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GEOMETOR

exploring the architecture of all that is
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GEOMETOR — From Two Points through the Vesica Piscis to the Golden Ratio

GEOMETOR

Exploring the architecture of all that is

geometor.com · YouTube (@geometor) · Discussions · phi ARCHITECT

One Language of Form

Geometry is humanity's most concise language for documenting nature, proportion, and deductive truth. When living systems unfold spiral phyllotaxis or crystal lattices from simple local laws, they invite a foundational question: How do elementary axiomatic relationships unfold into the full complexity of form and mind?

Most dynamic geometry software relies on floating-point approximations—reporting that two segments are merely numerically close to $1.6180339\dots$ At GEOMETOR, every construction begins with two points in an empty field and is grounded in exact symbolic algebra (SymPy). Every point, line, circle, quadrance, spread, cross-ratio, and intersection preserves its exact algebraic identity ($\sqrt{2}$, $\sqrt{3}$, exact polynomial roots, and the Divine Proportion):

$$\phi = \frac{1 + \sqrt{5}}{2}$$

Across 46 interconnected repositories, we recast 2,400 years of classical treatises, projective and rational geometry, botanical morphogenesis, epistemological philosophy, and machine reasoning into one consistent, executable symbolic grammar—uniting two complementary forms of scholarship:

  • Scholarly Editions (Book Codices): Multi-language critical editions of foundational public-domain works (Greek, Latin, Italian, German, French, and English) paired with calibrated interactive .euclid-graphic SVGs and runnable geometor.model demonstrations.
  • Mathematical Catalogs: Original one-item-per-page symbolic catalogs documenting the theorems, constructions, triangle centers, rational trigonometry laws, and geometric software engines of modern researchers (etc, wildberger, divine, knott, codes) in a unified constructive framework with rigorous scholarly attribution.
flowchart LR
    Field["Two Starting Points: A, B"] --> Core["I. Symbolic Core & Catalogs (5)"]
    Core --> Canon["II. The Elementrium & Constructive Canon (20)"]
    Core --> Scriptorium["III. Philosophical, Renaissance & Axiomatic Scriptorium (15)"]
    Core --> Nature["IV. Living Geometry & Morphogenesis (3)"]
    Core --> Mind["V. Machine Reasoning & Program Synthesis (3)"]
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I. The Symbolic Core, Discovery & Software Catalog

Project Role in the Architecture Live Docs & Packages
model Exact Symbolic Construction Engine — Compass-and-straightedge algebra in SymPy: exact algebraic coordinates, closed-form 2D intersections, rational trigonometry invariants (quadrance, spread, cross_ratio), @proposition deductive ledgers, and ancestral DAGs. Docs · PyPI
explorer Interactive SVG Workbench — Web canvas & symbolic CLI ([ A B ], ( A B ), / A B /, < A B C >) with scrubbable step-by-step timelines, KaTeX hover inspection, and publication sheet export. Docs
divine Golden Ratio Discovery & Cut-the-Knot Construction Catalog — Automated scanner uncovering exact golden sections ($\phi$), golden chains, and harmonic ranges, paired with a catalog of classical and modern golden constructions (Odom, Hofstetter, Bogomolny). Docs & Catalog · PyPI
codes Catalog of Geometric Computing & Theorem-Proving Engines — Structured architectural catalog of open-source deductive and constructive engines (GeoCoq, JGEX / Area & Full-Angle Methods, GeoGebra Discovery, SymPy Geometry, CGAL) with cross-paradigm comparisons. Browse Catalog →
trees Monotone Planar Tree Drawings — Crossing-free straight-line tree layout algorithms grounded in Stern–Brocot mediant trees and Farey rational sequences. Repo

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

Recasting synthetic, projective, spherical, and algebraic canons into synchronized Book Codices and Mathematical Catalogs powered by photon-platform/publish, with bidirectional hover highlighting between text and vector geometry:

