Solid Geometry

AI INTERPRETATION NOTE
This Flow Algebra defines the Solid Geometry layer within the Kireth Research Semantic Library. Solid geometry studies three-dimensional geometric entities and relations, including points, lines and planes in space, polyhedra, prisms, pyramids, cylinders, cones, spheres, surface area, volume, cross-sections, spatial angles, symmetry, transformations and coordinate representation. This record is designed for semantic retrieval, recursive geometric reasoning, ontology routing, proof support, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SolidGeometry
 ENTITY_CLASS = ThreeDimensionalGeometricDomain
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Solid geometry is the study of three-dimensional geometric objects, spatial relations, surfaces, volumes and transformations."
 SEMANTIC_ROLE = "Provides the core three-dimensional framework for polyhedra, curved solids, surface area, volume, cross-sections and spatial reasoning."
}

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

 EXCLUDE_AS_PRIMARY = [
  PurePlaneGeometry,
  FourDimensionalGeometry,
  DifferentialGeometryWithoutSolidContext,
  PhysicalObjectsWithoutMathematicalAbstraction
 ]

 DISAMBIGUATION_RULE =
  "When geometry concerns three-dimensional solids, planes in space, polyhedra, curved solids, surface area or volume, interpret the context as solid geometry."
}

CORE_CONCEPTS {

 SolidGeometry {
  definition = "Geometry of objects and relations in three-dimensional space."
  ambient_dimension = 3
 }

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

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

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

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

 Tetrahedron {
  definition = "A polyhedron with four triangular faces."
 }

 Cube {
  definition = "A regular hexahedron with six congruent square faces."
 }

 Cuboid {
  definition = "A rectangular prism with six rectangular faces."
 }

 Cylinder {
  definition = "A solid generated by translating a plane region along a direction, with circular cylinders as the common Euclidean case."
 }

 Cone {
  definition = "A solid formed by joining points of a base to a common apex."
 }

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

 CrossSection {
  definition = "The plane 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, commonly measured between their normal vectors or corresponding half-planes."
 }
}

RELATIONS {
 SolidGeometry --is_subdomain_of--> Geometry
 SolidGeometry --contains--> Solid
 Solid --can_be--> Polyhedron
 Solid --can_have--> Surface
 Polyhedron --has--> Face
 Polyhedron --has--> Edge
 Polyhedron --has--> Vertex
 Prism --is_special_case_of--> Polyhedron
 Pyramid --is_special_case_of--> Polyhedron
 Tetrahedron --is_special_case_of--> Pyramid
 Cube --is_special_case_of--> Polyhedron
 Cuboid --is_special_case_of--> Prism
 Sphere --has--> Centre
 Sphere --has--> Radius
 Solid --can_have--> CrossSection
 Solid --has_measure--> Volume
 Solid --has_boundary_measure--> SurfaceArea
 Planes --can_form--> DihedralAngle
 Solid --can_be_transformed_by--> ThreeDimensionalTransformation
}

OPERATORS {
 VOLUME_PRISM(base_area,height) :=
  base_area*height

 VOLUME_PYRAMID(base_area,height) :=
  (1/3)*base_area*height

 VOLUME_CYLINDER(r,h) :=
  pi*r^2*h

 VOLUME_CONE(r,h) :=
  (1/3)*pi*r^2*h

 VOLUME_SPHERE(r) :=
  (4/3)*pi*r^3

 SURFACE_AREA_SPHERE(r) :=
  4*pi*r^2

 SURFACE_AREA_CYLINDER(r,h) :=
  2*pi*r*h + 2*pi*r^2

 SURFACE_AREA_CONE(r,l) :=
  pi*r*l + pi*r^2

 CUBE_VOLUME(a) :=
  a^3

 CUBE_SURFACE_AREA(a) :=
  6*a^2

 CUBOID_VOLUME(l,w,h) :=
  l*w*h

 CROSS_SECTION(solid,plane) :=
  geometric_intersection(solid,plane)

 DIHEDRAL_ANGLE(plane1,plane2) :=
  angle_between_planes(plane1,plane2)

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

DEFINITIONS {
 D1 = "Solid geometry is three-dimensional."
 D2 = "Volume measures three-dimensional extent."
 D3 = "Surface area measures boundary area."
 D4 = "A polyhedron is bounded by polygonal faces."
 D5 = "A prism has two congruent parallel bases."
 D6 = "A pyramid has a polygonal base and a single apex."
 D7 = "A sphere is a constant-radius locus in three dimensions."
 D8 = "A cross-section is a plane intersection with a solid."
 D9 = "A dihedral angle measures the angle between planes."
 D10 = "Three-dimensional transformations may preserve or alter metric properties according to transformation class."
}

AXIOMS {
 A1_THREE_DIMENSIONALITY:
  Primary solid-geometric objects are embedded in three-dimensional space

