometriae (1954) on intuitive formal abstraction and the origin of geometric certainty.

  • russell (GitHub): Digital edition and 3D cube-sphere wave-field reconstruction of Walter & Lao Russell’s 48-lesson Home Study Course (180 vector figures).

IV. Living Geometry & Morphogenesis

  • phyllotaxis (GitHub): Symbolic golden-angle divergence (\(\Phi = 2\pi(1 - 1/\phi) \approx 137.50776^\circ\)), opposing Fibonacci parastichy curves, and live 2D Apical Meristem & 3D Recursive Propagation WebGL laboratories.

  • thompson (GitHub): Sir D’Arcy Wentworth Thompson’s On Growth and Form (1917 Cambridge 1st ed.)—Plateau soap-film angles, Maraldi honeycomb rhombic apex, gnomonic logarithmic shells, and Cartesian transformation grids.

  • knott (GitHub): Catalog of Ron Knott’s Fibonacci Numbers, the Golden Section & Wythoff Geometry—Pisano periods, Zeckendorf representations, golden gnomons, continued fractions, and Wythoff’s Nim.

V. Machine Reasoning & Program Synthesis

  • seer (GitHub): Multi-agent neuro-symbolic architecture (Dreamer + Coder) synthesizing verifiable Python transform() programs for abstract spatial reasoning benchmarks (ARC-AGI).

  • seer_navigator (GitHub) & seer_sessions (GitHub): Textual terminal trajectory browser and static observatory indexing 8,514 task runs across 2,939 puzzles and 475 sessions.

Start Exploring

  1. Walk the Genesis Spine: Open G.17 (Book I, Prop. 1 — Construct an Equilateral Triangle) in elements or compare Oliver Byrne’s chromatic pseudocode in byrne.

  2. Test Invariance Across Four Triangles: Browse X₁ through X₄₀₀₀+ in etc across Acute (5,6,7), Egyptian Right (3,4,5), Kepler Golden (1,√φ,φ), and Kimberling (6,9,13) reference triangles.

  3. Explore Projective, Classical & Rational Canons: Trace Ptolemy’s single-circle pentagon in ptolemy, Pascal’s 60-line Hexagrammum Mysticum in pappus, and algebraic quadrance/spread laws in wildberger.

  4. Grow a Living Meristem in 2D & 3D: Launch the interactive 2D Meristem Studio and 3D Propagation WebGL Lab in phyllotaxis, paired with D’Arcy Thompson’s On Growth and Form.

  5. Watch Visual Broadcasts on YouTube (`@geometor`): Experience 135+ dual-format QHD propositions and vertical Shorts across Euclid’s Elements and the Encyclopedia of Triangle Centers on @geometor, or visit our Broadcasts showcase.

  6. Construct in Exact Symbolic Python:

    from geometor.model import Model

        model = Model("vesica")
        A = model.set_point(0, 0, classes=["given"], ID="A")
        B = model.set_point(1, 0, classes=["given"], ID="B")
        model.construct_line(A, B)
        model.construct_circle(A, B)
        model.construct_circle(B, A)

        for pt in model.points:
            print(f"{model[pt].ID}: ({pt.x}, {pt.y})")

Recent Chronicle

Scholarly Editions & Mathematical Catalogs: The 46-Repository Canon

November 12, 2026 — With release v0.3.0, GEOMETOR completes its Wave 2 expansion to 46 interconnected repositories, establishing a clear architectural complementarity between Scholarly Editions (*Book Codices*) for classical treatises and Mathematical Catalogs for contemporary mathematical corpora.

26.316-143500

The Flowering of the Elementrium & Digital Scriptorium

November 2026 — Across 2025 and 2026, GEOMETOR has grown from a modular Python prototype into a comprehensive symbolic geometry platform, multi-language digital scriptorium, living morphogenesis laboratory, and visual broadcast studio.

26.315-143000

Modularizing the Symbolic Engine

November 2023 — This log entry marks the architectural transition from the original prototype of GEOMETOR Explorer into a modular Python ecosystem built on SymPy.

23.324-124954