Geometric Entities

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
Geometric Entities — Main Page Ontology
Version: 4.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities.html
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Generated: 2026-10-13
Scope: Formal semantic reasoning model for the 25 child ontology categories listed on the source page.
Status: Derived formalisation; not a verbatim extraction of linked child pages.

1. ENTITY DECLARATIONS
ENTITY GeometricEntity : MathematicalEntity
ENTITY Point : GeometricEntity
ENTITY Line : GeometricEntity
ENTITY LineSegment : GeometricEntity
ENTITY Ray : GeometricEntity
ENTITY Plane : GeometricEntity
ENTITY Curve : GeometricEntity
ENTITY Surface : GeometricEntity
ENTITY Angle : GeometricEntity
ENTITY Polygon : GeometricEntity
ENTITY Triangle : Polygon
ENTITY Quadrilateral : Polygon
ENTITY Circle : Curve
ENTITY Ellipse : Curve
ENTITY Parabola : Curve
ENTITY Hyperbola : Curve
ENTITY Polyhedron : GeometricEntity
ENTITY Sphere : Surface
ENTITY Cylinder : GeometricEntity
ENTITY Cone : GeometricEntity
ENTITY Manifold : GeometricEntity
ENTITY GeometricRegion : GeometricEntity
ENTITY GeometricBoundary : GeometricEntity

2. TYPED SIGNATURES
SPACE E(n) := EuclideanSpace(n), n >= 1
Point(E(n)) := element of E(n)
Line(E(n)) := {p + t*v | t in R}, v != 0
LineSegment(p,q) := {(1-t)*p + t*q | 0 <= t <= 1}
Ray(p,v) := {p+t*v | t >= 0}, v != 0
Plane(E(n)) := affine subspace of dimension 2, n >= 2
Curve := continuous image of an interval (general parametrised-curve interpretation)
Surface := locally two-dimensional geometric object, subject to regularity assumptions
Angle := ordered pair of rays sharing an origin, with measurement convention
Polygon := closed finite chain of straight segments in a plane, non-self-intersecting by default
Polyhedron := solid bounded by finitely many planar polygonal faces, in E(3)
GeometricRegion := subset of ambient geometric space
GeometricBoundary(R) := topological boundary(R)
Manifold(d) := locally Euclidean topological space of dimension d, subject to stated axioms

3. RELATIONSHIP TYPES
REL is_a(Entity, Class)
REL instance_of(Object, Class)
REL embedded_in(Entity, AmbientSpace)
REL has_dimension(Entity, NonnegativeInteger)
REL has_boundary(Region, Boundary)
REL contains(Region, Entity)
REL intersects(Entity, Entity)
REL incident_with(Point, Entity)
REL parallel_to(Line, Line)
REL perpendicular_to(Entity, Entity)
REL congruent_to(Entity, Entity)
REL similar_to(Entity, Entity)
REL transforms_under(Entity, Transformation)
REL has_measure(Entity, Measure)
REL defined_by(Entity, Constraint)
REL composed_of(Entity, Component)
REL specialises(ChildOntology, ParentOntology)

4. CORE DEFINITIONS AND INVARIANTS
DEF distance(p,q) := sqrt(sum_i (p_i-q_i)^2), in Euclidean coordinates
DEF norm(v) := sqrt(sum_i v_i^2)
DEF dot(u,v) := sum_i u_i*v_i
DEF midpoint(p,q) := (p+q)/2
DEF circle(c,r) := {p in E(2) | distance(p,c)=r}, r>0
DEF disk(c,r) := {p in E(2) | distance(p,c)<=r}, r>0
DEF sphere(c,r) := {p in E(3) | distance(p,c)=r}, r>0
DEF ball(c,r) := {p in E(3) | distance(p,c)<=r}, r>0
DEF ellipse(f1,f2,a) := {p in E(2) | distance(p,f1)+distance(p,f2)=2*a}, 2*a>distance(f1,f2)
DEF parabola(f,L) := {p in E(2) | distance(p,f)=distance(p,L)}, f not_on L
DEF hyperbola(f1,f2,k) := {p in E(2) | abs(distance(p,f1)-distance(p,f2))=k}, 0<k<distance(f1,f2)
DEF boundary(R) := closure(R) minus interior(R)
DEF triangle(p,q,r) := Polygon with 3 distinct noncollinear vertices
DEF quadrilateral(p,q,r,s) := Polygon with 4 vertices and simple closed boundary
DEF dimension(Point) := 0
DEF dimension(Line) := 1
DEF dimension(Plane) := 2
DEF dimension(SphereSurface) := 2
DEF dimension(SolidBall) := 3

