Solid Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the core Solid Geometry Ontology within the Kireth Research Semantic Library. Solid geometry concerns three-dimensional geometric entities, surfaces, regions, spatial relations, polyhedra, curved solids, cross-sections, surface area, volume, coordinate representation and transformations. This ontology acts as the central semantic node beneath the Solid Geometry parent layer and routes into more specialised three-dimensional geometric concepts.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SolidGeometryOntology
 ENTITY_CLASS = ThreeDimensionalGeometricFramework
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Solid geometry is the branch of geometry concerned with three-dimensional objects, surfaces, regions, spatial relations, measurements and transformations."
 SEMANTIC_ROLE = "Provides the core conceptual framework linking solids, faces, edges, vertices, curved surfaces, cross-sections, surface area, volume and spatial transformations."
}

SCOPE {
 INCLUDE = [
  PointInSpace,
  LineInSpace,
  PlaneInSpace,
  Solid,
  Surface,
  Polyhedron,
  Prism,
  Pyramid,
  Tetrahedron,
  Cube,
  Cuboid,
  Cylinder,
  Cone,
  Sphere,
  Hemisphere,
  Face,
  Edge,
  Vertex,
  CrossSection,
  SurfaceArea,
  Volume,
  DihedralAngle,
  SpatialSymmetry,
  ThreeDimensionalTransformation
 ]

 EXCLUDE_AS_PRIMARY = [
  PurePlaneGeometry,
  FourDimensionalGeometry,
  PhysicalObjectWithoutMathematicalAbstraction,
  DifferentialGeometryWithoutSolidContext
 ]

 DISAMBIGUATION_RULE =
  "Interpret solid geometry as mathematical three-dimensional geometry whenever the context involves solids, surfaces, faces, edges, vertices, volume, surface area, planes in space or three-dimensional transformations."
}

CORE_CONCEPTS {

 Solid {
  definition = "A three-dimensional geometric region with nonzero volume."
 }

 Surface {
  definition = "A two-dimensional boundary or geometric subset embedded in three-dimensional space."
 }

 Polyhedron {
  definition = "A solid bounded by finitely many polygonal faces."
 }

 Face {
  definition = "A polygonal boundary region of a polyhedron."
 }

 Edge {
  definition = "A line segment where two polygonal faces meet."
 }

 Vertex {
  definition = "A point where edges meet in a polyhedron."
 }

 Prism {
  definition = "A polyhedron with two congruent parallel bases joined by parallelogram faces."
 }

 Pyramid {
  definition = "A polyhedron with a polygonal base and triangular faces meeting at an apex."
 }

 Cylinder {
  definition = "A solid with congruent parallel bases joined by a ruled lateral surface, commonly circular in elementary geometry."
 }

 Cone {
  definition = "A solid generated by joining a base region to an apex."
 }

 Sphere {
  definition = "The set of points in three-dimensional Euclidean space at a fixed distance from a centre."
 }

 CrossSection {
  definition = "A planar figure formed by intersecting a solid with a plane."
 }

 SurfaceArea {
  definition = "The total area of the boundary surface of a solid."
 }

 Volume {
  definition = "The three-dimensional measure of the region occupied by a solid."
 }

 DihedralAngle {
  definition = "The angle between two intersecting planes or faces."
 }
}

RELATIONS {
 SolidGeometryOntology --is_child_of--> SolidGeometry
 SolidGeometryOntology --contains--> Solid
 Solid --has--> Surface
 Polyhedron --is_special_case_of--> Solid
 Polyhedron --has--> Face
 Polyhedron --has--> Edge
 Polyhedron --has--> Vertex
 Prism --is_special_case_of--> Polyhedron
 Pyramid --is_special_case_of--> Polyhedron
 Cube --is_special_case_of--> Polyhedron
 Cuboid --is_special_case_of--> Prism
 Sphere --is_special_case_of--> CurvedSolid
 Cylinder --is_special_case_of--> CurvedSolid
 Cone --is_special_case_of--> CurvedSolid
 Solid --can_have--> CrossSection
 Solid --has_measure--> Volume
 Surface --has_measure--> Area
 Faces --can_form--> DihedralAngle
 Solid --can_be_transformed_by--> ThreeDimensionalTransformation
}