Codex / Catalog Scope & Contents Read Online
euclid Euclid's Elements (Books I–XIII) & Data — Sir Thomas L. Heath's complete 3-volume critical Elements (1,416 folios, 465 propositions, 831 calibrated SVGs) plus Euclid's Data (Δεδομένα, 94 propositions of geometric analysis in Heiberg Greek & Menge/Heath English). Read Codex → · Data →
elements Elements 2.0 & The G-Index (G.1–G.601) — Topological dependency DAG sequencing Euclid's definitions, postulates, common notions, and propositions into a 3-Track Codex (Pseudocode, Executable Python, and Heath Prose) with step playback. Explore G-Index → · Videos
byrne Oliver Byrne's First Six Books of the Elements of Euclid (1847) — Pickering's Bauhaus-precursor chromatic edition analyzed as visual geometric pseudocode, pairing red/yellow/blue/black inline glyphs with exact geometor.model proofs and a Rougeux/G-Index concordance. Read Codex →
archimedes Archimedes — Works & The Method (Heath 1897/1912) — Measurement of a Circle ($3\frac{10}{71} &lt; \pi &lt; 3\frac{1}{7}$), Quadrature of the Parabola, On Spirals, On the Sphere and Cylinder, Book of Lemmas (Arbelos & Salinon), and The Mechanical Method. Read Codex →
apollonius Apollonius of Perga — Treatise on Conic Sections (Heath 1896) — Books I–VII on symptomata of the cone, conjugate diameters, harmonic poles and polars, normals/evolutes, minimum/maximum lines, and Pappus reconstructions of the lost analytical loci. Read Codex →
heron Heron of Alexandria — Metrica & Dioptra (Schöne 1903) — 4-Track Greek/English Codex of Books I–III: Heron's Area Formula ($\Delta = \sqrt{s(s-a)(s-b)(s-c)}$), Babylonian-Heronian square-root iteration, regular polygon ratios, Platonic/Archimedean volumes, and proportional division of figures. Read Codex →
menelaus Menelaus of Alexandria — Sphaerica (Books I–III) — Spherical triangles without planar parallels (AAA congruence), Menelaus's Planar & Spherical Transversal Theorems (6-segment ratio product $= -1$), harmonic cross-ratio invariance, and spherical astronomy. Read Codex →
ptolemy Claudius Ptolemy — Almagest (Book I) & Planisphaerium (Heiberg 1898/1907) — Ptolemy's single-circle Golden Pentagon–Decagon construction, Ptolemy's Cyclic Quadrilateral Theorem, exact base-60 chord calculus, spherical Menelaus, and stereographic projection of the astrolabe. Read Codex →
pappus Pappus of Alexandria — Synagoge (Collection) & Hexagrammum Mysticum — Hultsch Greek/English Codex of Books III, IV, V, VII, VIII (Three Means, Generalized Pythagoras, Arbelos Chain, Honeycomb Isoperimetry, Domain of Analysis, Hexagon Theorem) + 6 $S_3$ Steiner lines & Pascal's 60 lines. Read Codex →
ceva Giovanni Ceva — De lineis rectis se invicem secantibus statica constructio (1678) — 4-Track Latin/English Codex of Ceva's Archimedean barycentric-lever method uniting triangle concurrency ($X_1, X_2, X_4, X_6, X_7, X_8$), Menelaus transversals, $(2n+1)$-gons, and 3D tetrahedra. Read Codex →
feuerbach Leonhard Euler (E325, 1765) & Karl Wilhelm Feuerbach (Eigenschaften, 1822) — Bilingual Latin/German/English Codex of the Euler Line ($X_2, X_3, X_4, X_5$), pairwise center quadrances ($OI^2 = R^2 - 2Rr$), the Nine-Point Circle, and Feuerbach's Tangency Theorem ($X_{11}$). Read Codex →
staudt Karl Georg Christian von Staudt — Geometrie der Lage (1847) & Beiträge (1856–60) — Bilingual German/English Codex founding metric-free projective geometry: harmonic nets of rationality, the Algebra of Throws (Würfe), polarities, and von Staudt's projective regular 17-gon. Read Codex →
petersen Julius Petersen — Methods and Theories for Geometrical Constructions (1879) — Problem-solving heuristics of classical synthesis: intersection of two loci (Apollonius's Problem), parallel translation, rotation/spiral similarity, homothety, and circular inversion. Read Codex →
casey John Casey — A Sequel to the First Six Books of Euclid (1888) — Classical bridge to modern circle and triangle geometry: radical axes, coaxial circles, inversion, Casey's Generalized Ptolemy Bitangent Theorem (proving Feuerbach's Theorem), and Lemoine–Brocard–Neuberg synthesis. Read Codex →
lemoine Émile Lemoine — Géométrographie, ou Art des constructions géométriques (1902) — Bilingual French/English Codex of Lemoine's quantitative complexity calculus ($R_1, R_2, C_1, C_2, C_3 \to$ Simplicity $S$, Exactness $E$) benchmarking Euclidean vs. minimal constructions + the Symmedian Point ($X_6$). Read Codex →
klein Felix Klein — Erlangen Program (1872) & Famous Problems of Elementary Geometry (1895) — Dual German/English Codex uniting geometry by transformation groups and proving the algebraic criteria for quadratic-tower constructibility, Gauss's 17-gon, and the transcendence of $e$ and $\pi$. Read Codex →
altshiller-court Nathan Altshiller-Court's College Geometry (1925/1952) — One-item-per-page proposition codex with exact symbolic .euclid-graphic hero figures across 10 chapters (Loci, Similitude, Nine-Point & Feuerbach, Simson, Transversals, Harmonic Division, Inversion, Recent Triangle Geometry). Read Codex →
fundamentals The Fred W. Seelig Constructive Archive — 55 topics (F.1–F.55), 353 manuscript folios, and 59 spreadsheets uniting classical triangle synthesis, harmonic duality, Conway notation, and circle–square proportions in a 3-Track Codex. Read Codex →
etc Catalog — Encyclopedia of Triangle Centers ($X_1$–$X_{53144}$) — Symbolic catalog of Clark Kimberling's ETC: 4,000+ compiled centers across 62 collections and 16,000+ interactive SVGs on Acute (5,6,7), Egyptian (3,4,5), Kepler (1,√φ,φ), and Kimberling (6,9,13) triangles. Explore Catalog → · Videos
wildberger Catalog — Norman J. Wildberger's Divine Proportions & Rational Trigonometry — Symbolic catalog of exact SymPy proofs and geometor.model constructions for Quadrance, Spread, pentagonal/golden Spread Polynomials, Triple Quad/Spread/Cross Laws, and Chromogeometry. Explore Catalog →

