Fractal Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of fractal geometry within the Kireth Research Semantic Library. Fractal geometry studies sets, curves, measures and dynamical constructions exhibiting scale-dependent structure, self-similarity, irregularity and non-integer dimensional behaviour. This ontology supports Hausdorff dimension, box-counting dimension, self-similar sets, iterated function systems, Cantor-type constructions, Julia and Mandelbrot sets, recursive subdivision, multifractals, fractal measures, scaling laws, lacunarity, escape-time dynamics and relationships with topology, measure theory, dynamical systems, complex analysis and advanced geometry.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = FractalGeometry
 SYMBOLIC_FORMS = [
  "dim_H(F)",
  "dim_B(F)",
  "F = union_i S_i(F)",
  "z_{n+1}=f(z_n)"
 ]
 ENTITY_CLASS = ScaleStructuredGeometryFramework
 PRIMARY_CONTEXT = AdvancedGeometry
 DEFINITION = "Fractal geometry studies geometric sets and processes characterised by recursive structure, scale dependence, irregularity and dimensions that may be non-integer."
 SEMANTIC_ROLE = "Provides the geometric framework for self-similarity, fractal dimension, recursive constructions, iterated function systems, complex dynamical fractals and scale-sensitive spatial structure."
}

SCOPE {
 INCLUDE = [
  FractalGeometry,
  Fractal,
  SelfSimilarity,
  StatisticalSelfSimilarity,
  HausdorffDimension,
  BoxCountingDimension,
  MinkowskiDimension,
  SimilarityDimension,
  FractalDimension,
  IteratedFunctionSystem,
  ContractionMapping,
  Attractor,
  CantorSet,
  SierpinskiTriangle,
  SierpinskiCarpet,
  KochCurve,
  KochSnowflake,
  MengerSponge,
  JuliaSet,
  MandelbrotSet,
  EscapeTime,
  RecursiveSubdivision,
  ScalingLaw,
  Lacunarity,
  Multifractal,
  FractalMeasure,
  HausdorffMeasure,
  LocalDimension,
  CapacityDimension,
  StrangeAttractor
 ]

 EXCLUDE_AS_PRIMARY = [
  ArbitraryIrregularShapeWithoutScaleStructure,
  PureChaosWithoutGeometricSet,
  GeneralTopologyWithoutDimensionOrScalingContext,
  TextureAnalysisWithoutMathematicalFractalModel
 ]

 DISAMBIGUATION_RULE =
  "When geometry concerns recursive construction, self-similarity, non-integer dimension, Hausdorff or box-counting dimension, iterated function systems, Julia sets, Mandelbrot sets or scale-dependent structure, interpret the context as fractal geometry."
}

CORE_CONCEPTS {

 FractalGeometry {
  definition = "The study of geometric structure across scales, including recursively generated and dimensionally nonclassical sets."
 }

 Fractal {
  definition = "A geometric object exhibiting scale-sensitive structure, often with self-similarity or non-integer dimension."
 }

 SelfSimilarity {
  definition = "A structure in which parts reproduce the whole under scaling transformations."
 }

 StatisticalSelfSimilarity {
  definition = "Self-similarity expressed in distribution rather than exact deterministic replication."
 }

 HausdorffDimension {
  symbol = "dim_H"
  definition = "The critical exponent at which Hausdorff measure transitions from infinity to zero."
 }

 BoxCountingDimension {
  symbol = "dim_B"
  definition = "A dimension estimated from the scaling of the number of boxes required to cover a set."
 }

 SimilarityDimension {
  definition = "For self-similar sets with contraction ratios r_i, the value s satisfying sum_i r_i^s=1 under suitable separation conditions."
 }

 IteratedFunctionSystem {
  symbol = "IFS"
  definition = "A finite family of contraction mappings whose invariant compact set is an attractor."
 }

 Attractor {
  definition = "An invariant set approached under repeated application of a dynamical or iterative system."
 }

 Lacunarity {
  definition = "A scale-dependent measure of gap distribution or texture heterogeneity in a fractal set."
 }

 Multifractal {
  definition = "A structure requiring a spectrum of scaling exponents rather than a single fractal dimension."
 }
}

