Discrete Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of discrete geometry within the Kireth Research Semantic Library. Discrete geometry studies combinatorial and finite or locally finite geometric structures, including point sets, lattices, packings, coverings, tilings, polytopes, convexity, incidence structures and extremal geometric configurations. It connects geometry with combinatorics, number theory, optimisation, topology and computational geometry. This ontology supports discrete point configurations, lattice geometry, packing and covering density, tessellations, polytopes, arrangements, incidence bounds, distance sets, geometric graphs and extremal problems.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = DiscreteGeometry
 SYMBOLIC_FORMS = [
  "S subset R^d finite_or_discrete",
  "Lambda lattice",
  "packing density",
  "covering density"
 ]
 ENTITY_CLASS = CombinatorialGeometricFramework
 PRIMARY_CONTEXT = AdvancedGeometry
 DEFINITION = "Discrete geometry studies finite, countable and locally finite geometric configurations and the combinatorial properties of their spatial arrangement."
 SEMANTIC_ROLE = "Provides the geometric framework for point sets, lattices, packings, coverings, tilings, polytopes, incidence problems and extremal geometric structures."
}

SCOPE {
 INCLUDE = [
  DiscreteGeometry,
  FinitePointSet,
  DiscretePointSet,
  Lattice,
  LatticePoint,
  Packing,
  SpherePacking,
  CirclePacking,
  Covering,
  SphereCovering,
  Tiling,
  Tessellation,
  Polytope,
  ConvexPolytope,
  LatticePolytope,
  Arrangement,
  HyperplaneArrangement,
  IncidenceGeometry,
  GeometricGraph,
  UnitDistanceGraph,
  DistanceSet,
  DiameterGraph,
  VoronoiDiagram,
  DelaunayTriangulation,
  ConvexHull,
  HellyTheorem,
  CaratheodoryTheorem,
  RadonTheorem,
  ExtremalGeometry,
  KissingNumber,
  PackingDensity,
  CoveringDensity
 ]

 EXCLUDE_AS_PRIMARY = [
  ContinuousDifferentialGeometryWithoutDiscreteStructure,
  GeneralGraphTheoryWithoutGeometricEmbedding,
  PureCombinatoricsWithoutSpatialMeaning,
  ComputationalGeometryWithoutDiscreteStructuralQuestion
 ]

 DISAMBIGUATION_RULE =
  "When geometry concerns finite or locally finite point configurations, lattices, packings, tilings, polytopes, incidences or extremal spatial arrangements, interpret the context as discrete geometry."
}

CORE_CONCEPTS {

 DiscreteGeometry {
  definition = "The study of geometric structures with finite, countable or locally finite combinatorial form."
 }

 FinitePointSet {
  definition = "A finite collection of points embedded in a geometric space."
 }

 DiscretePointSet {
  definition = "A point set with no accumulation points in the ambient local geometry."
 }

 Lattice {
  definition = "A discrete additive subgroup generated by a basis of a finite-dimensional vector space."
 }

 Packing {
  definition = "A family of geometric objects with pairwise disjoint interiors."
 }

 Covering {
  definition = "A family of geometric objects whose union contains the target space or region."
 }

 Tiling {
  definition = "A covering by tiles with disjoint interiors under the adopted boundary convention."
 }

 Polytope {
  definition = "A bounded convex hull of finitely many points, equivalently a bounded intersection of finitely many half-spaces."
 }

 IncidenceGeometry {
  definition = "The study of point-line, point-hyperplane and related containment relations."
 }

 GeometricGraph {
  definition = "A graph whose vertices and edges are associated with geometric positions or relations."
 }

 KissingNumber {
  definition = "The maximum number of nonoverlapping equal spheres tangent to a central equal sphere in a given dimension."
 }
}