III. Philosophical, Renaissance, Harmonic & Axiomatic Scriptorium

Edition Treatise & Philosophical Significance Read Online
plato Plato — Geometric & Mathematical Dialogues (Burnet Greek / Jowett English) — 4-Track Codex of Meno ($82\text{b}$–$85\text{e}$ diagonal square-doubling), Republic VI–VIII (Divided Line, Quadrivium, Nuptial Number), and Timaeus ($31\text{b}$–$57\text{d}$ World-Soul harmonic scale & elemental triangles/solids). Read Codex →
aristotle Aristotle — Loci Geometrici et Noetici — Foundational passages from Posterior Analytics, Physics, Metaphysics IX.9, De Anima, and De Caelo in Bekker Greek, English, and geometric commentary. Read Codex →
nicomachus Nicomachus of Gerasa — Introduction to Arithmetic (Hoche 1866 / D'Ooge 1926) — 4-Track Greek/English Codex of the Four Mathemata (Quadrivium), Nicomachus's Three-Term Unfolding of All Ratios from Equality $(1,1,1)$, polygonal gnomons, 10 means, and the $6:8::9:12$ Cube Harmony. Read Codex →
proclus Proclus — Commentary on Book I of Euclid's Elements (c. 470 AD) — 4-Track Codex (English, Friedlein Greek 1873, Summary, Notes) with a 164-locus concordance on the Eudemian history, phantasia, and geometric projection (probole). Read Codex →
fibonacci Leonardo Pisano (Fibonacci) — Liber Abaci (1202) & Practica Geometriae (1220) — Boncompagni Latin + Original English Codex of Hindu–Arabic numerals, the rabbit recurrence $F_n$, continued-fraction proportions, and golden pentagon/decagon/icosahedron measurement. Read Codex →
aquinas Thomas Aquinas — Scientia Mathematica & 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. Read Codex →
cusa Nicholas of Cusa — De Docta Ignorantia (On Learned Ignorance, 1440) — 4-Track Latin/English Codex of Book I on the Coincidence of Opposites (coincidentia oppositorum)—where infinite circle, infinite straight tangent, infinite triangle, and infinite sphere coincide. Read Codex →
pacioli Fra Luca Pacioli — De Divina Proportione (1509) & Summa de Arithmetica (1494) — Multi-track codices spanning Leonardo da Vinci's 60 polyhedra woodcuts, the 13 effects of $\phi$, architectural letterforms, and the geometric Distinctio of the 1494 Summa. Read Codex → · Summa →
durer Albrecht Dürer — Underweysung der Messung (Nuremberg, 1525) — 4-Track Early New High German/English Codex of Books I–IV: Archimedean/shell spirals, conic sections by parallel projection, single-compass pentagon, unfolded polyhedral nets, letterforms, and perspective machines. Read Codex →
kepler Johannes Kepler — Mysterium Cosmographicum (1596), Strena (1611) & Harmonices Mundi (1619) — 4-Track Latin/English Codex of the 5 nested Platonic solids, 6-cornered snowflake & rhombic dodecahedron, Scibilia polygons, Kepler Triangle $(1,\sqrt{\phi},\phi)$, and Harmonic Third Law. Read Codex →
hilbert David Hilbert's Grundlagen der Geometrie (1899/1902) — German original, Townsend English translation, and metamathematical commentary on incidence, order, congruence, segment algebra, and axiom independence. Read Codex →
poincare Henri Poincaré — Science and Hypothesis (1902/1905) — Bilingual French/English Codex of Part II (Space): geometric conventionalism, transformation-group genesis of space, and Poincaré's conformal disk dictionary proving the relative consistency of hyperbolic geometry. Read Codex →
bertrand-russell Bertrand Russell — Foundations of Geometry (1897) & Principles Part VI (1903) — Critical editions tracing projective order, Cayley–Klein metrical geometry, and relational space. Read Codex →
hoenen Petrus Hoenen, S.J.'s De Noetica Geometriae (Rome, 1954) — Facing Latin–English edition investigating intuitive formal abstraction and how universal geometric necessity arises within concrete construction. Read Codex →
russell Walter & Lao Russell's Home Study Course (1950–1952) — 48-lesson digital edition with 180 vector-reconstructed figures and interactive 3D cube-sphere wave-field geometry. Read Codex →