RELATIONS {
 FractalGeometry --is_branch_of--> AdvancedGeometry
 Fractal --may_have--> SelfSimilarity
 Fractal --has_measure--> FractalDimension
 HausdorffDimension --is_type_of--> FractalDimension
 BoxCountingDimension --is_type_of--> FractalDimension
 SimilarityDimension --is_type_of--> FractalDimension
 IteratedFunctionSystem --has--> ContractionMapping
 IteratedFunctionSystem --generates--> Attractor
 CantorSet --is_example_of--> Fractal
 SierpinskiTriangle --is_example_of--> Fractal
 KochCurve --is_example_of--> Fractal
 JuliaSet --is_generated_by--> ComplexDynamics
 MandelbrotSet --classifies--> QuadraticJuliaDynamics
 HausdorffMeasure --supports--> HausdorffDimension
 Multifractal --has--> SpectrumOfLocalDimensions
 Lacunarity --complements--> FractalDimension
}

OPERATORS {
 BOX_COUNTING_DIMENSION(F) :=
  limit_{epsilon->0} log(N_epsilon(F))/log(1/epsilon)
  WHEN limit_exists

 HAUSDORFF_DIMENSION(F) :=
  inf{s : H^s(F)=0}
  equivalent_to_sup{s : H^s(F)=infinity}
  under standard conditions

 SIMILARITY_DIMENSION({r_i}) :=
  unique_s such_that sum_i r_i^s=1
  WHEN 0<r_i<1

 IFS_ATTRACTOR({S_i}) :=
  unique_nonempty_compact_F such_that
  F = union_i S_i(F)
  WHEN S_i_are_contractions

 KOCH_DIMENSION :=
  log(4)/log(3)

 CANTOR_DIMENSION :=
  log(2)/log(3)

 SIERPINSKI_TRIANGLE_DIMENSION :=
  log(3)/log(2)

 MANDELBROT_MEMBERSHIP(c) :=
  orbit_of_0_under z->z^2+c remains_bounded

 JULIA_FILLED_MEMBERSHIP(z,c) :=
  orbit_of_z_under z->z^2+c remains_bounded

 ESCAPE_TIME(z,c,R,Nmax) :=
  first_n<=Nmax such_that abs(z_n)>R
  ELSE bounded_within_iteration_window

 LOCAL_DIMENSION(mu,x) :=
  limit_{r->0} log(mu(B(x,r)))/log(r)
  WHEN limit_exists
}

DEFINITIONS {
 D1 = "Fractal dimension need not be an integer."
 D2 = "Different fractal-dimension definitions may disagree on the same set."
 D3 = "Exact self-similarity is stronger than statistical self-similarity."
 D4 = "IFS attractors arise as fixed points of set-valued contraction operators."
 D5 = "Hausdorff dimension is intrinsic and theoretically robust but often difficult to compute."
 D6 = "Box-counting dimension is computationally accessible but can depend on limiting behaviour."
 D7 = "Similarity dimension equals Hausdorff dimension only under suitable separation hypotheses."
 D8 = "Julia and Mandelbrot sets arise from complex dynamical iteration."
 D9 = "Lacunarity measures gap structure beyond a single dimension value."
 D10 = "Multifractals require a spectrum of local scaling exponents."
}

AXIOMS {
 A1_SCALE_STRUCTURE:
  Fractal analysis concerns behaviour across changing observation scales

 A2_HAUSDORFF_MEASURE:
  Hausdorff dimension is defined through s-dimensional Hausdorff measures

 A3_BOX_SCALING:
  Box dimension is derived from covering-number scaling

 A4_IFS_FIXED_POINT:
  A finite contraction IFS has a unique nonempty compact invariant attractor

 A5_SIMILARITY_DIMENSION:
  Self-similar contraction ratios determine similarity dimension through sum_i r_i^s=1

 A6_COMPLEX_DYNAMICS:
  Julia and Mandelbrot structures are determined by orbit behaviour under iteration

 A7_DIMENSION_MONOTONICITY:
  If A subset B then dim_H(A)<=dim_H(B)

 A8_BILIPSCHITZ_INVARIANCE:
  Hausdorff dimension is preserved under bi-Lipschitz mappings
}

