Cube Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of a cube within the Kireth Research Semantic Library. A cube is a regular hexahedron with six congruent square faces, twelve equal edges and eight vertices. This ontology supports face-edge-vertex incidence, symmetry, diagonals, surface area, volume, coordinate representation, transformations, nets, cross-sections and relationships with cuboids, prisms and regular polyhedra.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Cube
 SYMBOLIC_FORMS = ["a×a×a", "RegularHexahedron"]
 ENTITY_CLASS = RegularPolyhedron
 PRIMARY_CONTEXT = SolidGeometry
 DEFINITION = "A cube is a regular hexahedron bounded by six congruent square faces, with twelve equal edges and eight vertices."
 SEMANTIC_ROLE = "Provides the canonical equal-edge orthogonal polyhedron for volume, surface area, diagonal, symmetry, coordinate and transformation reasoning."
}

SCOPE {
 INCLUDE = [
  Cube,
  RegularHexahedron,
  Face,
  SquareFace,
  Edge,
  Vertex,
  FaceDiagonal,
  SpaceDiagonal,
  SurfaceArea,
  Volume,
  Net,
  Symmetry,
  CoordinateCube,
  CrossSection,
  Circumsphere,
  Insphere,
  DihedralAngle
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralCuboid,
  RectangularPrismWithUnequalEdges,
  Hypercube,
  PhysicalCubeWithoutMathematicalAbstraction
 ]

 DISAMBIGUATION_RULE =
  "When cube occurs with six square faces, equal edges, volume, surface area, diagonals, nets, symmetry or coordinates, interpret it as a three-dimensional geometric cube."
}

CORE_CONCEPTS {

 Cube {
  definition = "A regular polyhedron with six congruent square faces."
  face_count = 6
  edge_count = 12
  vertex_count = 8
 }

 EdgeLength {
  symbol = "a"
  definition = "The common length of every edge."
 }

 Face {
  definition = "One of six congruent square boundary regions."
 }

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

 Vertex {
  definition = "A point where three mutually perpendicular edges meet."
 }

 FaceDiagonal {
  definition = "A diagonal of a square face."
  length = "a*sqrt(2)"
 }

 SpaceDiagonal {
  definition = "A segment joining opposite vertices through the interior of the cube."
  length = "a*sqrt(3)"
 }

 SurfaceArea {
  formula = "6a^2"
 }

 Volume {
  formula = "a^3"
 }

 DihedralAngle {
  definition = "The angle between adjacent faces."
  measure = "90 degrees"
 }

 Insphere {
  definition = "Sphere tangent to all six faces."
  radius = "a/2"
 }

 Circumsphere {
  definition = "Sphere passing through all eight vertices."
  radius = "a*sqrt(3)/2"
 }
}

RELATIONS {
 Cube --is_special_case_of--> Cuboid
 Cube --is_special_case_of--> Prism
 Cube --is_special_case_of--> Polyhedron
 Cube --is--> RegularHexahedron
 Cube --has--> SixSquareFaces
 Cube --has--> TwelveEdges
 Cube --has--> EightVertices
 Cube --has--> FaceDiagonal
 Cube --has--> SpaceDiagonal
 Cube --has--> Insphere
 Cube --has--> Circumsphere
 Face --is--> Square
 AdjacentFaces --meet_at--> Edge
 ThreeFaces --meet_at--> Vertex
 AdjacentFaces --are_perpendicular_to--> EachOther
 OppositeFaces --are_parallel_to--> EachOther
}

OPERATORS {
 SURFACE_AREA(a) :=
  6*a^2

 VOLUME(a) :=
  a^3

 FACE_DIAGONAL(a) :=
  a*sqrt(2)

 SPACE_DIAGONAL(a) :=
  a*sqrt(3)

 INSPHERE_RADIUS(a) :=
  a/2

 CIRCUMSPHERE_RADIUS(a) :=
  a*sqrt(3)/2

 EDGE_LENGTH_FROM_VOLUME(V) :=
  cube_root(V)

 EDGE_LENGTH_FROM_SURFACE_AREA(S) :=
  sqrt(S/6)

 IS_CUBE(SOLID) :=
  six_congruent_square_faces
  AND twelve_equal_edges
  AND eight_vertices
  AND right_dihedral_angles

 TRANSFORM_CUBE(C,T) :=
  image_of_cube_under_3D_transformation(C,T)
}

DEFINITIONS {
 D1 = "A cube has six faces."
 D2 = "Every face is a square."
 D3 = "A cube has twelve equal edges."
 D4 = "A cube has eight vertices."
 D5 = "Three mutually perpendicular edges meet at each vertex."
 D6 = "Opposite faces are parallel."
 D7 = "Adjacent faces meet at right dihedral angles."
 D8 = "Face diagonal length is a*sqrt(2)."
 D9 = "Space diagonal length is a*sqrt(3)."
 D10 = "A cube is a special cuboid with all three edge dimensions equal."
}