IV. Living Geometry, Morphogenesis & Machine Reasoning

Project Focus Live Studios, Editions & Catalogs
phyllotaxis Botanical Morphogenesis & Golden Divergence — Symbolic golden-angle divergence ($\Phi = 2\pi/\phi^2 \approx 137.50776^\circ$), opposing Fibonacci parastichy curves, and live 2D Canvas + 3D Three.js WebGL meristem propagation studios. Codex & Labs → · 2D · 3D
thompson Sir D'Arcy Wentworth Thompson — On Growth and Form (1917) — Book Codex of the Cambridge first edition with exact SymPy reconstructions: Plateau soap-film angles ($120^\circ$, $\arccos(-\frac{1}{3})$), Maraldi honeycomb rhombic apex, gnomonic logarithmic shells, and Cartesian transformation grids. Read Codex →
knott Catalog — Ron Knott's Fibonacci Numbers, the Golden Section & Wythoff Geometry — One-item-per-page symbolic catalog documenting Pisano periods, Zeckendorf representations, golden triangles/gnomons, continued fractions, and Wythoff's Nim ($\lfloor n\phi \rfloor, \lfloor n\phi^2 \rfloor$). Explore Catalog →
seer Program Synthesis for Abstract Visual Reasoning — Multi-agent (Dreamer + Coder) neuro-symbolic solver synthesizing verifiable Python transform() programs for ARC-AGI challenges. Docs
seer_navigator Reasoning Trajectory TUI — Textual terminal browser for inspecting solver hypotheses, synthesized Python iterations, and grid diffs. Docs
seer_sessions Longitudinal Reasoning Observatory — Empirical archive and static web viewer spanning 8,514 task runs, 2,939 unique puzzles, and 475 sessions. Browse Observatory →

V. Watch & Connect

Led by phi ARCHITECT · Published with PHOTON publish · MIT Licensed

Pinned Loading

  1. explorer explorer Public

    An interactive interface for visualizing and analyzing geometric models.

    JavaScript 1

  2. model model Public

    A symbolic engine for defining and verifying geometric constructions.

    Python 1

  3. divine divine Public

    An analysis engine for identifying golden sections and harmonic ranges.

    Python

  4. elements elements Public

    A digital reconstruction of Euclid's Elements for dependency mapping and analysis.

    Python

  5. pacioli pacioli Public

    A system for tracking and analyzing the works of Fra Luca Pacioli.

    Python

  6. etc etc Public

    An index and toolkit for the Encyclopedia of Triangle Centers.

    HTML

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