RELATIONS {
 DiscreteGeometry --is_branch_of--> AdvancedGeometry
 FinitePointSet --has--> ConvexHull
 Lattice --contains--> LatticePoint
 Packing --has_measure--> PackingDensity
 Covering --has_measure--> CoveringDensity
 Tiling --is_special_case_of--> Covering
 ConvexPolytope --is_special_case_of--> Polytope
 LatticePolytope --has_vertices_in--> Lattice
 VoronoiDiagram --is_associated_with--> DiscretePointSet
 DelaunayTriangulation --is_dual_to--> VoronoiDiagram
 IncidenceGeometry --studies--> IncidenceRelation
 GeometricGraph --encodes--> GeometricAdjacency
 UnitDistanceGraph --is_special_case_of--> GeometricGraph
 HellyTheorem --constrains--> ConvexSetIntersections
 CaratheodoryTheorem --constrains--> ConvexHullRepresentation
 RadonTheorem --partitions--> PointSet
 KissingNumber --is_extremal_quantity_of--> SpherePacking
}

OPERATORS {
 LATTICE(B) :=
  {sum_i n_i*b_i : n_i in Z}

 IS_DISCRETE_POINT_SET(S) :=
  every_point_has_neighbourhood_containing_only_finitely_many_points_of_S

 PACKING_DENSITY(P,R) :=
  occupied_volume_fraction_in_large_region_limit
  under declared asymptotic convention

 COVERING_DENSITY(C,R) :=
  total_covering_measure_relative_to_region
  under declared averaging convention

 CONVEX_HULL(S) :=
  smallest_convex_set_containing_S

 UNIT_DISTANCE_GRAPH(S) :=
  graph_with_vertices_S_and_edges_between_pairs_at_distance_1

 DISTANCE_SET(S) :=
  {d(p,q): p,q in S, p!=q}

 KISSING_NUMBER(d) :=
  maximum_number_of_unit_spheres_tangent_to_one_unit_sphere_without_overlap

 VORONOI(S) :=
  nearest_site_partition_of_space

 DELAUNAY(S) :=
  Voronoi_dual_triangulation_or_cell_complex

 IS_TILING({T_i},Omega) :=
  union_i T_i = Omega
  AND interiors_pairwise_disjoint
}

DEFINITIONS {
 D1 = "Discrete geometry emphasises combinatorial structure in geometric configurations."
 D2 = "A lattice is discrete and generated by integer combinations of basis vectors."
 D3 = "Packing prohibits interior overlap."
 D4 = "Covering requires complete spatial coverage."
 D5 = "Tilings combine coverage with nonoverlapping interiors."
 D6 = "Polytopes have finite combinatorial face structure."
 D7 = "Incidence geometry studies containment relations such as point-on-line."
 D8 = "Voronoi and Delaunay structures encode discrete spatial proximity."
 D9 = "Extremal discrete geometry seeks maxima or minima under geometric constraints."
 D10 = "Dimension strongly affects packing, covering and incidence behaviour."
}

AXIOMS {
 A1_LOCAL_FINITE_STRUCTURE:
  Discrete configurations have finite local complexity under the adopted model

 A2_PACKING_NONOVERLAP:
  Packing elements have disjoint interiors

 A3_COVERING_COMPLETENESS:
  Covering elements collectively contain the target domain

 A4_TILING:
  Tilings satisfy both coverage and interior nonoverlap

 A5_LATTICE_DISCRETENESS:
  Full-rank lattices are discrete subsets of Euclidean space

 A6_POLYTOPE_FINITE_STRUCTURE:
  Polytopes have finitely many faces

 A7_DIMENSION_DEPENDENCE:
  Extremal constants and structural theorems may depend on ambient dimension

 A8_INCIDENCE:
  Incidence relations are combinatorial-geometric invariants under suitable transformations
}

REASONING {

 R1_CONFIGURATION_CLASSIFICATION {
  INPUT = geometric_configuration
  IDENTIFY = finite_point_set_lattice_packing_covering_tiling_polytope_or_incidence structure
  RETURN = discrete_geometry_type
 }