REASONING {

 R1_FRACTAL_CLASSIFICATION {
  INPUT = geometric_object_or_process
  IDENTIFY = recursive_selfsimilar_statistical_dynamical_or_measure_based structure
  RETURN = fractal_context
 }

 R2_BOX_DIMENSION {
  INPUT = set_F_and_cover_scales
  COMPUTE = covering_counts_N_epsilon
  FIT_OR_LIMIT = log_N_vs_log_inverse_scale
  RETURN = box_dimension_estimate_or_exact_value
 }

 R3_HAUSDORFF_DIMENSION {
  INPUT = set_F
  IDENTIFY = coverings_and_scaling_bounds
  ESTABLISH = critical_measure_exponent
  RETURN = Hausdorff_dimension
 }

 R4_SIMILARITY_DIMENSION {
  INPUT = contraction_ratios_r_i
  SOLVE = sum_i r_i^s=1
  VERIFY = suitable_separation_condition_if_equating_with_Hausdorff_dimension
  RETURN = similarity_dimension
 }

 R5_IFS_ATTRACTOR {
  INPUT = contractions_S_i
  VERIFY = contraction_property
  ITERATE = Hutchinson_operator_on_compact_sets
  CONVERGE = invariant_attractor
  RETURN = fractal_attractor
 }

 R6_COMPLEX_ESCAPE {
  INPUT = parameter_c_or_initial_point_z
  ITERATE = z_{n+1}=z_n^2+c
  TEST = escape_radius_condition
  RETURN = bounded_or_escape_classification
 }

 R7_MULTIFRACTAL_ANALYSIS {
  INPUT = measure_mu
  ESTIMATE = local_scaling_exponents
  BUILD = singularity_spectrum
  RETURN = multifractal_spectrum
 }

 R8_LACUNARITY {
  INPUT = fractal_set_and_scale_window
  COMPUTE = occupancy_distribution
  DERIVE = gap_heterogeneity_measure
  RETURN = lacunarity_profile
 }

 R9_DIMENSION_COMPARISON {
  INPUT = candidate_dimensions
  IDENTIFY = Hausdorff_box_similarity_information_or_correlation dimension
  COMPARE = only_under_shared_definitions_and_hypotheses
  RETURN = valid_relation_or_noncomparability
 }
}

CONSTRAINTS {
 C1 = "Fractal dimension must specify which definition is used."
 C2 = "Finite-resolution box-count estimates are approximations, not exact dimensions."
 C3 = "Similarity dimension does not automatically equal Hausdorff dimension."
 C4 = "Exact self-similarity must not be inferred from visual resemblance."
 C5 = "Escape-time iteration counts depend on bailout radius and iteration limit."
 C6 = "Julia and Mandelbrot computations approximate sets numerically."
 C7 = "Multifractal spectra depend on the chosen measure."
 C8 = "Hausdorff dimension can be non-integer, integer or even zero."
 C9 = "Topological dimension and fractal dimension must not be conflated."
 C10 = "Scaling ranges in empirical data must be justified before declaring fractality."
}

SEMANTIC_EQUIVALENCE {
 "FractalGeometry" <=> "GeometryOfScaleStructuredSets"
 "FractalDimension" <=> "ScaleDependentDimension"
 "IteratedFunctionSystem" <=> "FiniteContractionSystem"
 "IFSAttractor" <=> "InvariantCompactSetOfIFS"
 "BoxCountingDimension" <=> "CapacityDimension" WHEN conventional_usage
 "SelfSimilarity" <=> "ScaleReplicatingStructure"
}

VALIDATION {
 V1 = "Dimension definition is identified."
 V2 = "IFS maps are contractions where fixed-point theorem is invoked."
 V3 = "Similarity-dimension equality uses required separation assumptions."
 V4 = "Escape-time computations state iteration limit."
 V5 = "Empirical scaling uses multiple scales."
 V6 = "Topological dimension is distinguished from fractal dimension."
 V7 = "Multifractal analysis specifies the measure."
 V8 = "Numerical estimates are labelled approximate."
 V9 = "Recursive construction rules are internally consistent."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FRACTAL_GEOMETRY_DEFINITION = 0.99
 BOX_DIMENSION_REASONING = 0.98
 HAUSDORFF_DIMENSION_REASONING = 0.97
 SIMILARITY_DIMENSION_REASONING = 0.99
 IFS_REASONING = 0.99
 COMPLEX_DYNAMICS_REASONING = 0.98
 MULTIFRACTAL_REASONING = 0.96
 LACUNARITY_REASONING = 0.97

