Geometric Entity Ontology

FLOW ALGEBRA
TITLE: Geometric Entity Ontology
VERSION: 4.0
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/geometric-entity-ontology.html
SOURCE_ALIGNMENT: Independently constructed; source page not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
DATE: 2026-10-13

01. ONTOLOGY IDENTITY
ontology_id: geometric_entity_ontology
entity_type: MathematicalEntity > GeometricEntity
scope: Euclidean, affine, projective, metric, topological and differential geometry, with explicit context distinctions.

02. ENTITY TYPES
Point: position without intrinsic extent.
Line: one-dimensional straight affine set.
Ray: half-line with endpoint.
Segment: bounded part of a line between endpoints.
Plane: two-dimensional affine flat.
Curve: continuous or differentiable map/image, depending on context.
Surface: two-dimensional geometric object, possibly embedded.
Polygon: closed planar polygonal chain with interior under suitable simplicity assumptions.
Polyhedron: solid or surface bounded by polygonal faces, convention specified.
Circle: planar locus at fixed distance from centre.
Sphere: locus at fixed distance from centre in three-dimensional Euclidean space.
Angle: ordered or unoriented pair of rays with common endpoint, convention specified.
Vector: displacement or tangent element in an associated vector space.
Transformation: map between geometric spaces.
Manifold: locally Euclidean topological space, with additional structure when specified.
Boundary: context-dependent topological or manifold boundary.

03. TYPED ATTRIBUTES
id: unique string
label: string
geometry_context: enum(Euclidean,Affine,Projective,Metric,Topological,Differential)
ambient_space: Space
intrinsic_dimension: nonnegative integer or unknown
embedding_dimension: nonnegative integer or unknown
coordinates: optional ordered tuple
coordinate_system: optional CoordinateSystem
metric: optional Metric
orientation: optional Orientation
units: optional MeasurementUnit
regularity: optional enum(continuous,smooth,piecewise_smooth,singular)
representation: optional enum(implicit,parametric,explicit,combinatorial)
provenance: SourceReference
confidence: number in [0,1]

04. RELATIONSHIP SIGNATURES
is_a(GeometricEntity, EntityType)
embedded_in(GeometricEntity, Space)
subset_of(GeometricEntity, GeometricEntity)
incident_with(GeometricEntity, GeometricEntity)
intersects(GeometricEntity, GeometricEntity)
contains(GeometricEntity, GeometricEntity)
boundary_of(GeometricEntity, GeometricEntity)
parallel_to(Line, Line) [Euclidean/affine context]
perpendicular_to(GeometricEntity, GeometricEntity) [metric context]
congruent_to(GeometricEntity, GeometricEntity) [specified isometry class]
similar_to(GeometricEntity, GeometricEntity) [Euclidean similarity context]
transformed_by(GeometricEntity, Transformation)
has_dimension(GeometricEntity, Dimension)
has_measure(GeometricEntity, Measurement)

05. FORMAL AXIOMS
AX01: Every geometric entity belongs to at least one explicitly specified geometry context.
AX02: Coordinate descriptions are representations, not intrinsic identities.
AX03: Euclidean distance requires a metric; affine incidence does not.
AX04: Dimension must state whether intrinsic, affine, topological or embedding dimension.
AX05: A Euclidean line through distinct points p,q is {p+t(q-p):t in R}.
AX06: A Euclidean segment [p,q] is {p+t(q-p):0<=t<=1}.
AX07: A Euclidean circle C(c,r) in R^2 is {x:||x-c||=r}, r>0.
AX08: A Euclidean sphere S(c,r) in R^3 is {x:||x-c||=r}, r>0.
AX09: Incidence and intersection are distinct: incidence is typed participation; intersection is shared points.
AX10: A geometric object's boundary depends on its ambient topology.
AX11: Congruence requires an appropriate structure-preserving isometry.
AX12: Similarity permits uniform scale with Euclidean rigid motion (including reflections where allowed).
AX13: A transformation must declare its domain, codomain and preserved structures.
AX14: Empty sets and degenerate cases require explicit handling.