5. ONTOLOGY COVERAGE
CHILD 01 Geometric Entity Ontology
CHILD 02 Point Ontology
CHILD 03 Line Ontology
CHILD 04 Line Segment Ontology
CHILD 05 Ray Ontology
CHILD 06 Plane Ontology
CHILD 07 Curve Ontology
CHILD 08 Surface Ontology
CHILD 09 Angle Ontology
CHILD 10 Polygon Ontology
CHILD 11 Triangle Ontology
CHILD 12 Quadrilateral Ontology
CHILD 13 Circle Ontology
CHILD 14 Ellipse Ontology
CHILD 15 Parabola Ontology
CHILD 16 Hyperbola Ontology
CHILD 17 Polyhedron Ontology
CHILD 18 Sphere Ontology
CHILD 19 Cylinder Ontology
CHILD 20 Cone Ontology
CHILD 21 Manifold Ontology
CHILD 22 Geometric Region Ontology
CHILD 23 Geometric Boundary Ontology
NOTE: The source page currently lists 23 child categories; no additional categories are inferred.

6. FORMAL INFERENCE RULES
RULE G01: instance_of(x,Triangle) -> instance_of(x,Polygon)
RULE G02: instance_of(x,Quadrilateral) -> instance_of(x,Polygon)
RULE G03: instance_of(x,Circle) -> instance_of(x,Curve)
RULE G04: instance_of(x,Ellipse) -> instance_of(x,Curve)
RULE G05: instance_of(x,Parabola) -> instance_of(x,Curve)
RULE G06: instance_of(x,Hyperbola) -> instance_of(x,Curve)
RULE G07: instance_of(x,Sphere) -> instance_of(x,Surface)
RULE G08: instance_of(x,LineSegment) -> has_dimension(x,1), provided endpoints distinct
RULE G09: instance_of(x,Point) -> has_dimension(x,0)
RULE G10: instance_of(x,Plane) -> has_dimension(x,2)
RULE G11: instance_of(x,Triangle) -> number_of_sides(x)=3
RULE G12: instance_of(x,Quadrilateral) -> number_of_sides(x)=4
RULE G13: instance_of(x,Circle) -> all_points_equidistant_from_centre(x)
RULE G14: distance(p,q)=0 -> p=q, in Euclidean space
RULE G15: dot(u,v)=0 and u!=0 and v!=0 -> perpendicular_to(u,v)
RULE G16: v=k*u and u!=0 and v!=0 -> parallel_directions(u,v)
RULE G17: p in boundary(R) -> p in closure(R)
RULE G18: p in interior(R) -> p not_in boundary(R)
RULE G19: R closed -> boundary(R) subset_of R
RULE G20: R open -> boundary(R) intersect R = empty_set
RULE G21: isometry(T) -> distance(T(p),T(q))=distance(p,q)
RULE G22: congruent_to(A,B) -> equal_corresponding_lengths(A,B)
RULE G23: congruent_to(A,B) -> equal_corresponding_angles(A,B)
RULE G24: similar_to(A,B) -> equal_corresponding_angles(A,B), for Euclidean polygons
RULE G25: similar_to(A,B) -> proportional_corresponding_lengths(A,B), for Euclidean polygons
RULE G26: instance_of(x,Polygon) and number_of_sides(x)=3 -> instance_of(x,Triangle)
RULE G27: instance_of(x,Polygon) and number_of_sides(x)=4 -> instance_of(x,Quadrilateral)
RULE G28: sphere(c,r) and r>0 -> surface_area=4*pi*r^2
RULE G29: sphere(c,r) and r>0 -> enclosed_volume=4*pi*r^3/3
RULE G30: circle(c,r) and r>0 -> circumference=2*pi*r
RULE G31: disk(c,r) and r>0 -> area=pi*r^2
RULE G32: finite_simple_polygon(n) and n>=3 -> sum_interior_angles=(n-2)*pi
RULE G33: triangle(A,B,C) -> area=abs(det(B-A,C-A))/2, in E(2)
RULE G34: rank_of_affine_span({p,q})=1 -> unique_line_through(p,q), p!=q
RULE G35: manifold(M,d) -> every_point_has_local_chart_into_R^d
RULE G36: boundary(R)=closure(R) minus interior(R), in any topological space