 POLICY =
  "Reduce confidence when dimension definition, separation condition, empirical scaling range, iteration limit, measure choice or exact-versus-numerical construction is underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FractalGeometryReasoning,
  FractalDimensionReasoning,
  HausdorffDimensionReasoning,
  BoxCountingReasoning,
  SelfSimilarityReasoning,
  IteratedFunctionSystemReasoning,
  ComplexDynamicsReasoning,
  MandelbrotReasoning,
  JuliaSetReasoning,
  MultifractalReasoning,
  LacunarityReasoning,
  ScalingLawReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "fractal geometry" -> FractalGeometry
  "fractal" -> Fractal
  "self-similar" -> SelfSimilarity
  "Hausdorff dimension" -> HausdorffDimension
  "box-counting dimension" -> BoxCountingDimension
  "Minkowski dimension" -> MinkowskiDimension
  "similarity dimension" -> SimilarityDimension
  "iterated function system" -> IteratedFunctionSystem
  "IFS" -> IteratedFunctionSystem
  "Julia set" -> JuliaSet
  "Mandelbrot set" -> MandelbrotSet
  "lacunarity" -> Lacunarity
  "multifractal" -> Multifractal
 }

 INTERPRETATION_ORDER = [
  detect_fractal_geometry_context,
  identify deterministic_statistical_dynamical_or_measure_based construction,
  identify dimension_definition_scaling_rule_or_iterative_process,
  identify exact_theoretical_or_empirical_numerical goal,
  apply fractal_specific scaling_dimension_or_dynamics reasoning,
  verify separation_scale_range_iteration_and_measure assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("fractal geometry")
  OR self_similarity_noninteger_dimension_context
  THEN activate(FractalGeometryOntology)

 E2:
  IF box_counting_dimension_requested
  THEN require covering_counts_across_scales

 E3:
  IF Hausdorff_dimension_requested
  THEN use Hausdorff_measure critical_exponent semantics

 E4:
  IF self_similar_IFS_with_ratios
  THEN solve sum_i r_i^s=1
  AND check separation_before equating_with_Hausdorff_dimension

 E5:
  IF IFS_attractor_requested
  THEN verify all_maps_are_contractions

 E6:
  IF Mandelbrot_membership_requested
  THEN iterate orbit_of_zero

 E7:
  IF Julia_membership_requested
  THEN iterate orbit_of_given_point

 E8:
  IF empirical_fractal_claim
  THEN require justified scaling_range

 E9:
  IF multifractal_context
  THEN specify measure_and_local_scaling exponents
}

RECURSION {
 ENTRY = FractalGeometry

 EXPAND(FractalGeometry) -> [
  Fractal,
  SelfSimilarity,
  StatisticalSelfSimilarity,
  HausdorffDimension,
  BoxCountingDimension,
  MinkowskiDimension,
  SimilarityDimension,
  IteratedFunctionSystem,
  ContractionMapping,
  Attractor,
  CantorSet,
  SierpinskiTriangle,
  SierpinskiCarpet,
  KochCurve,
  KochSnowflake,
  MengerSponge,
  JuliaSet,
  MandelbrotSet,
  ScalingLaw,
  Lacunarity,
  Multifractal,
  FractalMeasure,
  HausdorffMeasure,
  LocalDimension,
  StrangeAttractor
 ]

 EXPAND(FractalDimension) -> [
  HausdorffDimension,
  BoxCountingDimension,
  MinkowskiDimension,
  SimilarityDimension,
  CorrelationDimension,
  InformationDimension
 ]

 EXPAND(IteratedFunctionSystem) -> [
  ContractionMapping,
  HutchinsonOperator,
  FixedPointAttractor,
  OpenSetCondition,
  SimilarityDimension
 ]