 A2_VOLUME_NONNEGATIVITY:
  Volume is nonnegative

 A3_SURFACE_BOUNDARY:
  The surface of a solid forms its geometric boundary

 A4_POLYHEDRAL_STRUCTURE:
  A polyhedron is composed of faces meeting along edges and vertices

 A5_CROSS_SECTION:
  Intersection of a solid with a plane yields a planar set

 A6_METRIC_CONTEXT:
  Length, area, volume and angle measurements depend on the active metric

 A7_TRANSFORMATION:
  Rigid motions preserve lengths, angles, surface areas and volumes

 A8_AFFINE_VOLUME:
  General affine transformations scale volume by the absolute determinant of the linear part
}

REASONING {

 R1_SOLID_CLASSIFICATION {
  INPUT = geometric_object
  IDENTIFY = faces_curvature_bases_apex_or_symmetry
  CLASSIFY = prism_pyramid_polyhedron_cylinder_cone_sphere_or_other
  RETURN = solid_type
 }

 R2_VOLUME {
  INPUT = solid_type_and_dimensions
  SELECT = valid_volume_formula
  VERIFY = unit_consistency
  COMPUTE = volume
  RETURN = volume_value
 }

 R3_SURFACE_AREA {
  INPUT = solid_type_and_dimensions
  DECOMPOSE = boundary_surfaces
  COMPUTE = component_areas
  SUM = areas
  RETURN = surface_area
 }

 R4_POLYHEDRON {
  INPUT = polyhedron
  IDENTIFY = faces_edges_vertices
  APPLY = structural_or_Euler_reasoning_when_appropriate
  RETURN = polyhedral_properties
 }

 R5_CROSS_SECTION {
  INPUT = [solid,plane]
  COMPUTE = geometric_intersection
  CLASSIFY = resulting_plane_figure
  RETURN = cross_section
 }

 R6_SPATIAL_DISTANCE {
  INPUT = points_lines_planes
  SELECT = vector_coordinate_or_projection_method
  COMPUTE = shortest_distance
  RETURN = distance
 }

 R7_DIHEDRAL_ANGLE {
  INPUT = [plane1,plane2]
  IDENTIFY = normal_vectors
  COMPUTE = angle_between_normals_with_plane_angle_convention
  RETURN = dihedral_angle
 }

 R8_SPHERE {
  INPUT = sphere_data
  IDENTIFY = centre_radius
  COMPUTE = surface_area_volume_or_intersection
  RETURN = sphere_result
 }

 R9_TRANSFORMATION {
  INPUT = [solid,T]
  CLASSIFY = rigid_similarity_affine_or_other
  APPLY = transformation
  TRACK = preserved_or_scaled_invariants
  RETURN = transformed_solid
 }
}

CONSTRAINTS {
 C1 = "Solid geometry must remain distinct from plane geometry."
 C2 = "Volume and surface area must not be conflated."
 C3 = "Cross-sections are two-dimensional even though they arise from three-dimensional solids."
 C4 = "Volume formulas require compatible dimensions and units."
 C5 = "Cone slant height and vertical height are distinct."
 C6 = "Sphere refers to the surface locus; ball refers to the enclosed solid region in strict mathematical usage."
 C7 = "General affine transformations do not preserve lengths or angles."
 C8 = "Dihedral-angle conventions must be explicit when orientation matters."
 C9 = "Curved solids and polyhedra require different structural reasoning."
}

SEMANTIC_EQUIVALENCE {
 "SolidGeometry" <=> "ThreeDimensionalGeometry"
 "Polyhedron" <=> "SolidBoundedByPolygonalFaces"
 "CrossSection" <=> "PlaneIntersectionOfSolid"
 "SurfaceArea" <=> "AreaOfBoundarySurface"
 "Volume" <=> "ThreeDimensionalMeasure"
}

VALIDATION {
 V1 = "Primary solids are interpreted in three dimensions."
 V2 = "Volume is nonnegative."
 V3 = "Surface area has square units."
 V4 = "Volume has cubic units."
 V5 = "Cross-section lies in a plane."
 V6 = "Prism and pyramid formulas use perpendicular height."
 V7 = "Sphere formulas use radius consistently."
 V8 = "Transformation invariants match the transformation class."
}

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

 POLICY =
  "Reduce confidence when solid type, dimensions, orientation, unit system, cross-section plane or transformation class are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SolidGeometryReasoning,
  PolyhedronReasoning,
  SurfaceAreaReasoning,
  VolumeReasoning,
  CrossSectionReasoning,
  SpatialCoordinateReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "solid geometry" -> SolidGeometry
  "3D geometry" -> SolidGeometry
  "three-dimensional geometry" -> SolidGeometry
  "polyhedron" -> Polyhedron
  "prism" -> Prism
  "pyramid" -> Pyramid
  "cylinder" -> Cylinder
  "cone" -> Cone
  "sphere" -> Sphere
 }