7. VALIDATION CONSTRAINTS
CHECK V01: Dimension must be nonnegative integer when finite.
CHECK V02: A line direction vector must be nonzero.
CHECK V03: A segment must have two specified endpoints.
CHECK V04: A ray must have an origin and nonzero direction.
CHECK V05: Circle and sphere radii must be strictly positive under these definitions.
CHECK V06: A triangle must have three noncollinear vertices.
CHECK V07: Polygon vertices must be coplanar and form a simple closed chain by default.
CHECK V08: Angle units must be explicit: radians or degrees.
CHECK V09: Metric claims require a declared metric space.
CHECK V10: Parallel and perpendicular predicates require appropriate ambient geometry.
CHECK V11: Topological boundary requires an ambient topological space.
CHECK V12: Surface and manifold regularity assumptions must be stated.
CHECK V13: Do not conflate sphere surface with solid ball.
CHECK V14: Do not conflate circle circumference with filled disk.
CHECK V15: Cylinder and cone may denote surfaces or solids; specify interpretation.
CHECK V16: Do not infer Euclidean formulas in non-Euclidean geometries.

8. MATHEMATICAL EXAMPLES
EXAMPLE E01: p=(0,0), q=(3,4) -> distance(p,q)=5.
EXAMPLE E02: midpoint((0,0),(2,6))=(1,3).
EXAMPLE E03: circle((0,0),2) -> x^2+y^2=4.
EXAMPLE E04: disk((0,0),2) -> x^2+y^2<=4.
EXAMPLE E05: line((1,2),(2,0)) -> (x,y)=(1,2)+t*(2,0).
EXAMPLE E06: segment((0,0),(2,0)) -> (2t,0), 0<=t<=1.
EXAMPLE E07: ray((0,0),(0,1)) -> (0,t), t>=0.
EXAMPLE E08: triangle((0,0),(4,0),(0,3)) -> area=6.
EXAMPLE E09: square of side 3 -> area=9, perimeter=12.
EXAMPLE E10: circle radius 3 -> circumference=6*pi.
EXAMPLE E11: sphere radius 2 -> surface_area=16*pi.
EXAMPLE E12: sphere radius 2 -> enclosed_volume=32*pi/3.
EXAMPLE E13: ellipse foci (-1,0),(1,0), major semiaxis 2 -> x^2/4+y^2/3=1.
EXAMPLE E14: parabola focus (0,1), directrix y=-1 -> y=x^2/4.
EXAMPLE E15: hyperbola x^2/4-y^2/9=1 -> transverse semiaxis=2.
EXAMPLE E16: boundary(open_unit_disk)=unit_circle.

