Polyhedron Ontology

FLOW ALGEBRA
TITLE: Polyhedron Ontology
VERSION: 4.0
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/polyhedron-ontology-2.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
STATUS: Independently formalised; source-page content not verified

01. ONTOLOGY IDENTITY
ENTITY Polyhedron : GeometricEntity
DOMAIN Geometry / Polyhedral Geometry / Topology
DEFINITION A polyhedron is a three-dimensional geometric solid whose boundary consists of finitely many polygonal faces, joined along edges and vertices, subject to an explicitly stated polyhedral convention.
CONTEXT The term may denote the solid, its boundary complex, or a possibly nonconvex polyhedral set; assertions must declare the convention.

02. TYPES
TYPE Point := element of EuclideanSpace(3)
TYPE Vertex := Point
TYPE Edge := Segment(Vertex,Vertex)
TYPE Face := PolygonalRegion
TYPE Polyhedron := {vertices:Set(Vertex), edges:Set(Edge), faces:Set(Face), solid:Region3D?}
TYPE ConvexPolyhedron <: Polyhedron
TYPE NonconvexPolyhedron <: Polyhedron
TYPE RegularPolyhedron <: ConvexPolyhedron
TYPE Prism <: Polyhedron
TYPE Pyramid <: Polyhedron
TYPE PolyhedralSurface := finite polygonal face complex
TYPE FaceAdjacencyGraph := Graph(Face,SharedEdge)
TYPE EulerCharacteristic := Integer
TYPE Volume := NonnegativeReal
TYPE SurfaceArea := NonnegativeReal

03. ENTITY ATTRIBUTES
ATTRIBUTE dimension: Integer = 3 (for full-dimensional solids)
ATTRIBUTE vertex_count: NonnegativeInteger
ATTRIBUTE edge_count: NonnegativeInteger
ATTRIBUTE face_count: NonnegativeInteger
ATTRIBUTE bounded: Boolean
ATTRIBUTE convex: Boolean
ATTRIBUTE manifold_boundary: Boolean
ATTRIBUTE closed_boundary: Boolean
ATTRIBUTE connected_boundary: Boolean
ATTRIBUTE orientable_boundary: Boolean
ATTRIBUTE genus: NonnegativeInteger? (when connected closed orientable surface)
ATTRIBUTE volume: Volume? (when bounded solid is well-defined)
ATTRIBUTE surface_area: SurfaceArea? (when finite face geometry is defined)
ATTRIBUTE coordinate_system: CoordinateSystem?
ATTRIBUTE units: UnitSystem?

04. CORE RELATIONSHIPS
RELATION HAS_VERTEX(Polyhedron,Vertex)
RELATION HAS_EDGE(Polyhedron,Edge)
RELATION HAS_FACE(Polyhedron,Face)
RELATION EDGE_CONNECTS(Edge,Vertex,Vertex)
RELATION FACE_BOUNDED_BY(Face,Edge)
RELATION FACES_ADJACENT(Face,Face)
RELATION VERTEX_INCIDENT_TO(Vertex,Edge)
RELATION EDGE_INCIDENT_TO(Edge,Face)
RELATION IS_CONVEX(Polyhedron)
RELATION IS_REGULAR(Polyhedron)
RELATION IS_COMBINATORIALLY_EQUIVALENT(Polyhedron,Polyhedron)
RELATION IS_GEOMETRICALLY_CONGRUENT(Polyhedron,Polyhedron)
RELATION IS_DUAL_TO(Polyhedron,Polyhedron)
RELATION HAS_EULER_CHARACTERISTIC(PolyhedralSurface,Integer)
RELATION HAS_VOLUME(Polyhedron,Volume)
RELATION HAS_SURFACE_AREA(Polyhedron,SurfaceArea)