06. EXECUTABLE FLOW RULES
R01 IF is_a(x,Point) AND embedded_in(x,S) THEN validate_coordinate_arity(x,S) WHEN coordinates_present.
R02 IF is_a(x,Line) AND Euclidean(S) THEN intrinsic_dimension(x)=1.
R03 IF is_a(x,Plane) AND Euclidean(S) THEN intrinsic_dimension(x)=2.
R04 IF is_a(x,Circle) AND radius(x)>0 THEN intrinsic_dimension(x)=1.
R05 IF is_a(x,Sphere) AND radius(x)>0 THEN intrinsic_dimension(x)=2.
R06 IF is_a(x,Segment) THEN requires(endpoints(x)).
R07 IF is_a(x,Polygon) THEN requires(ordered_vertices(x)).
R08 IF distance(p,q)=0 IN metric_space THEN p=q.
R09 IF perpendicular_to(a,b) THEN requires(metric_context).
R10 IF congruent_to(a,b) THEN equal_corresponding_distances(a,b).
R11 IF similar_to(a,b) THEN proportional_corresponding_lengths(a,b).
R12 IF boundary_of(b,x) THEN b subset_of closure(x) IN same ambient_topology.
R13 IF A subset_of B AND B subset_of C THEN A subset_of C.
R14 IF A intersects B THEN B intersects A.
R15 IF T is bijective_isometry THEN T preserves distances.
R16 IF T is affine_map THEN T preserves collinearity.
R17 IF T is homeomorphism THEN T preserves topological invariants.
R18 IF T is diffeomorphism THEN T preserves differentiable-manifold structure.
R19 IF point p belongs_to line L THEN incident_with(p,L).
R20 IF entities share no points THEN disjoint(x,y).
R21 IF two distinct Euclidean lines in R^2 intersect THEN their intersection is exactly one point.
R22 IF two distinct lines in R^3 are nonparallel THEN do_not_infer_intersection (skew lines possible).
R23 IF circle radius r>0 THEN circumference=2*pi*r AND enclosed_disk_area=pi*r^2.
R24 IF sphere radius r>0 THEN surface_area=4*pi*r^2 AND enclosed_ball_volume=(4/3)*pi*r^3.
R25 IF transformation preserves metric THEN classify_candidate(isometry).
R26 IF affine coordinates change THEN intrinsic_geometry_remains_unchanged.
R27 IF a property depends on units THEN require(unit_consistency).
R28 IF dimension_claim lacks dimension_type THEN flag(ambiguous_dimension).
R29 IF curve is closed THEN do_not_infer(simple).
R30 IF polygon is simple THEN Jordan_boundary_property_applies IN Euclidean_plane.
R31 IF object is a smooth embedded manifold THEN local_tangent_space_defined.
R32 IF entity is finite point set THEN cardinality_is_not_intrinsic_dimension.
R33 IF polygon vertices are collinear THEN flag(degenerate_polygon).
R34 IF line endpoints are identical THEN flag(degenerate_line_definition).
R35 IF transformation lacks codomain THEN flag(incomplete_transformation).
R36 IF statement requires geometry_context AND context_missing THEN status=undetermined.

07. VALIDATION CONSTRAINTS
V01: Coordinates must match the ambient coordinate-system arity.
V02: Length, area and volume must carry dimensionally compatible units.
V03: Radius must be strictly positive for nondegenerate circles and spheres.
V04: A polygon's vertices must form a closed chain; simple polygon requires non-self-intersection.
V05: Do not equate a sphere (surface) with a ball (solid).
V06: Do not equate a circle (boundary) with a disk (filled region).
V07: Never infer orthogonality from incidence alone.
V08: Never assume all manifolds admit a global coordinate chart.
V09: Distinguish metric, affine and projective invariants.
V10: Record degeneracy, uncertainty and missing structure explicitly.