 R2_LATTICE_ANALYSIS {
  INPUT = basis_vectors
  FORM = integer_span
  COMPUTE = fundamental_parallelepiped
  COMPUTE = determinant_or_covolume
  RETURN = lattice_structure
 }

 R3_PACKING_ANALYSIS {
  INPUT = object_family
  TEST = pairwise_interior_disjointness
  COMPUTE = local_or_asymptotic_density
  RETURN = valid_packing_and_density
 }

 R4_COVERING_ANALYSIS {
  INPUT = covering_family_and_domain
  TEST = complete_coverage
  COMPUTE = covering_density_if_requested
  RETURN = valid_covering_and_density
 }

 R5_TILING_ANALYSIS {
  INPUT = tile_family
  TEST = coverage
  TEST = interior_nonoverlap
  ANALYSE = periodicity_symmetry_and_adjacency
  RETURN = tiling_structure
 }

 R6_POLYTOPE_ANALYSIS {
  INPUT = vertices_or_halfspaces
  CONSTRUCT = convex_polytope
  DERIVE = faces_edges_vertices_and_face_lattice
  RETURN = polytope_structure
 }

 R7_INCIDENCE_REASONING {
  INPUT = points_lines_hyperplanes_or_other_objects
  COUNT = incidences
  APPLY = combinatorial_and_geometric_constraints
  RETURN = incidence_structure_or_bound
 }

 R8_DISTANCE_GRAPH {
  INPUT = point_set_and_distance_rule
  BUILD = geometric_graph
  ANALYSE = degree_cliques_colouring_or_extremal quantity
  RETURN = geometric_graph_result
 }

 R9_EXTREMAL_REASONING {
  INPUT = discrete_geometric_constraint
  IDENTIFY = quantity_to_maximise_or_minimise
  APPLY = known_bounds_symmetry_convexity_or_combinatorial arguments
  RETURN = extremal_value_or_bound
 }
}

CONSTRAINTS {
 C1 = "Discrete must not be conflated with finite; infinite discrete sets are allowed."
 C2 = "Packing and covering are distinct optimisation problems."
 C3 = "Density definitions require explicit limiting or periodic conventions."
 C4 = "Tilings may share boundaries while interiors remain disjoint."
 C5 = "Lattice rank and ambient dimension must be identified."
 C6 = "Polytope terminology may distinguish abstract and geometric polytopes."
 C7 = "Incidence bounds depend strongly on dimension and degeneracy assumptions."
 C8 = "Geometric graph edges depend on the declared spatial rule."
 C9 = "Extremal results may require general-position assumptions."
 C10 = "Computational approximations must not be presented as exact extremal results."
}

SEMANTIC_EQUIVALENCE {
 "DiscreteGeometry" <=> "CombinatorialGeometryOfDiscreteConfigurations"
 "Lattice" <=> "IntegerSpanOfBasisVectors"
 "Packing" <=> "InteriorDisjointObjectArrangement"
 "Tiling" <=> "NonoverlappingCovering"
 "GeometricGraph" <=> "GraphDefinedByGeometricRelations"
 "KissingNumber" <=> "MaximumEqualSphereTangencyCount"
}

VALIDATION {
 V1 = "Discrete or finite status is correctly classified."
 V2 = "Packing objects do not overlap in interiors."
 V3 = "Coverings fully cover the intended domain."
 V4 = "Tiling boundary conventions are explicit."
 V5 = "Lattice basis has declared rank."
 V6 = "Polytope face incidences are combinatorially consistent."
 V7 = "Distance-graph edges satisfy the declared metric rule."
 V8 = "Density definitions match the asymptotic or periodic model."
 V9 = "Extremal claims state dimensional assumptions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DISCRETE_GEOMETRY_DEFINITION = 0.99
 LATTICE_REASONING = 0.99
 PACKING_REASONING = 0.98
 COVERING_REASONING = 0.98
 TILING_REASONING = 0.99
 POLYTOPE_REASONING = 0.99
 INCIDENCE_REASONING = 0.97
 EXTREMAL_REASONING = 0.96