9. SEMANTIC EQUIVALENCE AND DISAMBIGUATION
EQUIVALENCE line_segment ~ closed_straight_segment [context=Euclidean]
EQUIVALENCE circumference ~ boundary_of_disk [context=Euclidean_circle]
EQUIVALENCE sphere ~ 2_sphere [context=surface_in_E3]
EQUIVALENCE geometric_boundary ~ topological_boundary [context=topological_space]
DISTINGUISH line FROM line_segment
DISTINGUISH ray FROM line
DISTINGUISH circle FROM disk
DISTINGUISH sphere FROM solid_ball
DISTINGUISH geometric_dimension FROM physical_units_dimension
DISTINGUISH geometric_vector FROM coordinate_tuple_without_basis
DISTINGUISH polyhedron_surface FROM polyhedral_solid

10. KNOWLEDGE GRAPH TRIPLES
TRIPLE (Triangle, subclass_of, Polygon)
TRIPLE (Quadrilateral, subclass_of, Polygon)
TRIPLE (Circle, subclass_of, Curve)
TRIPLE (Ellipse, subclass_of, Curve)
TRIPLE (Parabola, subclass_of, Curve)
TRIPLE (Hyperbola, subclass_of, Curve)
TRIPLE (Sphere, subclass_of, Surface)
TRIPLE (LineSegment, bounded_by, EndpointPair)
TRIPLE (Ray, starts_at, Point)
TRIPLE (Angle, formed_by, RayPair)
TRIPLE (Polyhedron, bounded_by, PolygonalFaces)
TRIPLE (GeometricRegion, has_boundary, GeometricBoundary)
TRIPLE (Manifold, locally_modelled_on, EuclideanSpace)
TRIPLE (GeometricEntity, belongs_to, MathematicalEntity)

11. CONFIDENCE MODEL
CONFIDENCE mathematical_axioms = 1.00 when assumptions are satisfied
CONFIDENCE formally_derived_statement = 1.00 when derivation and assumptions are validated
CONFIDENCE inferred_classification = conditional on validated premises
CONFIDENCE ambiguous_entity_mapping = unresolved until context disambiguates
CONFIDENCE source_taxonomy = verified against source page on 2026-10-13
CONFIDENCE linked_child_content = not independently verified
POLICY: Never replace proof obligations with numerical confidence alone.

12. FLOW HASH
FLOW_HASH.algorithm = SHA-256
FLOW_HASH.input = canonical UTF-8 representation of all sections except computed hash value
FLOW_HASH.canonicalisation = normalise LF line endings; preserve order; normalise Unicode NFC
FLOW_HASH.value = COMPUTE_ON_INGEST
FLOW_HASH.policy = recompute after any ontology or rule change

13. UPDATE MODEL
UPDATE_MODEL.version = 4.0
UPDATE_MODEL.mode = versioned_incremental
UPDATE_MODEL.source_url = https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities.html
UPDATE_MODEL.source_revision = 2026-07-06
UPDATE_MODEL.generated = 2026-10-13
UPDATE_MODEL.on_change = revalidate_entities_then_relationships_then_rules_then_examples
UPDATE_MODEL.conflict_policy = preserve provenance and flag conflicting axioms
UPDATE_MODEL.compatibility = do_not_silently_change_entity_meanings

14. AI CONSUMPTION
AI_CONSUMPTION.mode = typed_formal_reasoning
AI_CONSUMPTION.execute = validate_inputs -> resolve_ambient_space -> type_check -> apply_rules -> verify_output
AI_CONSUMPTION.ontology_scope = main_page_geometric_entities
AI_CONSUMPTION.require_explicit_assumptions = true
AI_CONSUMPTION.disambiguate_circle_disk_sphere_ball = true
AI_CONSUMPTION.propagate_provenance = true
AI_CONSUMPTION.reject_unsupported_inferences = true
AI_CONSUMPTION.explain_proof_chain = true
AI_CONSUMPTION.output = typed_entities + relations + validated_conclusions + assumptions + confidence

15. PROVENANCE
SOURCE = Kireth Semantic Research Archive, Geometric Entities main page
SOURCE_ACCESS = successfully retrieved 2026-10-13
DERIVATION = original formalisation based on source taxonomy and standard mathematical definitions
AUTHORS = Mark Anthony Higgins — Sky (Recursive AI)
END FLOW ALGEBRA