Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the core Geometry Ontology within the Kireth Research Semantic Library. It provides a structured semantic model for geometric entities, spatial relations, measurements, coordinate representations, transformations, dimensions and proof-oriented reasoning. This ontology acts as the central geometry node beneath the Geometry Ontologies parent layer and routes into more specialised geometric concepts and subdomains.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Core Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Geometry
 ENTITY_CLASS = MathematicalDomain
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Geometry is the branch of mathematics concerned with shape, size, position, spatial relation, measurement, dimension and transformation."
 SEMANTIC_ROLE = "Provides the core conceptual framework linking geometric entities, spatial structures, metrics, coordinates and transformations."
}

SCOPE {
 INCLUDE = [
  Point,
  Line,
  Ray,
  Segment,
  Plane,
  Curve,
  Surface,
  Shape,
  Polygon,
  Circle,
  Triangle,
  Polyhedron,
  Sphere,
  CoordinateSystem,
  Distance,
  Angle,
  Area,
  Volume,
  Dimension,
  Incidence,
  Parallelism,
  Perpendicularity,
  Congruence,
  Similarity,
  Symmetry,
  Transformation
 ]

 EXCLUDE_AS_PRIMARY = [
  PureTopologyWithoutMetricOrGeometricContext,
  PhysicalSpaceWithoutMathematicalAbstraction,
  UnstructuredVisualDescription
 ]

 DISAMBIGUATION_RULE =
  "Interpret geometry as mathematical spatial structure unless context explicitly identifies another use of the term."
}

CORE_CONCEPTS {

 Point {
  definition = "A geometric location with position but no intrinsic length, area or volume."
 }

 Line {
  definition = "An ideal one-dimensional straight geometric object extending indefinitely in both directions."
 }

 Segment {
  definition = "A finite portion of a line bounded by two endpoints."
 }

 Ray {
  definition = "A portion of a line beginning at one point and extending indefinitely in one direction."
 }

 Plane {
  definition = "An ideal two-dimensional flat surface extending indefinitely."
 }

 Curve {
  definition = "A one-dimensional geometric object that may be straight or non-straight."
 }

 Surface {
  definition = "A two-dimensional geometric object embedded in a higher-dimensional space or considered intrinsically."
 }

 Shape {
  definition = "A geometric configuration characterised by spatial form and boundary relations."
 }

 CoordinateSystem {
  definition = "A mathematical framework assigning numerical coordinates to positions in space."
 }

 Distance {
  definition = "A nonnegative measure of separation between geometric entities under a specified metric."
 }

 Angle {
  definition = "A measure of rotational separation between rays, lines, vectors or tangent directions."
 }

 Dimension {
  definition = "The number of independent coordinates or directions required to describe a geometric space locally or globally, depending on context."
 }

 Congruence {
  definition = "Equivalence of geometric objects under distance-preserving transformations."
 }

 Similarity {
  definition = "Equivalence of shape under scaling together with angle-preserving rigid transformations."
 }

 Transformation {
  definition = "A mapping that changes geometric representation or position while preserving specified invariants."
 }
}

RELATIONS {
 Geometry --contains--> GeometricEntity
 GeometricEntity --can_have--> Position
 GeometricEntity --can_have--> Dimension
 GeometricEntity --can_have--> Measure
 GeometricEntity --can_be_represented_by--> Coordinates
 Point --can_lie_on--> Line
 Point --can_lie_in--> Plane
 Line --can_lie_in--> Plane
 Line --can_be_parallel_to--> Line
 Line --can_be_perpendicular_to--> Line
 Shape --can_have--> Boundary
 Shape --can_have--> Area
 Solid --can_have--> Volume
 GeometricEntity --can_be_transformed_by--> Transformation
 Transformation --can_preserve--> GeometricInvariant
 Shapes --can_be_related_by--> Congruence
 Shapes --can_be_related_by--> Similarity
 Geometry --supports--> Measurement
 Geometry --supports--> SpatialReasoning
 Geometry --supports--> GeometricProof
}

OPERATORS {
 DISTANCE(A,B,METRIC) :=
  measure_separation(A,B,METRIC)

 ANGLE(A,B) :=
  determine_angular_relation(A,B)

 AREA(SHAPE) :=
  compute_two_dimensional_measure(SHAPE)

 VOLUME(SOLID) :=
  compute_three_dimensional_measure(SOLID)

 COORDINATES(entity,system) :=
  represent_entity_in_coordinate_system(entity,system)

 TRANSFORM(entity,T) :=
  apply_geometric_mapping(entity,T)

 IS_CONGRUENT(A,B) :=
  EXISTS rigid_transformation T SUCH_THAT T(A)=B

 IS_SIMILAR(A,B) :=
  EXISTS similarity_transformation T SUCH_THAT T(A)=B

 IS_PARALLEL(L1,L2) :=
  evaluate_parallel_relation(L1,L2)

