GEOMETOR

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?

Every repository in GEOMETOR lays a stone in that architecture—recasting 2,300 years of classical treatises, projective and rational geometry, natural morphogenesis, epistemological philosophy, and machine reasoning into one consistent, exact symbolic grammar so we can witness the whole edifice unfold step by step.

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:

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 Aristotle, Proclus, Thomas Aquinas (materia intelligibilis), 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.
The Elementrium & Multi-Track Book Codices: Across millennia, geometry fractured into incompatible idioms—Greek synthetic prose, Renaissance perspective woodcuts, projective cross-ratios, homogeneous trilinears, and imperative code. Powered by photon-platform/publish, GEOMETOR unites classical sources, symbolic pseudocode, interactive SVGs, and executable Python side by side.

The Five Pillars of Exploration¶

I. The Symbolic Core & Discovery Engine¶

model (GitHub · PyPI): Exact SymPy compass-and-straightedge construction engine with rational trigonometry invariants (quadrance, spread, cross_ratio) and ancestral DAGs.
explorer (GitHub): Interactive SVG workbench, symbolic CLI, scrubbable construction timeline, and publication sheet exporter.
divine (GitHub · PyPI): Automated discovery engine scanning constructed models to uncover golden sections, golden chains, and harmonic ranges.
trees: Monotone planar tree drawing algorithms grounded in Stern–Brocot trees and Farey rational sequences.

II. The Unfolding Canon (The Elementrium)¶

euclid (GitHub): Complete 13-Book critical edition of Sir Thomas L. Heath’s Euclid’s Elements—465 propositions, 1,416 archival folios, and 831 recalibrated SVGs in a synchronized 2-Track Book Codex.
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).
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.
altshiller-court (GitHub): Nathan Altshiller-Court’s College Geometry (1952)—334 folios and 699 articles bridging Euclid to modern triangle and circle synthesis (Method of Loci, Similitude, Nine-Point & Feuerbach, Simson, Ceva/Menelaus, Harmonic Division, Inversion, Lemoine & Brocard).
etc (GitHub): Semantic transformation of Clark Kimberling’s Encyclopedia of Triangle Centers (53,144 centers parsed; 4,000+ centers compiled across 62 collections with 16,000+ interactive SVGs across Acute, Egyptian Right, Kepler Golden, and Kimberling triangles).
pappus (GitHub): Exact SymPy projective geometry—all six \(S_3\) Pappus hexagon configurations, Steiner star points, Pascal’s Hexagrammum Mysticum (60 Pascal lines, 20 Steiner points, 60 Kirkman points, 15 Plücker lines), straightedge harmonic fourths, and Golden Conics over \(\mathbb{Q}(\phi)\).
wildberger (GitHub): Constructive companion to Norman J. Wildberger’s Divine Proportions: Rational Trigonometry to Universal Geometry—exact SymPy proofs for Quadrance, Spread, pentagonal/golden Spread Polynomials, Triple Quad/Spread/Cross Laws, and Chromogeometry.

III. Historical, Epistemological & Axiomatic Scriptorium¶

aristotle (GitHub): Loci Geometrici et Noetici—critical selections from Posterior Analytics, Physics, Metaphysics, De Anima, and De Caelo in Bekker Greek, English, and geometric commentary.
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 the Eudemian history, phantasia, and geometric projection (probole).
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.
pacioli (GitHub): 5-Track Book Codex of Fra Luca Pacioli’s De Divina Proportione (Venice, 1509 / Vienna, 1889)—318 folios and 482 woodcuts including Leonardo da Vinci’s 60 polyhedra plates.
hilbert (GitHub): David Hilbert’s Grundlagen der Geometrie / Foundations of Geometry (1899/1902)—German text, Townsend English translation, and metamathematical independence models.
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.

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¶

Walk the Genesis Spine: Open G.17 (Book I, Prop. 1 — Construct an Equilateral Triangle) in elements and step through the interactive construction, Python script, and Heath commentary.
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.
Explore Projective & Rational Geometry: Trace Pascal’s 60-line Hexagrammum Mysticum in pappus and algebraic quadrance/spread laws in wildberger.
Grow a Living Meristem in 2D & 3D: Launch the interactive 2D Meristem Studio and 3D Propagation WebGL Lab in phyllotaxis.
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.

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})")
        