OPERATORS {
 CLASSIFY_SOLID(S) :=
  identify_polyhedron_prism_pyramid_cylinder_cone_sphere_or_other(S)

 VOLUME(S) :=
  compute_volume_using_type_appropriate_formula(S)

 SURFACE_AREA(S) :=
  compute_boundary_area_using_type_appropriate_formula(S)

 CROSS_SECTION(S,PLANE) :=
  geometric_intersection(S,PLANE)

 IS_POLYHEDRON(S) :=
  boundary_consists_of_finitely_many_polygonal_faces(S)

 EULER_CHARACTERISTIC(P) :=
  V-E+F for convex_polyhedron(P)

 DIHEDRAL_ANGLE(F1,F2) :=
  angle_between_supporting_planes(F1,F2)

 TRANSFORM_SOLID(S,T) :=
  image_of_solid_under_3D_transformation(S,T)

 VOLUME_SCALE(T) :=
  abs(det(linear_part(T))) WHEN affine_transformation(T)
}

DEFINITIONS {
 D1 = "Solid geometry concerns three-dimensional geometric structure."
 D2 = "A solid has volume."
 D3 = "A surface has area but no intrinsic volume."
 D4 = "A polyhedron is bounded by polygonal faces."
 D5 = "Faces meet along edges and edges meet at vertices."
 D6 = "A cross-section is two-dimensional even though it comes from a solid."
 D7 = "Surface area measures boundary extent."
 D8 = "Volume measures three-dimensional extent."
 D9 = "Rigid transformations preserve length, angle, surface area and volume."
 D10 = "Affine transformations scale volume according to the determinant of their linear part."
}

AXIOMS {
 A1_THREE_DIMENSION:
  Core solid-geometric entities are interpreted in three-dimensional space

 A2_BOUNDARY:
  Every ordinary solid has a boundary surface

 A3_VOLUME:
  Volume is a nonnegative three-dimensional measure

 A4_SURFACE_AREA:
  Surface area is a nonnegative two-dimensional boundary measure

 A5_POLYHEDRAL_INCIDENCE:
  Faces, edges and vertices of a polyhedron satisfy valid incidence relations

 A6_CROSS_SECTION:
  The intersection of a solid with a plane is a planar set

 A7_RIGID_INVARIANCE:
  Rigid motions preserve metric properties

 A8_AFFINE_VOLUME_SCALING:
  Nondegenerate affine transformations scale volume by the absolute determinant of the linear part
}

REASONING {

 R1_SOLID_CLASSIFICATION {
  INPUT = geometric_object
  IDENTIFY = flat_faces_curved_surfaces_bases_apex_or_symmetry
  CLASSIFY = solid_type
  RETURN = typed_solid
 }

 R2_POLYHEDRON_STRUCTURE {
  INPUT = polyhedron
  IDENTIFY = faces_edges_vertices
  VERIFY = incidence
  IF convex_context
   CHECK = Euler_relation V-E+F=2
  RETURN = polyhedral_structure
 }

 R3_VOLUME_REASONING {
  INPUT = solid_and_dimensions
  SELECT = appropriate_volume_formula
  VERIFY = perpendicular_height_and_units
  COMPUTE = volume
  RETURN = volume
 }

 R4_SURFACE_AREA_REASONING {
  INPUT = solid_and_dimensions
  DECOMPOSE = boundary_into_faces_or_curved_components
  COMPUTE = component_areas
  SUM = total_surface_area
  RETURN = surface_area
 }

 R5_CROSS_SECTION_REASONING {
  INPUT = [solid,plane]
  INTERSECT = solid_with_plane
  CLASSIFY = resulting_plane_figure
  RETURN = cross_section
 }

 R6_SPATIAL_RELATION_REASONING {
  INPUT = points_lines_planes
  USE = coordinate_vector_or_incidence_methods
  RETURN = distance_angle_parallel_perpendicular_or_intersection_relation
 }