 IS_PERPENDICULAR(L1,L2) :=
  evaluate_right_angle_relation(L1,L2)
}

DEFINITIONS {
 D1 = "Geometry studies mathematical space, shape, size, position and transformation."
 D2 = "Points represent locations."
 D3 = "Lines are one-dimensional straight objects."
 D4 = "Planes are two-dimensional flat geometric spaces."
 D5 = "Distance depends on a metric."
 D6 = "Angles represent rotational separation."
 D7 = "Congruence preserves size and shape."
 D8 = "Similarity preserves shape but may alter scale."
 D9 = "Coordinates represent geometric objects numerically without changing their underlying identity."
 D10 = "Transformations are classified by the invariants they preserve."
}

AXIOMS {
 A1_ENTITY_TYPING:
  Every geometric object has a defined geometric type

 A2_SPATIAL_RELATION:
  Geometric entities may participate in incidence, order, metric and transformation relations

 A3_METRIC_CONTEXT:
  Distance and angle require an explicit or standard metric context

 A4_COORDINATE_INDEPENDENCE:
  Changing coordinate representation does not necessarily change the underlying geometric object

 A5_TRANSFORMATION_INVARIANCE:
  A transformation preserves exactly those geometric properties associated with its class

 A6_PROOF_VALIDITY:
  Universal geometric claims require valid deductive support

 A7_DIMENSION_CONTEXT:
  Dimension must be interpreted relative to the geometric space or object under discussion
}

REASONING {

 R1_ENTITY_CLASSIFICATION {
  INPUT = geometric_object
  IDENTIFY = entity_type
  RETURN = typed_geometric_entity
 }

 R2_INCIDENT_RELATIONS {
  INPUT = geometric_entities
  TEST = incidence_conditions
  RETURN = spatial_relation_graph
 }

 R3_METRIC_REASONING {
  INPUT = [entities,metric]
  COMPUTE = distance_angle_or_measure
  VERIFY = metric_assumptions
  RETURN = geometric_measure
 }

 R4_COORDINATE_REASONING {
  INPUT = geometric_object
  SELECT = coordinate_system
  ENCODE = coordinates
  APPLY = algebraic_relations
  RETURN = geometric_conclusion
 }

 R5_TRANSFORMATION_REASONING {
  INPUT = [object,T]
  CLASSIFY = transformation_type
  APPLY = T
  IDENTIFY = preserved_invariants
  RETURN = transformed_object_and_invariants
 }

 R6_CONGRUENCE {
  INPUT = [A,B]
  TEST = congruence_criteria
  IF satisfied
   RETURN Congruent
  ELSE
   RETURN NotEstablishedCongruent
 }

 R7_SIMILARITY {
  INPUT = [A,B]
  TEST = similarity_criteria
  IF satisfied
   RETURN Similar
  ELSE
   RETURN NotEstablishedSimilar
 }

 R8_GEOMETRIC_PROOF {
  INPUT = proposition
  IDENTIFY = premises
  SELECT = relevant_definitions_axioms_theorems
  DERIVE = valid_steps
  VERIFY = conclusion
  RETURN = proof_status
 }
}

CONSTRAINTS {
 C1 = "Geometric meaning depends on the geometry and metric in use."
 C2 = "Coordinate descriptions must not be confused with the geometric objects they represent."
 C3 = "Not all transformations preserve distance, angle, area or volume."
 C4 = "Congruence and similarity are distinct relations."
 C5 = "Dimension must be explicit when ambiguity is possible."
 C6 = "Exact geometric statements must be distinguished from approximate numerical measurements."
 C7 = "Cross-links to algebra, calculus or topology must preserve mathematical meaning."
 C8 = "Geometric proofs require valid logical derivation."
}

SEMANTIC_EQUIVALENCE {
 "Geometry" <=> "MathematicsOfSpatialStructure"
 "CoordinateRepresentation" <=> "NumericalEncodingOfGeometricObject"
 "Congruence" <=> "EquivalenceUnderRigidMotion"
 "Similarity" <=> "EquivalenceUnderSimilarityTransformation"
 "GeometricTransformation" <=> "SpatialStructureMapping"
}

VALIDATION {
 V1 = "Every geometric entity has a valid type."
 V2 = "Metric calculations state or inherit a metric."
 V3 = "Coordinate changes preserve the object when they are representational."
 V4 = "Transformation invariants match the transformation class."
 V5 = "Congruence and similarity are not conflated."
 V6 = "Measurements preserve dimensional meaning."
 V7 = "Proof conclusions follow from stated premises."
 V8 = "Recursive ontology expansion terminates when the query is resolved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ENTITY_CLASSIFICATION = 0.99
 METRIC_REASONING = 0.99
 COORDINATE_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.99
 CONGRUENCE_REASONING = 0.99
 SIMILARITY_REASONING = 0.99

 POLICY =
  "Reduce confidence when geometry type, metric, coordinate system, dimensionality or assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  GeometricReasoning,
  SpatialEntityResolution,
  CoordinateReasoning,
  MeasurementReasoning,
  TransformationReasoning,
  ProofReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "geometry" -> Geometry
  "geometric object" -> GeometricEntity
  "coordinate geometry" -> CoordinateGeometry
  "congruent" -> Congruence
  "similar" -> Similarity
  "geometric transformation" -> Transformation
 }