05. FORMAL DEFINITIONS
DEFINE convex(P) := forall x,y in P, forall t in [0,1], t*x+(1-t)*y in P
DEFINE boundary(P) := closure(P) \ interior(P)
DEFINE face_count(P) := |faces(P)|
DEFINE edge_count(P) := |edges(P)|
DEFINE vertex_count(P) := |vertices(P)|
DEFINE surface_area(P) := sum_{f in faces(P)} area(f) [interiors disjoint]
DEFINE euler_characteristic(S) := V(S)-E(S)+F(S) [finite cell decomposition]
DEFINE volume(P) := integral_{x in P} 1 dV [measurable bounded solid]
DEFINE regular(P) := convex(P) and all faces congruent regular polygons and identical vertex configuration

06. MATHEMATICAL LAWS
LAW EulerConvex: bounded convex 3-polyhedron with spherical boundary => V-E+F=2
LAW EulerSphere: closed connected polyhedral surface homeomorphic to sphere => V-E+F=2
LAW EulerGenus: closed connected orientable polyhedral surface of genus g => V-E+F=2-2*g
LAW HandshakeEdges: sum_{v in vertices} degree(v)=2*E [finite graph, loops counted twice]
LAW FaceEdgeIncidence: sum_{f in faces} sides(f)=2*E [closed 2-manifold polygonal surface]
LAW ConvexVertexMinimum: convex 3-polyhedron => V>=4, E>=6, F>=4
LAW ConvexFaceMinimum: every face has >=3 edges
LAW ConvexGraph: 1-skeleton of a convex 3-polyhedron is planar and 3-vertex-connected
LAW VolumeNonnegative: volume(P)>=0
LAW AreaNonnegative: surface_area(P)>=0
LAW ConvexDualCounts: for combinatorial dual P*, V(P*)=F(P), E(P*)=E(P), F(P*)=V(P)
LAW CongruenceInvariance: congruent solids have equal volume and surface area
LAW ScaleArea: scale(P,k), k>0 => area(scale(P,k))=k^2*area(P)
LAW ScaleVolume: scale(P,k), k>0 => volume(scale(P,k))=k^3*volume(P)

07. INFERENCE RULES
RULE R01 IF is_polyhedron(P) THEN is_geometric_entity(P)
RULE R02 IF is_convex_polyhedron(P) THEN is_polyhedron(P)
RULE R03 IF is_regular_polyhedron(P) THEN is_convex_polyhedron(P)
RULE R04 IF has_face(P,f) THEN is_polygonal_region(f)
RULE R05 IF has_edge(P,e) THEN has_two_endpoints(e)
RULE R06 IF bounded_convex_3_polyhedron(P) THEN euler(P)=2
RULE R07 IF closed_connected_orientable_boundary(P) AND genus(P)=g THEN euler(P)=2-2*g
RULE R08 IF euler(P)!=2 AND boundary_is_spherical(P) THEN reject_geometry(P)
RULE R09 IF closed_manifold_boundary(P) THEN every_edge_incident_to_exactly_two_faces(P)
RULE R10 IF convex(P) AND x_in(P) AND y_in(P) THEN segment(x,y)_subset(P)
RULE R11 IF regular(P) THEN all_faces_regular_and_congruent(P)
RULE R12 IF regular(P) THEN same_number_of_faces_at_each_vertex(P)
RULE R13 IF cube(P) THEN vertex_count(P)=8
RULE R14 IF cube(P) THEN edge_count(P)=12
RULE R15 IF cube(P) THEN face_count(P)=6
RULE R16 IF tetrahedron(P) THEN vertex_count(P)=4
RULE R17 IF tetrahedron(P) THEN edge_count(P)=6
RULE R18 IF tetrahedron(P) THEN face_count(P)=4
RULE R19 IF octahedron(P) THEN (V,E,F)=(6,12,8)
RULE R20 IF dodecahedron(P) THEN (V,E,F)=(20,30,12)
RULE R21 IF icosahedron(P) THEN (V,E,F)=(12,30,20)
RULE R22 IF closed_polygonal_manifold(P) THEN sum_face_sides(P)=2*E(P)
RULE R23 IF finite_simple_graph(P) THEN sum_vertex_degrees(P)=2*E(P)
RULE R24 IF congruent(P,Q) THEN volume(P)=volume(Q)
RULE R25 IF congruent(P,Q) THEN surface_area(P)=surface_area(Q)
RULE R26 IF scale_factor(P,Q)=k AND k>0 THEN volume(Q)=k^3*volume(P)
RULE R27 IF scale_factor(P,Q)=k AND k>0 THEN surface_area(Q)=k^2*surface_area(P)
RULE R28 IF dual(P,Q) THEN V(P)=F(Q)
RULE R29 IF dual(P,Q) THEN E(P)=E(Q)
RULE R30 IF dual(P,Q) THEN F(P)=V(Q)
RULE R31 IF finite_polyhedron(P) THEN counts(P) are nonnegative integers
RULE R32 IF solid(P) AND bounded(P) THEN volume(P) is finite
RULE R33 IF closed_manifold_boundary(P) AND orientable(P) AND connected(P) THEN genus(P) is nonnegative integer
RULE R34 IF prism(P) AND base_n_gon(P,n) THEN (V,E,F)=(2*n,3*n,n+2) [nondegenerate prism]
RULE R35 IF pyramid(P) AND base_n_gon(P,n) THEN (V,E,F)=(n+1,2*n,n+1) [nondegenerate pyramid]
RULE R36 IF polyhedron(P) AND edge_face_incidence_invalid(P) THEN flag_topology_for_review(P)