AXIOMS {
 A1_EQUAL_EDGES:
  All twelve cube edges have equal length

 A2_SQUARE_FACES:
  All six faces are congruent squares

 A3_ORTHOGONAL_EDGE_STRUCTURE:
  The three edges incident at each vertex are pairwise perpendicular

 A4_OPPOSITE_FACE_PARALLELISM:
  Opposite faces are parallel

 A5_RIGHT_DIHEDRAL_ANGLES:
  Adjacent faces meet at 90 degrees

 A6_VOLUME:
  Cube volume equals a^3

 A7_SURFACE_AREA:
  Cube surface area equals 6a^2

 A8_REGULARITY:
  A cube is a regular polyhedron
}

REASONING {

 R1_CUBE_CLASSIFICATION {
  INPUT = solid_properties
  TEST = six_faces
  TEST = all_faces_square_and_congruent
  TEST = twelve_equal_edges
  TEST = eight_vertices
  IF all_true
   RETURN Cube
  ELSE
   RETURN NotEstablishedCube
 }

 R2_SURFACE_AREA {
  INPUT = edge_length a
  COMPUTE = six_face_areas
  RETURN = 6a^2
 }

 R3_VOLUME {
  INPUT = edge_length a
  COMPUTE = a*a*a
  RETURN = a^3
 }

 R4_FACE_DIAGONAL {
  INPUT = edge_length a
  APPLY = Pythagorean_theorem_on_square_face
  RETURN = a*sqrt(2)
 }

 R5_SPACE_DIAGONAL {
  INPUT = edge_length a
  APPLY = Pythagorean_theorem_in_three_dimensions
  RETURN = a*sqrt(3)
 }

 R6_COORDINATE_CUBE {
  INPUT = edge_length a_and_anchor_point
  ASSIGN = orthogonal_coordinate_axes
  GENERATE = eight_vertex_coordinates
  VERIFY = equal_edge_distances
  RETURN = coordinate_representation
 }

 R7_SPHERE_RELATIONS {
  INPUT = cube_edge_length a
  COMPUTE = insphere_radius=a/2
  COMPUTE = circumsphere_radius=a*sqrt(3)/2
  RETURN = sphere_parameters
 }

 R8_CROSS_SECTION {
  INPUT = [cube,plane]
  INTERSECT = cube_with_plane
  CLASSIFY = resulting_polygon
  RETURN = cross_section
 }
}

CONSTRAINTS {
 C1 = "A cube must not be confused with a general cuboid."
 C2 = "All cube edges are equal."
 C3 = "Every face is square."
 C4 = "Surface area and volume use square and cubic units respectively."
 C5 = "Face diagonal and space diagonal are distinct."
 C6 = "A general affine transformation may map a cube to a parallelepiped rather than a cube."
 C7 = "Rigid motions preserve cube structure."
 C8 = "Similarity transformations preserve cubic shape but scale lengths, areas and volumes."
}

SEMANTIC_EQUIVALENCE {
 "Cube" <=> "RegularHexahedron"
 "Cube" <=> "CuboidWithEqualEdgeLengths"
 "FaceDiagonal" <=> "a*sqrt(2)"
 "SpaceDiagonal" <=> "a*sqrt(3)"
 "InsphereRadius" <=> "a/2"
 "CircumsphereRadius" <=> "a*sqrt(3)/2"
}

VALIDATION {
 V1 = "Cube has six square faces."
 V2 = "Cube has twelve equal edges."
 V3 = "Cube has eight vertices."
 V4 = "Surface area equals 6a^2."
 V5 = "Volume equals a^3."
 V6 = "Face diagonal equals a*sqrt(2)."
 V7 = "Space diagonal equals a*sqrt(3)."
 V8 = "Adjacent faces are perpendicular."
 V9 = "Opposite faces are parallel."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CUBE_DEFINITION = 0.99
 FACE_EDGE_VERTEX_STRUCTURE = 0.99
 SURFACE_AREA_REASONING = 0.99
 VOLUME_REASONING = 0.99
 DIAGONAL_REASONING = 0.99
 SPHERE_RELATIONS = 0.99
 CROSS_SECTION_REASONING = 0.98

 POLICY =
  "Reduce confidence when edge equality, face type, transformation class or cross-section plane are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  CubeReasoning,
  PolyhedralReasoning,
  SurfaceAreaReasoning,
  VolumeReasoning,
  DiagonalReasoning,
  SymmetryReasoning,
  CoordinateReasoning,
  CrossSectionReasoning,
  TransformationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "cube" -> Cube
  "regular hexahedron" -> Cube
  "cube edge" -> EdgeLength
  "face diagonal" -> FaceDiagonal
  "space diagonal" -> SpaceDiagonal
  "body diagonal" -> SpaceDiagonal
  "cube net" -> Net
 }