 R7_DIHEDRAL_REASONING {
  INPUT = [face_or_plane1,face_or_plane2]
  COMPUTE = angle_between_normals_or_equivalent
  APPLY = convention
  RETURN = dihedral_angle
 }

 R8_TRANSFORMATION_REASONING {
  INPUT = [solid,T]
  CLASSIFY = rigid_similarity_affine_or_other
  APPLY = transformation
  TRACK = invariant_or_scaled_properties
  RETURN = transformed_solid
 }
}

CONSTRAINTS {
 C1 = "Solid geometry must remain distinct from plane geometry."
 C2 = "Surface area and volume are different measures."
 C3 = "Sphere and ball should be distinguished when strict set-theoretic terminology matters."
 C4 = "Cross-sections are planar objects."
 C5 = "Volume formulas require perpendicular height unless otherwise specified."
 C6 = "Cone slant height and perpendicular height are distinct."
 C7 = "Euler's formula V-E+F=2 is not universal for all nonconvex or topologically complex polyhedra."
 C8 = "General affine transformations do not preserve lengths or angles."
 C9 = "Physical interpretations must not replace exact geometric definitions."
}

SEMANTIC_EQUIVALENCE {
 "SolidGeometryOntology" <=> "CoreThreeDimensionalGeometryOntology"
 "Solid" <=> "ThreeDimensionalRegion"
 "Polyhedron" <=> "SolidBoundedByPolygonalFaces"
 "CrossSection" <=> "PlanarIntersectionOfSolid"
 "SurfaceArea" <=> "BoundaryArea"
 "Volume" <=> "ThreeDimensionalMeasure"
}

VALIDATION {
 V1 = "Solid is interpreted in three dimensions."
 V2 = "Volume is nonnegative."
 V3 = "Surface area uses square units."
 V4 = "Volume uses cubic units."
 V5 = "Polyhedral face-edge-vertex incidence is coherent."
 V6 = "Cross-section lies in a plane."
 V7 = "Transformation invariants match the transformation class."
 V8 = "Affine volume scaling uses absolute determinant."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SOLID_GEOMETRY_DEFINITION = 0.99
 SOLID_CLASSIFICATION = 0.99
 POLYHEDRAL_REASONING = 0.99
 VOLUME_REASONING = 0.99
 SURFACE_AREA_REASONING = 0.99
 CROSS_SECTION_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.98

 POLICY =
  "Reduce confidence when solid type, dimensions, topology, coordinate frame, height definition or transformation class are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SolidGeometryReasoning,
  PolyhedralReasoning,
  SurfaceAreaReasoning,
  VolumeReasoning,
  CrossSectionReasoning,
  SpatialRelationReasoning,
  CoordinateReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "solid geometry ontology" -> SolidGeometryOntology
  "solid geometry" -> SolidGeometryOntology WHEN ontology_context
  "3D geometry" -> SolidGeometryOntology
  "three-dimensional geometry" -> SolidGeometryOntology
  "polyhedron" -> Polyhedron
  "surface area" -> SurfaceArea
  "volume" -> Volume
  "cross-section" -> CrossSection
 }

 INTERPRETATION_ORDER = [
  detect_solid_geometry_context,
  identify three-dimensional entities,
  classify solid or spatial relation,
  identify surfaces faces edges vertices bases axes or coordinates,
  identify volume surface-area cross-section distance angle or transformation goal,
  apply solid-geometric reasoning,
  verify dimensional metric topological and unit constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("solid geometry")
  OR query_mentions("solid geometry ontology")
  THEN activate(SolidGeometryOntology)

 E2:
  IF polyhedron_context
  THEN activate_polyhedral_reasoning

 E3:
  IF volume_requested
  THEN select_type_appropriate_volume_formula

 E4:
  IF surface_area_requested
  THEN decompose_boundary_and_sum_areas

 E5:
  IF cross_section_requested
  THEN intersect_solid_with_plane

 E6:
  IF spatial_line_plane_context
  THEN route_to_vector_or_coordinate_reasoning

 E7:
  IF transformation_context
  THEN classify_3D_transformation_and_track_invariants