 EXPAND(ComplexFractals) -> [
  JuliaSet,
  FilledJuliaSet,
  MandelbrotSet,
  EscapeTime,
  CriticalOrbit
 ]

 STOP_CONDITION =
  "Stop when the fractal construction, scaling law, fractal dimension, IFS attractor, dynamical orbit, lacunarity or multifractal structure is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Advanced Geometry"

 IMMEDIATE_CONTEXTS = [
  "Advanced Geometry",
  "Topology Ontology",
  "Metric Space Ontologies",
  "Complex Geometry Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Advanced Geometry",
  "Topology Ontology",
  "Metric Space Ontologies",
  "Measure Theory Ontologies",
  "Dynamical Systems Ontologies",
  "Complex Analysis Ontologies",
  "Complex Geometry Ontologies",
  "Probability & Statistics Ontologies",
  "Computational Geometry Ontology",
  "Calculus & Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Fractal Geometry -> Advanced Geometry for parent structural semantics",
  "Fractal Geometry -> Topology for connectedness and topological dimension",
  "Fractal Geometry -> Metric Spaces for coverings and dimensions",
  "Fractal Geometry -> Measure Theory for Hausdorff measure and multifractals",
  "Fractal Geometry -> Dynamical Systems for strange attractors and iterative fractals",
  "Fractal Geometry -> Complex Analysis for Julia and Mandelbrot sets",
  "Fractal Geometry -> Probability & Statistics for statistical self-similarity",
  "Fractal Geometry -> Computational Geometry for numerical approximation and rendering"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible fractal dimensions, constructions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve scale-structured geometry semantics.",
  "Preserve distinction among fractal-dimension definitions.",
  "Preserve exact-versus-statistical self-similarity distinction.",
  "Preserve IFS contraction requirements.",
  "Preserve similarity-dimension separation-condition caveat.",
  "Preserve numerical-versus-exact distinction for dynamical fractals.",
  "Preserve topological-versus-fractal dimension distinction.",
  "Preserve machine-readable fractal-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Fractal geometry studies geometric structures whose complexity and organisation persist or vary systematically across scale."

 DIMENSION_SUMMARY =
  "Fractal dimension must specify its definition, such as Hausdorff or box-counting dimension."

 HAUSDORFF_SUMMARY =
  "Hausdorff dimension is the critical exponent of Hausdorff measure."

 BOX_SUMMARY =
  "Box-counting dimension is obtained from covering-number scaling."

 IFS_SUMMARY =
  "A contraction IFS has a unique compact attractor satisfying F=union_i S_i(F)."

 SIMILARITY_SUMMARY =
  "Similarity dimension solves sum_i r_i^s=1 and may equal Hausdorff dimension under suitable separation conditions."

 COMPLEX_DYNAMICS_SUMMARY =
  "Julia and Mandelbrot sets are defined through boundedness and escape behaviour of iterated complex maps."

 MULTIFRACTAL_SUMMARY =
  "Multifractals require a spectrum of local scaling exponents rather than a single global dimension."

 LACUNARITY_SUMMARY =
  "Lacunarity describes the distribution of gaps and complements fractal-dimension information."
}

FLOW {
 INPUT
 -> detect_fractal_geometry_context
 -> identify fractal_set_process_metric_measure_scaling_rule_and_iteration data
 -> identify Hausdorff_box_similarity_IFS_complex_dynamics_lacunarity_or_multifractal goal
 -> distinguish exact_theoretical_from_empirical_or_numerical pathway
 -> apply fractal_dimension_scaling_fixed_point_or_escape reasoning
 -> verify dimension_definition_separation_scale_range_iteration_measure_and_precision constraints
 -> route to advanced_geometry topology metric_space measure_theory dynamical_systems complex_analysis statistics_or_computational_geometry ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Fractal Geometry Ontology defines scale-structured and recursively generated geometry. It supports exact and statistical self-similarity, Hausdorff, box-counting and similarity dimensions, iterated function systems, classical fractals, Julia and Mandelbrot sets, lacunarity, multifractals, fractal measures and scale-law reasoning while preserving distinctions between theoretical, numerical and empirical notions of fractality."
}

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

}

END_FLOW