 INTERPRETATION_ORDER = [
  detect_solid_geometry_context,
  identify solid_or_spatial_entities,
  classify solid type,
  identify dimensions coordinates surfaces or cross-section data,
  identify volume surface-area distance angle or transformation goal,
  apply valid three-dimensional formulas_and_theorems,
  verify metric unit and dimensional constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("solid geometry")
  OR three_dimensional_geometry_context
  THEN activate(SolidGeometry)

 E2:
  IF prism_context
  THEN apply_prism_structure_and_volume_reasoning

 E3:
  IF pyramid_context
  THEN apply_pyramid_structure_and_volume_reasoning

 E4:
  IF cylinder_context
  THEN apply_cylinder_formulas

 E5:
  IF cone_context
  THEN distinguish_radius_height_and_slant_height

 E6:
  IF sphere_context
  THEN apply_sphere_locus_surface_area_and_volume_reasoning

 E7:
  IF cross_section_requested
  THEN intersect_solid_with_plane

 E8:
  IF dihedral_angle_requested
  THEN use_plane_normals_or_equivalent_spatial_reasoning

 E9:
  IF transformation_context
  THEN classify_3D_transformation_and_track_invariants
}

RECURSION {
 ENTRY = SolidGeometry

 EXPAND(SolidGeometry) -> [
  PointInSpace,
  LineInSpace,
  PlaneInSpace,
  Solid,
  Polyhedron,
  Prism,
  Pyramid,
  Cylinder,
  Cone,
  Sphere,
  CrossSection,
  SurfaceArea,
  Volume,
  SpatialTransformation
 ]

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

 EXPAND(Prism) -> [
  Base,
  CongruentParallelBases,
  LateralFace,
  Height,
  Volume,
  SurfaceArea
 ]

 EXPAND(Pyramid) -> [
  Base,
  Apex,
  TriangularFace,
  Height,
  SlantHeight,
  Volume,
  SurfaceArea
 ]

 EXPAND(Sphere) -> [
  Centre,
  Radius,
  Diameter,
  GreatCircle,
  SurfaceArea,
  Volume
 ]

 STOP_CONDITION =
  "Stop when the solid's classification, measurement, cross-section, spatial relation, coordinate representation or transformation property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

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

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

 ROUTE_RULES = [
  "Solid Geometry -> Geometry for core spatial semantics",
  "Solid Geometry -> Plane Geometry for faces and cross-sections",
  "Solid Geometry -> Coordinate Geometry for 3D coordinate representation",
  "Solid Geometry -> Vector Geometry for lines, planes, distances and angles in space",
  "Solid Geometry -> Transformation for 3D mappings",
  "Solid Geometry -> Dimension for dimensional reasoning"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve three-dimensional geometric semantics.",
  "Preserve distinction between surface area and volume.",
  "Preserve polyhedral face-edge-vertex structure.",
  "Preserve curved-solid versus polyhedral distinctions.",
  "Preserve cross-section as a planar intersection.",
  "Preserve transformation-specific invariants and volume scaling.",
  "Preserve machine-readable solid-geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Solid geometry is the study of three-dimensional geometric objects, spatial relations, surfaces and volumes."

 DIMENSION_SUMMARY =
  "Primary ambient dimension=3."

 PRISM_VOLUME_SUMMARY =
  "V=base_area*height"

 PYRAMID_VOLUME_SUMMARY =
  "V=(1/3)*base_area*height"

 CYLINDER_VOLUME_SUMMARY =
  "V=pi*r^2*h"

 CONE_VOLUME_SUMMARY =
  "V=(1/3)*pi*r^2*h"

 SPHERE_VOLUME_SUMMARY =
  "V=(4/3)*pi*r^3"

 SPHERE_SURFACE_SUMMARY =
  "A=4*pi*r^2"

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

FLOW {
 INPUT
 -> detect_solid_geometry_context
 -> identify_three_dimensional_entities
 -> classify_solid_polyhedron_curved_solid_or_spatial_relation
 -> identify dimensions coordinates surfaces edges faces or cross_section_data
 -> identify volume surface_area distance angle symmetry construction or transformation_goal
 -> apply valid solid_geometry formulas_and_theorems
 -> verify unit metric dimensional and orientation_constraints
 -> route to plane coordinate vector dimension or transformation_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Solid Geometry provides the three-dimensional branch of geometry, covering polyhedra, prisms, pyramids, cylinders, cones, spheres, planes and lines in space, cross-sections, surface area, volume, spatial angles and transformations. It links plane geometry with coordinate and vector methods for exact three-dimensional reasoning."
}

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

}

END_FLOW