 INTERPRETATION_ORDER = [
  detect_cube_context,
  identify edge_length_or_structural_data,
  verify cube classification if required,
  identify surface_area volume diagonal sphere symmetry coordinate or cross-section goal,
  apply cube formulas_and_theorems,
  verify units and transformation constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("cube")
  AND geometry_context
  THEN activate(CubeOntology)

 E2:
  IF edge_length_given
  THEN derive surface_area volume face_diagonal and space_diagonal as requested

 E3:
  IF volume_given
  THEN derive edge_length_by_cube_root

 E4:
  IF surface_area_given
  THEN derive edge_length_by_sqrt(S/6)

 E5:
  IF diagonal_requested
  THEN distinguish face_diagonal_from_space_diagonal

 E6:
  IF coordinate_representation_requested
  THEN generate orthogonal equal-edge vertex set

 E7:
  IF cross_section_requested
  THEN intersect cube_with_plane

 E8:
  IF transformation_context
  THEN classify transformation and determine whether cube structure is preserved
}

RECURSION {
 ENTRY = Cube

 EXPAND(Cube) -> [
  Face,
  Edge,
  Vertex,
  FaceDiagonal,
  SpaceDiagonal,
  SurfaceArea,
  Volume,
  Symmetry,
  Net,
  CrossSection,
  Insphere,
  Circumsphere
 ]

 EXPAND(Face) -> [
  Square,
  Edge,
  FaceDiagonal,
  Area
 ]

 EXPAND(Symmetry) -> [
  RotationalSymmetry,
  ReflectionSymmetry,
  CentralSymmetry
 ]

 EXPAND(CrossSection) -> [
  Triangle,
  Quadrilateral,
  Pentagon,
  Hexagon,
  RegularCrossSectionWhenApplicable
 ]

 STOP_CONDITION =
  "Stop when the cube's structure, measurement, diagonal, sphere relation, coordinate representation, symmetry or cross-section property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Solid Geometry"

 IMMEDIATE_CONTEXTS = [
  "Solid Geometry Ontology",
  "Cuboid Ontology",
  "Prism Ontology",
  "Polyhedron Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Solid Geometry Ontology",
  "Plane Geometry",
  "Polygon Ontology",
  "Square Ontology",
  "Coordinate Geometry Ontology",
  "Cartesian Coordinate Ontology",
  "Vector Geometry Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology",
  "Dimension Ontology"
 ]

 ROUTE_RULES = [
  "Cube -> Solid Geometry Ontology for general 3D semantics",
  "Cube -> Cuboid for rectangular-prism inheritance",
  "Cube -> Prism for prism structure",
  "Cube -> Plane Geometry for square faces and polygonal cross-sections",
  "Cube -> Coordinate Geometry for vertex coordinates",
  "Cube -> Vector Geometry for diagonals and spatial directions",
  "Cube -> Transformation for rigid, similarity and affine images"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve cube as a regular hexahedron.",
  "Preserve six-square-face structure.",
  "Preserve twelve equal edges and eight vertices.",
  "Preserve face-versus-space diagonal distinction.",
  "Preserve surface-area and volume formulas.",
  "Preserve transformation-specific shape invariance.",
  "Preserve machine-readable cube semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A cube is a regular hexahedron bounded by six congruent square faces."

 STRUCTURE_SUMMARY =
  "6 faces, 12 edges, 8 vertices."

 SURFACE_AREA_SUMMARY =
  "A=6a^2"

 VOLUME_SUMMARY =
  "V=a^3"

 FACE_DIAGONAL_SUMMARY =
  "d_face=a*sqrt(2)"

 SPACE_DIAGONAL_SUMMARY =
  "d_space=a*sqrt(3)"

 INSPHERE_SUMMARY =
  "r_in=a/2"

 CIRCUMSPHERE_SUMMARY =
  "r_out=a*sqrt(3)/2"
}

FLOW {
 INPUT
 -> detect_cube_context
 -> identify edge_length_faces_edges_vertices_or_coordinates
 -> verify regular_hexahedron_structure when classification is required
 -> identify surface_area volume diagonal sphere symmetry cross_section or transformation goal
 -> apply cube_specific geometry
 -> verify square_face equal_edge orthogonality unit and transformation constraints
 -> route to solid_geometry cuboid prism plane coordinate vector or transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Cube Ontology defines the regular hexahedron with six congruent square faces, twelve equal edges and eight vertices. It supports exact reasoning about surface area, volume, face and space diagonals, inscribed and circumscribed spheres, symmetry, coordinate representation, cross-sections and transformations."
}

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

}

END_FLOW