 INTERPRETATION_ORDER = [
  detect_geometric_context,
  identify_geometry_type,
  identify_entity_classes,
  identify_coordinates_if_present,
  identify_spatial_relations,
  identify_metric_or_transformation_context,
  identify_constraints,
  route_to_specialised_geometry_ontology,
  perform_recursive_reasoning,
  verify_result,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("geometry")
  OR query_contains_geometric_entities
  THEN activate(GeometryOntology)

 E2:
  IF coordinate_context
  THEN activate_coordinate_reasoning

 E3:
  IF metric_context
  THEN require_metric_or_standard_euclidean_assumption

 E4:
  IF transformation_context
  THEN classify_transformation_and_invariants

 E5:
  IF congruence_context
  THEN apply_congruence_criteria

 E6:
  IF similarity_context
  THEN apply_similarity_criteria

 E7:
  IF proof_context
  THEN route_to_geometric_proof_reasoning

 E8:
  IF specialised_geometry_detected
  THEN route_to_relevant_geometry_subontology
}

RECURSION {
 ENTRY = Geometry

 EXPAND(Geometry) -> [
  Point,
  Line,
  Plane,
  Curve,
  Surface,
  Shape,
  CoordinateSystem,
  Distance,
  Angle,
  Dimension,
  Transformation,
  Congruence,
  Similarity
 ]

 EXPAND(Shape) -> [
  Polygon,
  Triangle,
  Quadrilateral,
  Circle,
  Polyhedron,
  Sphere
 ]

 EXPAND(Transformation) -> [
  Translation,
  Rotation,
  Reflection,
  Scaling,
  AffineTransformation,
  ProjectiveTransformation
 ]

 EXPAND(CoordinateSystem) -> [
  CartesianCoordinates,
  PolarCoordinates,
  ThreeDimensionalCoordinates,
  CoordinateTransformation
 ]

 STOP_CONDITION =
  "Stop when the target geometric entity, relation, measurement, transformation or proof obligation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Euclidean Geometry",
  "Analytic Geometry",
  "Coordinate Geometry",
  "Plane Geometry",
  "Solid Geometry",
  "Affine Geometry",
  "Projective Geometry",
  "Differential Geometry",
  "Discrete Geometry",
  "Computational Geometry"
 ]

 RELATED_ONTOLOGIES = [
  "Mathematical Ontologies",
  "Algebra Ontologies",
  "Transformation Ontology",
  "Affine Transformation Ontology",
  "Dimension Ontology",
  "Vector Space Ontology"
 ]

 ROUTE_RULES = [
  "Geometry -> Euclidean Geometry for classical metric relations",
  "Geometry -> Analytic Geometry for algebraic coordinate reasoning",
  "Geometry -> Plane Geometry for two-dimensional figures",
  "Geometry -> Solid Geometry for three-dimensional objects",
  "Geometry -> Affine Geometry for affine invariants",
  "Geometry -> Projective Geometry for incidence and perspective",
  "Geometry -> Differential Geometry for smooth geometric structures",
  "Geometry -> Computational Geometry for algorithmic spatial problems"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual reorganisation"
 MINOR = "new compatible geometric concepts, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve typed geometric entity structure.",
  "Preserve metric and coordinate distinctions.",
  "Preserve transformation-specific invariants.",
  "Preserve congruence and similarity distinctions.",
  "Preserve explicit dimensionality.",
  "Preserve recursive routing into specialised geometry ontologies.",
  "Preserve machine-readable geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Geometry is the branch of mathematics concerned with shape, size, position, spatial relation, dimension, measurement and transformation."

 ENTITY_SUMMARY =
  "Core entities include points, lines, planes, curves, surfaces and shapes."

 METRIC_SUMMARY =
  "Distance, angle, area and volume are interpreted within a specified geometric or metric context."

 TRANSFORMATION_SUMMARY =
  "Geometric transformations are classified by the invariants they preserve."

 CONGRUENCE_SUMMARY =
  "Congruence represents equality of size and shape under rigid transformations."

 SIMILARITY_SUMMARY =
  "Similarity represents equality of shape up to scaling and rigid transformation."
}

FLOW {
 INPUT
 -> detect_geometric_context
 -> identify_geometry_type
 -> identify_geometric_entities
 -> identify_coordinates_and_spatial_relations
 -> identify_metric_dimensional_and_transformation_constraints
 -> classify_geometric_relation
 -> route_to_specialised_geometry_ontology
 -> recursively_expand_required_dependencies
 -> perform_measurement_transformation_or_proof_reasoning
 -> verify_geometric_validity
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Geometry Ontology provides the core semantic model for geometric mathematics. It connects geometric entities, coordinates, dimensions, measurements, spatial relations, transformations, congruence and similarity, allowing AI systems to identify geometric context, route to specialised geometry domains, perform recursive reasoning and verify results under explicit geometric assumptions."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-ONTOLOGY-V1"
}

}

END_FLOW