 POLICY =
  "Reduce confidence when dimension, lattice rank, density convention, periodicity, degeneracy assumptions, metric or exact-versus-computational status are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  DiscreteGeometryReasoning,
  LatticeReasoning,
  PackingReasoning,
  CoveringReasoning,
  TilingReasoning,
  PolytopeReasoning,
  IncidenceReasoning,
  GeometricGraphReasoning,
  DistanceSetReasoning,
  ExtremalGeometryReasoning,
  VoronoiDelaunayReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "discrete geometry" -> DiscreteGeometry
  "finite point set" -> FinitePointSet
  "lattice" -> Lattice WHEN geometry_context
  "sphere packing" -> SpherePacking
  "circle packing" -> CirclePacking
  "covering" -> Covering WHEN geometry_context
  "tiling" -> Tiling
  "tessellation" -> Tessellation
  "polytope" -> Polytope
  "incidence geometry" -> IncidenceGeometry
  "geometric graph" -> GeometricGraph
  "kissing number" -> KissingNumber
 }

 INTERPRETATION_ORDER = [
  detect_discrete_geometry_context,
  identify ambient_space_dimension_metric_and_configuration_type,
  identify lattice_packing_covering_tiling_polytope_incidence_graph_or_extremal goal,
  identify finite_infinite_periodic_or_asymptotic assumptions,
  apply combinatorial_geometric reasoning,
  verify discreteness_density_nonoverlap_coverage_dimension_and_degeneracy constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("discrete geometry")
  OR finite_lattice_packing_tiling_polytope_context
  THEN activate(DiscreteGeometryOntology)

 E2:
  IF lattice_context
  THEN identify basis_rank_and_covolume

 E3:
  IF packing_context
  THEN verify pairwise interior disjointness

 E4:
  IF covering_context
  THEN verify total domain coverage

 E5:
  IF tiling_context
  THEN require coverage_and_interior_nonoverlap

 E6:
  IF polytope_context
  THEN route to convex_hull_halfspace_and_face_lattice reasoning

 E7:
  IF incidence_problem
  THEN identify degeneracies_and_dimension before applying bounds

 E8:
  IF geometric_graph
  THEN define edge rule explicitly

 E9:
  IF extremal_quantity_requested
  THEN distinguish exact_value_upper_bound_lower_bound_and_computational estimate
}

RECURSION {
 ENTRY = DiscreteGeometry

 EXPAND(DiscreteGeometry) -> [
  FinitePointSet,
  DiscretePointSet,
  Lattice,
  LatticePoint,
  Packing,
  SpherePacking,
  Covering,
  Tiling,
  Tessellation,
  Polytope,
  ConvexPolytope,
  LatticePolytope,
  Arrangement,
  IncidenceGeometry,
  GeometricGraph,
  UnitDistanceGraph,
  DistanceSet,
  KissingNumber,
  PackingDensity,
  CoveringDensity,
  VoronoiDiagram,
  DelaunayTriangulation
 ]

 EXPAND(LatticeGeometry) -> [
  LatticeBasis,
  FundamentalDomain,
  Covolume,
  ShortestVector,
  VoronoiCell,
  LatticePacking,
  LatticeCovering
 ]

 EXPAND(PolytopeGeometry) -> [
  Vertex,
  Edge,
  Face,
  Facet,
  FaceLattice,
  ConvexHull,
  HalfSpaceRepresentation
 ]

 EXPAND(ExtremalGeometry) -> [
  KissingNumber,
  PackingDensity,
  CoveringDensity,
  UnitDistanceProblem,
  DiameterProblem,
  IncidenceBound
 ]