08. VALIDATION CONSTRAINTS
CONSTRAINT C01 All edge endpoints must reference declared vertices.
CONSTRAINT C02 All face boundary edges must reference declared edges.
CONSTRAINT C03 Polygonal faces must have at least three distinct vertices.
CONSTRAINT C04 For a closed manifold boundary, each edge has exactly two incident faces.
CONSTRAINT C05 Face interiors must not overlap for an embedded boundary mesh.
CONSTRAINT C06 Euler formula V-E+F=2 must not be applied to arbitrary non-spherical or disconnected surfaces.
CONSTRAINT C07 Convexity requires containment of every segment between any two points of the solid.
CONSTRAINT C08 A face-count tuple does not by itself establish geometric realisability.
CONSTRAINT C09 Units of volume are cubic and units of surface area are squared.
CONSTRAINT C10 Topological claims must declare manifold, connectedness and orientability assumptions.
CONSTRAINT C11 Self-intersecting or open meshes require separate interpretation; do not silently treat them as closed solids.
CONSTRAINT C12 Do not infer regularity solely from equal edge lengths.

09. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE "polyhedron" <=> "polyhedral solid" [when solid convention is explicitly selected]
SEMANTIC_EQUIVALENCE "polyhedral surface" <=> "polygonal boundary complex" [when boundary convention selected]
SEMANTIC_EQUIVALENCE "vertex" <=> "corner point" [informal contextual synonym]
SEMANTIC_EQUIVALENCE "face" <=> "polygonal facet" [for 3-dimensional polyhedra]
SEMANTIC_DIFFERENCE "polyhedron" != "polygon" [3D versus 2D]
SEMANTIC_DIFFERENCE "convex polyhedron" != "arbitrary polyhedron"
SEMANTIC_DIFFERENCE "topological equivalence" != "geometric congruence"
SEMANTIC_DIFFERENCE "face" != "supporting plane"

10. KNOWLEDGE GRAPH TRIPLES
TRIPLE (Polyhedron,subClassOf,GeometricEntity)
TRIPLE (ConvexPolyhedron,subClassOf,Polyhedron)
TRIPLE (RegularPolyhedron,subClassOf,ConvexPolyhedron)
TRIPLE (Polyhedron,hasPart,Vertex)
TRIPLE (Polyhedron,hasPart,Edge)
TRIPLE (Polyhedron,hasPart,Face)
TRIPLE (Cube,instanceOf,RegularPolyhedron)
TRIPLE (Tetrahedron,instanceOf,RegularPolyhedron)
TRIPLE (Octahedron,instanceOf,RegularPolyhedron)
TRIPLE (Dodecahedron,instanceOf,RegularPolyhedron)
TRIPLE (Icosahedron,instanceOf,RegularPolyhedron)
TRIPLE (EulerCharacteristic,appliesTo,PolyhedralSurface)