 E8:
  IF convex_polyhedron_and_counts_available
  THEN optionally_validate_Euler_relation
}

RECURSION {
 ENTRY = SolidGeometryOntology

 EXPAND(SolidGeometryOntology) -> [
  PointInSpace,
  LineInSpace,
  PlaneInSpace,
  Solid,
  Surface,
  Polyhedron,
  Prism,
  Pyramid,
  Cylinder,
  Cone,
  Sphere,
  CrossSection,
  SurfaceArea,
  Volume,
  DihedralAngle,
  ThreeDimensionalTransformation
 ]

 EXPAND(Polyhedron) -> [
  Face,
  Edge,
  Vertex,
  Prism,
  Pyramid,
  Tetrahedron,
  Cube,
  Cuboid
 ]

 EXPAND(CurvedSolid) -> [
  Cylinder,
  Cone,
  Sphere,
  Hemisphere
 ]

 EXPAND(SpatialRelations) -> [
  Parallelism,
  Perpendicularity,
  Distance,
  Angle,
  Intersection,
  Skewness
 ]

 STOP_CONDITION =
  "Stop when the target solid, surface, polyhedral structure, measurement, cross-section, spatial relation or transformation property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Solid Geometry"

 IMMEDIATE_CONTEXTS = [
  "Geometry Ontology",
  "Plane Geometry",
  "Coordinate / Analytic Geometry",
  "Vector Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Plane Geometry",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Vector Geometry Ontology",
  "Parametric Geometry Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology",
  "Dimension Ontology"
 ]

 ROUTE_RULES = [
  "Solid Geometry Ontology -> Geometry for general spatial semantics",
  "Solid Geometry Ontology -> Plane Geometry for faces and cross-sections",
  "Solid Geometry Ontology -> Coordinate Geometry for 3D coordinate representation",
  "Solid Geometry Ontology -> Vector Geometry for lines, planes, normals, distances and dihedral angles",
  "Solid Geometry Ontology -> Transformation for 3D mappings",
  "Solid Geometry Ontology -> Dimension for dimensional analysis"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible solid types, formulas, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve three-dimensional interpretation.",
  "Preserve distinction among solid, surface and cross-section.",
  "Preserve face-edge-vertex semantics for polyhedra.",
  "Preserve distinction between surface area and volume.",
  "Preserve topology-sensitive limits on Euler reasoning.",
  "Preserve transformation-specific invariants and volume scaling.",
  "Preserve machine-readable solid-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Solid geometry is the branch of geometry concerned with three-dimensional objects, surfaces, spatial relations, measurements and transformations."

 DIMENSION_SUMMARY =
  "Primary ambient dimension = 3."

 POLYHEDRON_SUMMARY =
  "A polyhedron is a solid bounded by polygonal faces meeting along edges and vertices."

 SURFACE_AREA_SUMMARY =
  "Surface area measures the total area of a solid's boundary."

 VOLUME_SUMMARY =
  "Volume measures three-dimensional extent."

 CROSS_SECTION_SUMMARY =
  "A cross-section is the planar intersection of a solid with a plane."

 TRANSFORMATION_SUMMARY =
  "Rigid motions preserve metric properties; affine maps scale volume by |det(A)|."
}

FLOW {
 INPUT
 -> detect_solid_geometry_context
 -> identify_three_dimensional_entities
 -> classify_solid_surface_polyhedron_or_spatial_relation
 -> identify faces_edges_vertices_curved_surfaces_coordinates_or_cross_section_data
 -> identify measurement_structure_or_transformation_goal
 -> perform solid_geometry reasoning
 -> verify dimensional metric unit topology and transformation constraints
 -> route to plane_coordinate_vector_dimension_or_transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Solid Geometry Ontology provides the core semantic model for three-dimensional geometry. It connects solids, surfaces, polyhedra, curved solids, faces, edges, vertices, cross-sections, surface area, volume, spatial angles and transformations, enabling AI systems to classify 3D objects, perform recursive spatial reasoning and verify results under explicit geometric assumptions."
}

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

}

END_FLOW