 STOP_CONDITION =
  "Stop when the discrete configuration, lattice, packing, covering, tiling, polytope, incidence structure, geometric graph or extremal quantity is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Advanced Geometry"

 IMMEDIATE_CONTEXTS = [
  "Advanced Geometry",
  "Computational Geometry Ontology",
  "Convex Hull Ontology",
  "Voronoi Diagram Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Advanced Geometry",
  "Computational Geometry Ontology",
  "Convex Hull Ontology",
  "Voronoi Diagram Ontology",
  "Delaunay Triangulation Ontology",
  "Topology Ontology",
  "Linear Algebra Ontologies",
  "Combinatorics Ontologies",
  "Graph Theory Ontologies",
  "Number Theory Ontologies",
  "Optimisation Ontologies"
 ]

 ROUTE_RULES = [
  "Discrete Geometry -> Advanced Geometry for parent structural semantics",
  "Discrete Geometry -> Computational Geometry for algorithmic realisation",
  "Discrete Geometry -> Convex Hull for finite point-set envelopes",
  "Discrete Geometry -> Voronoi and Delaunay for discrete proximity structures",
  "Discrete Geometry -> Linear Algebra for lattice bases and determinants",
  "Discrete Geometry -> Combinatorics for incidence and extremal counting",
  "Discrete Geometry -> Graph Theory for geometric graphs",
  "Discrete Geometry -> Number Theory for lattice-point structure",
  "Discrete Geometry -> Optimisation for packing and covering extrema"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible discrete structures, extremal quantities or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve finite-versus-discrete distinction.",
  "Preserve packing-versus-covering distinction.",
  "Preserve tiling coverage-and-nonoverlap semantics.",
  "Preserve lattice integer-span semantics.",
  "Preserve polytope finite-face semantics.",
  "Preserve dimension-dependent extremal reasoning.",
  "Preserve exact-versus-approximate result distinction.",
  "Preserve machine-readable discrete-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Discrete geometry studies finite and locally finite geometric configurations and their combinatorial spatial structure."

 LATTICE_SUMMARY =
  "A lattice is the integer span of a linearly independent generating set."

 PACKING_SUMMARY =
  "A packing is a family of geometric objects with pairwise disjoint interiors."

 COVERING_SUMMARY =
  "A covering is a family whose union contains the target region."

 TILING_SUMMARY =
  "A tiling covers the domain with pairwise disjoint interiors."

 POLYTOPE_SUMMARY =
  "A polytope is a bounded convex hull of finitely many points, equivalently a bounded finite half-space intersection."

 INCIDENCE_SUMMARY =
  "Incidence geometry studies containment relations such as point-on-line and point-on-hyperplane."

 EXTREMAL_SUMMARY =
  "Discrete geometry includes extremal questions such as maximal packing density and kissing number."

 PROXIMITY_SUMMARY =
  "Voronoi and Delaunay structures encode proximity relationships within discrete point configurations."
}

FLOW {
 INPUT
 -> detect_discrete_geometry_context
 -> identify ambient_space_dimension_metric_configuration_and_finiteness
 -> classify lattice_packing_covering_tiling_polytope_incidence_graph_distance_or_extremal problem
 -> identify periodicity_density_degeneracy_and_exactness assumptions
 -> apply combinatorial_convex_metric_or_lattice reasoning
 -> verify discreteness_nonoverlap_coverage_dimension_density_and_incidence constraints
 -> route to advanced_geometry computational_geometry convex_hull Voronoi Delaunay linear_algebra combinatorics graph_theory number_theory_or_optimisation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Discrete Geometry Ontology defines the geometry of finite, countable and locally finite configurations. It supports point sets, lattices, packings, coverings, tilings, polytopes, Voronoi and Delaunay structures, incidences, geometric graphs, distance sets, packing and covering densities, kissing numbers and other extremal geometric problems while preserving dimensional and combinatorial structure."
}

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

}

END_FLOW