08. SEMANTIC EQUIVALENCE
Euclidean point ~ location (only when 'location' denotes a mathematical position).
Line segment ~ segment (in Euclidean geometry).
Closed ball != sphere.
Disk != circle.
Geometric dimension != physical unit dimension.
Congruence != similarity.
Affine equivalence != isometric equivalence.
Coordinate tuple != geometric entity independently of reference frame.

09. KNOWLEDGE GRAPH TRIPLES
(GeometricEntity, subclassOf, MathematicalEntity)
(Point, subclassOf, GeometricEntity)
(Line, subclassOf, GeometricEntity)
(Plane, subclassOf, GeometricEntity)
(Circle, subclassOf, Curve)
(Sphere, subclassOf, Surface)
(Polygon, subclassOf, GeometricEntity)
(Transformation, actsOn, GeometricEntity)
(Isometry, subclassOf, Transformation)
(CoordinateSystem, represents, GeometricEntity)
(Metric, defines, Distance)
(Boundary, dependsOn, AmbientTopology)

10. WORKED EXAMPLES
E01: p=(0,0), q=(3,4) in R^2; distance(p,q)=5.
E02: Segment from p to q: (3t,4t), 0<=t<=1.
E03: Circle centre (0,0), radius 2: x^2+y^2=4; circumference=4*pi.
E04: Sphere centre origin, radius 3: x^2+y^2+z^2=9; surface area=36*pi.
E05: Lines y=x and y=-x in R^2 intersect at (0,0) and are perpendicular.
E06: Lines {(t,0,0)} and {(0,s,1)} in R^3 are skew.
E07: Triangle vertices (0,0),(2,0),(0,2) has area 2.
E08: Translation T(x,y)=(x+3,y-1) is a Euclidean isometry.
E09: Scaling S(x,y)=(2x,2y) is a similarity but not an isometry.
E10: A circle is one-dimensional intrinsically although embedded in R^2.
E11: Boundary of an open unit disk in R^2 is the unit circle.
E12: The same unit circle considered as a topological space has empty boundary relative to itself.
E13: A square and rectangle may be affine-equivalent without being similar.
E14: An affine map preserves collinearity but need not preserve angles.
E15: A nonzero vector specifies direction; it is not identical to a point without an origin choice.
E16: A torus is locally two-dimensional but cannot be covered by one global Euclidean chart.

11. CONFIDENCE
confidence_scale: 0.0..1.0
axiomatic_mathematical_claim: 1.0 when assumptions verified
proved_derived_claim: 1.0 when proof and context verified
numeric_approximation: attach error bound and numerical confidence
ambiguous_or_missing_context: undetermined, never forced true
source_alignment_confidence: unverified

12. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8; Unicode NFC; LF line endings; stable field order; exclude FLOW_HASH value itself
value: COMPUTE_AT_PUBLICATION
purpose: detect ontology content changes; not a proof of mathematical truth

13. UPDATE_MODEL
version: 4.0
update_trigger: new theorem, corrected axiom, improved terminology, taxonomy revision or source-page change
process: validate -> compare semantic diff -> increment version -> recompute hash -> publish provenance
compatibility: retain entity IDs and record deprecated aliases

14. AI_CONSUMPTION
parse: typed entities -> geometry context -> attributes -> relationships -> rules
reasoning: apply only rules whose preconditions are satisfied
uncertainty: preserve unknowns; avoid inferring geometry from labels alone
output: claim + assumptions + derivation + confidence + provenance
safety: never substitute visual resemblance for formal proof
interoperability: RDF/JSON-LD mappings possible; this TXT is a conceptual flow specification, not an executable interpreter

15. COMPLETION
status: COMPLETE
verification: mathematical consistency reviewed; source-page alignment not verified