What is Poly Geometry?
– A new branch of geometric thought –
Poly Geometry explores how geometric entities arise from structural dependencies — between points, lines, and polynomial curves. It is not a finished theory, but an evolving framework shaped by curiosity, collaboration, and care.
This page outlines the archive’s main divisions, its historical and algebraic foundations, and the contributors who helped shape it. It is both a reference and a reflection.
The project began as a personal endeavor and grew through the generous insights of many, especially members of the QPG Forum.
Navigate the archive via these thematic pathways:
n-Point / n-Line Constructs
Geometric entities defined solely by dependencies on n points or lines — triangles, quadrilaterals, and general n-figures.
Begin here
Polynomial Curves
Investigate how curve properties evolve as the polynomial degree n increases.
Begin here
Contents
The Encyclopedia of Poly Geometry (EPG) is a collaborative archive of geometric entities — points, lines, curves, transformations, and related constructs — defined by their dependencies on n points and/or n lines.
It explores two central questions:
- Which geometric constructs depend solely on n arbitrary points or lines, as n increases?
- What are the properties of polynomial curves, and how do they evolve as the degree n increases?
These questions give rise to two main branches: Configurations of Points and Lines and Polynomial Curves of Degree n. The first explores reference figures such as triangles, quadrilaterals, and general n-figures. The second investigates how curve behavior changes with increasing degree.
Structure of the Archive
The archive is organized into thematic sections, listed below and accessible via the left column on every page.
Figures of n-Points / n-Lines
- Mono-Geometry: Single-point or single-line and their foundational roles.
- Duo-Geometry: Configurations involving two points or lines, often defining axes or directions.
- Triangle Geometry: Classical and extended properties of triangles.
- Perspective Fields: Structural linear relationships between triangle points.
- Quadri-Geometry: Four-point/Four-line configurations and their combinatorial richness.
- QA-items: Quadrangle (Four-point) constructions.
- QL-items: Quadrilateral (Four-line) constructions.
- QG-items: Quadrigon (closed figure of four points/four lines) constructions.
- Penta-Geometry: Five-point/Five-line configurations and their combinatorial richness.
- Hexa-Geometry: Six-point/Six-line configurations and their combinatorial richness.
- Hepta-Geometry: Seven-point/Seven-line configurations and their combinatorial richness.
- Octa-Geometry: Eight-point/Eight-line configurations and their combinatorial richness.
- Nona-Geometry: Nine-point/Nine-line configurations and their combinatorial richness.
- Decaplus-Geometry: Ten+ point/Ten+ line configurations and their combinatorial richness.
- n-Geometry: General constructions for arbitrary n.
Polynomial Curves
- Conics: Degree 2 curves such as ellipses, parabolas, and hyperbolas.
- Cubics: Degree 3 curves with inflection points and complex intersections.
- Higher Degree Curves: Curves of degree ≥4, often arising from n-point dependencies.
History
The origins of this website trace back to 2011, with the introduction of Perspective Fields — a theory that revealed the linear relationships between central points within Triangle Geometry.
In 2012, the Encyclopedia of Quadri-Figures was launched, with substantial support from Eckart Schmidt.
In 2017, the Encyclopedia of Polygon Geometry followed, jointly developed with Eckart Schmidt and Bernard Keizer.
In 2018, a modest section was added for Conics (n-curves of degree 2).
In 2025, these encyclopedias have been unified into the present Encyclopedia of Poly Geometry, which also introduces a substantial new section dedicated to Cubics (n-curves of degree 3), jointly developed with Eckart Schmidt and Bernard Keizer.
Algebraic Foundations and Collaborative Proofs
In the EPG, many entries are supported by algebraic proofs, with calculated coordinates and equations provided wherever possible. Established proofs are cited, and new contributions — whether from forum members or external sources — are incorporated. Items without explicit attribution are assumed to originate from the encyclopedia’s author, who initially contributed the majority of content before others joined.
Unless otherwise indicated, individual findings often trace back to the QFG and QPG forums — longstanding mailing groups that function as informal yet vital spaces for collaborative inquiry. These forums are frequently referenced throughout the encyclopedia, and their activity is documented annually in the QFG Annual Journal and QPG Annual Journal, which preserve key discoveries, discussions, and developments.
Today, the QPG Forum continues to serve as a living environment for geometric exploration. All who share an interest in these topics are warmly invited to participate and contribute.
Acknowledgments
Two individuals merit distinct recognition: Eckart Schmidt (Germany) and Bernard Keizer (France), who year after year expanded the encyclopedia with insight and care. Their enduring commitment — and the collegial bond that grew alongside it — is hereby honored with deep gratitude.
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