11. WORKED EXAMPLES
EXAMPLE E01 Cube: V=8,E=12,F=6 => V-E+F=2.
EXAMPLE E02 Tetrahedron: V=4,E=6,F=4 => 4-6+4=2.
EXAMPLE E03 Octahedron: V=6,E=12,F=8 => 6-12+8=2.
EXAMPLE E04 Dodecahedron: V=20,E=30,F=12 => 20-30+12=2.
EXAMPLE E05 Icosahedron: V=12,E=30,F=20 => 12-30+20=2.
EXAMPLE E06 Triangular prism: V=6,E=9,F=5 => 6-9+5=2.
EXAMPLE E07 Square pyramid: V=5,E=8,F=5 => 5-8+5=2.
EXAMPLE E08 n-gonal prism: V=2n,E=3n,F=n+2 => 2.
EXAMPLE E09 n-gonal pyramid: V=n+1,E=2n,F=n+1 => 2.
EXAMPLE E10 Cube edge a>0: surface_area=6a^2, volume=a^3.
EXAMPLE E11 Rectangular box a,b,c>0: surface_area=2(ab+bc+ca), volume=abc.
EXAMPLE E12 Regular tetrahedron edge a>0: surface_area=sqrt(3)*a^2, volume=a^3/(6*sqrt(2)).
EXAMPLE E13 Convex dual cube <=> octahedron: vertex and face counts swap.
EXAMPLE E14 Scale cube edge from a to 2a: area factor 4; volume factor 8.
EXAMPLE E15 Connected orientable toroidal polyhedral surface genus 1: Euler characteristic 0.
EXAMPLE E16 Two disjoint spherical components: total Euler characteristic 4; connected-sphere formula not applicable.

12. PROOF AND EVIDENCE POLICY
PROOF_STATUS := {axiom, theorem, derived, example, conjecture, unverified}
CONFIDENCE mathematical_theorem := 1.00 [conditional on stated hypotheses]
CONFIDENCE validated_example := 1.00 [after arithmetic and assumptions verified]
CONFIDENCE ambiguous_geometry := 0.50 [requires convention and validation]
CONFIDENCE source_page_alignment := unverified
INFERENCE_POLICY Never elevate empirical confidence to formal proof.
INFERENCE_POLICY Store all theorem preconditions with conclusions.
INFERENCE_POLICY Distinguish geometric facts from graph-theoretic and topological facts.

13. FLOW HASH
FLOW_HASH.algorithm := SHA-256
FLOW_HASH.canonicalisation := UTF-8, Unicode NFC, LF newlines, stable key order
FLOW_HASH.input := canonical_flow_content_excluding_hash_field
FLOW_HASH.value := COMPUTE_AT_PUBLICATION
FLOW_HASH.note := No digest is asserted until canonical content is frozen and hashed.

14. UPDATE MODEL
UPDATE_MODEL.version := 4.0
UPDATE_MODEL.mode := versioned_append_and_validate
UPDATE_MODEL.trigger := source_revision | theorem_correction | taxonomy_change | example_revision
UPDATE_MODEL.actions := revalidate_types -> recheck_axioms -> recompute_inferences -> rebuild_graph -> recompute_hash
UPDATE_MODEL.history := preserve_prior_versions_and_change_log
UPDATE_MODEL.conflict := prefer formally proved result with explicit hypotheses; escalate unresolved cases

15. AI CONSUMPTION
AI_CONSUMPTION.entity := Polyhedron
AI_CONSUMPTION.purpose := semantic_search | mathematical_reasoning | knowledge_graph | geometry_validation
AI_CONSUMPTION.parse := typed_entities + relations + constraints + inference_rules
AI_CONSUMPTION.guardrails := require_geometry_convention; preserve_theorem_preconditions; reject_invalid_inferences
AI_CONSUMPTION.output := classification + evidence + assumptions + confidence + provenance
AI_CONSUMPTION.crosslinks := GeometricEntityOntology, PolygonOntology, SurfaceOntology, VertexOntology, EdgeOntology, TopologyOntology
AI_CONSUMPTION.provenance := Kireth Semantic Library / independently formalised draft

END FLOW ALGEBRA