Ordered geometry¶
Form of geometry without distances.
Core Idea¶
Ordered geometry is an axiomatic geometry organized around points and the ternary relation of betweenness rather than distance, angle measure, or coordinates. A statement [ABC] says that B lies between A and C. From this primitive relation one defines open segments, closed intervals, rays, lines, triangles, planes, and higher-dimensional incidence structures. Axioms enforce symmetry in the endpoints, exclude degenerate orderings, extend lines, provide points off a line, and regulate how lines entering a triangle must leave it, notably through Pasch-type conditions.
Scope of Application¶
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Axiomatic geometry. Symmetry, nondegeneracy, incidence, extension, and Pasch-type rules define admissible models.
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Foundations of affine and Euclidean geometry. Order-only results are isolated before parallel or metric axioms are added.
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Segments, rays, and lines. Linear figures are reconstructed from betweenness without coordinates.
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Plane separation and triangles. Crossing and sidedness arise through incidence and order principles.
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Model construction. Structures show consistency and variation among axiom systems.
Clarity¶
Ordered geometry builds geometric structure from incidence and the ternary relation ‘between’ rather than from distance, angle, coordinates, or metric congruence. Betweenness axioms create local line order, segments, rays, separation, and Pasch-type constraints; continuity and dimension require additional axioms. The term does not imply that all geometric magnitudes are already available.
Manages Complexity¶
Ordered geometry compresses geometric reasoning to points, incidence, and betweenness, with continuity and dimension added only when needed. Segments, rays, lines, separation, triangles, and convexity can be built from the ternary order relation without coordinates or metric magnitude. Linear, planar, higher-dimensional, Paschian, and complete branches reflect axiom choices. This organization reveals which theorems depend only on order and which require distance, angle, or congruence.
Abstract Reasoning¶
Betweenness move. Take a ternary relation stating that one point lies between two others and derive segments, rays, lines, separation, and convexity. Axiom move. Use endpoint symmetry, extension, nondegeneracy, incidence, and Pasch-type conditions to test candidate models. Order move. Recover a linear ordering along each line without assigning distances or angles. Independence move. Determine which familiar geometric theorems follow from order alone and which need congruence, parallels, coordinates, or continuity. Boundary move.
Knowledge Transfer¶
Within the home domain. Ordered geometry transfers across axiomatic geometry, foundations, model theory, and the study of affine, Euclidean, and hyperbolic systems as geometry built from betweenness and incidence before metric or congruence. Point, ternary order, segment, line, Pasch condition, dimension, and continuity retain formal roles. Beyond the home domain (C — axiomatic framework). It applies literally to models satisfying the axioms. Its boundary is expressive: it supplies no length, angle, perpendicularity, or distance by itself, and ordinary ranking or left–right layout is not ordered geometry. Projective models may lack compatible global betweenness.
Relationships to Other Abstractions¶
Current abstraction Ordered geometry Domain-specific
Parents (1) — more general patterns this builds on
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Ordered geometry is a kind of Theory Prime
Ordered geometry is a domain-specific kind of Theory: Form of geometry without distances.
Children (1) — more specific cases that build on this
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Neutral Geometry Domain-specific is a kind of Ordered geometry
Neutral geometry preserves ordered-geometry betweenness and adds congruence while withholding a parallel axiom.
Hierarchy paths (2) — routes to 2 parentless roots
- Ordered geometry → Theory → Formalization → Representation → Abstraction
- Ordered geometry → Theory → Formalization → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Ordered geometry sits in a sparse region of the domain-specific corpus (60th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Synthetic geometry — 0.85
- Neutral Geometry — 0.85
- Complete variety — 0.84
- Nine-Point Conic — 0.84
- Planar ternary ring — 0.84
Computed from structural-signature embeddings · 2026-10-08