Quadrilateral Ontology

FLOW ALGEBRA
TITLE: Quadrilateral Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Geometric Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/quadrilateral-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed; webpage wording not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

[ONTOLOGY]
ENTITY: Quadrilateral
TYPE: SimplePolygon
DIMENSION: 2
DEFINITION: A simple planar polygon with exactly four distinct vertices and four straight edges, forming a closed non-self-intersecting boundary.
CONTEXT: Euclidean plane unless explicitly qualified.

[TYPED ENTITIES]
Point: EuclideanPoint2D
Vertex: Point
Edge: ClosedLineSegment
Diagonal: ClosedLineSegment
Angle: InteriorAngle
Quadrilateral: OrderedTuple[Point,Point,Point,Point]
Length: NonnegativeReal
Area: NonnegativeReal
Perimeter: NonnegativeReal
Vector: RealVector2
Boolean: {true,false}

[CONSTRUCTION]
INPUT: A,B,C,D : Point
REQUIRE: pairwise_distinct(A,B,C,D)
REQUIRE: simple_closed_polygon(A,B,C,D)
OUTPUT: Q = Quadrilateral(A,B,C,D)
EDGES: AB, BC, CD, DA
DIAGONALS: AC, BD
INTERIOR_ANGLE_SUM: 2*pi radians = 360 degrees
PERIMETER: |AB| + |BC| + |CD| + |DA|
AREA: abs(shoelace(A,B,C,D))/2
SHOELACE: sum(x_i*y_(i+1)-y_i*x_(i+1)) with cyclic indexing, i=1..4

[CLASSIFICATION]
ConvexQuadrilateral: all interior angles strictly less than 180 degrees.
ConcaveQuadrilateral: exactly one reflex interior angle greater than 180 degrees.
Parallelogram: AB parallel CD and BC parallel DA.
Rectangle: parallelogram with four right angles.
Rhombus: parallelogram with four equal side lengths.
Square: rectangle and rhombus.
TrapezoidInclusive: at least one pair of opposite sides parallel.
TrapezoidExclusive: exactly one pair of opposite sides parallel.
IsoscelesTrapezoid: exclusive trapezoid with equal nonparallel sides (Euclidean convention).
KiteInclusive: two pairs of adjacent equal sides; permits rhombi.
KiteExclusive: two distinct adjacent equal-side pairs excluding rhombi.
CyclicQuadrilateral: all vertices lie on one circle.
TangentialQuadrilateral: admits an incircle tangent to all four sides.
NOTE: Trapezoid and kite conventions differ across curricula; record convention explicitly.

[FORMAL RELATIONSHIPS]
Quadrilateral IS_A Polygon
Quadrilateral HAS_VERTEX Point [cardinality=4]
Quadrilateral HAS_EDGE Edge [cardinality=4]
Quadrilateral HAS_DIAGONAL Diagonal [cardinality=2]
Square SUBCLASS_OF Rectangle
Square SUBCLASS_OF Rhombus
Rectangle SUBCLASS_OF Parallelogram
Rhombus SUBCLASS_OF Parallelogram
Parallelogram SUBCLASS_OF Quadrilateral
CyclicQuadrilateral SUBCLASS_OF Quadrilateral
TangentialQuadrilateral SUBCLASS_OF Quadrilateral
Quadrilateral HAS_PROPERTY InteriorAngleSum
Quadrilateral HAS_MEASURE Area
Quadrilateral HAS_MEASURE Perimeter

[GEOMETRIC LAWS]
LAW-01: For any simple quadrilateral, sum(interior_angles)=360 degrees.
LAW-02: For any parallelogram, opposite sides are equal and parallel.
LAW-03: For any parallelogram, diagonals bisect one another.
LAW-04: For any rectangle, diagonals are equal in length.
LAW-05: For any rhombus, diagonals are perpendicular.
LAW-06: For any square, diagonals are equal, perpendicular and mutually bisecting.
LAW-07: For any cyclic quadrilateral, opposite interior angles sum to 180 degrees.
LAW-08: For any convex quadrilateral, area = 0.5*d1*d2*sin(theta), where theta is the angle between diagonals.
LAW-09: For any parallelogram, area = base*corresponding_height.
LAW-10: For any trapezoid with parallel bases a,b, area = 0.5*(a+b)*height.
LAW-11: For any rectangle, area = length*width.
LAW-12: For any square with side s, area=s^2 and perimeter=4*s.
LAW-13: For any convex tangential quadrilateral with side lengths a,b,c,d in cyclic order, a+c=b+d (Pitot's theorem).
LAW-14: For any cyclic quadrilateral with side lengths a,b,c,d and semiperimeter p, area=sqrt((p-a)*(p-b)*(p-c)*(p-d)) (Brahmagupta).
LAW-15: For any convex quadrilateral, sum(exterior_turning_angles)=360 degrees.

[INFERENCE RULES]
RULE-01: IF Q is a square THEN Q is a rectangle.
RULE-02: IF Q is a square THEN Q is a rhombus.
RULE-03: IF Q is a rectangle THEN Q is a parallelogram.
RULE-04: IF Q is a rhombus THEN Q is a parallelogram.
RULE-05: IF Q is a parallelogram THEN both pairs of opposite sides are parallel.
RULE-06: IF Q is a parallelogram THEN opposite angles are equal.
RULE-07: IF Q is a parallelogram THEN adjacent angles are supplementary.
RULE-08: IF Q is a parallelogram THEN diagonals bisect each other.
RULE-09: IF Q is a parallelogram AND one angle is right THEN Q is a rectangle.
RULE-10: IF Q is a parallelogram AND adjacent sides are equal THEN Q is a rhombus.
RULE-11: IF Q is a rectangle AND adjacent sides are equal THEN Q is a square.
RULE-12: IF Q is a rhombus AND one angle is right THEN Q is a square.
RULE-13: IF Q is a parallelogram AND diagonals have equal lengths THEN Q is a rectangle.
RULE-14: IF Q is a parallelogram AND diagonals are perpendicular THEN Q is a rhombus.
RULE-15: IF Q is a convex quadrilateral AND opposite angles are supplementary THEN Q is cyclic.
RULE-16: IF Q is cyclic THEN opposite angles are supplementary.
RULE-17: IF Q is a simple quadrilateral THEN Q has four edges.
RULE-18: IF Q is a simple quadrilateral THEN Q has two diagonals.
RULE-19: IF Q is convex THEN all four interior angles are less than 180 degrees.
RULE-20: IF Q is concave THEN one interior angle exceeds 180 degrees.
RULE-21: IF Q is a rectangle THEN its diagonals bisect each other.
RULE-22: IF Q is a rhombus THEN its diagonals bisect each other.
RULE-23: IF Q is a square THEN its diagonals bisect the vertex angles.
RULE-24: IF Q is a square THEN its four sides are equal.
RULE-25: IF Q is a rectangle THEN its four angles are right.
RULE-26: IF Q is a trapezoid under inclusive convention THEN at least one opposite-side pair is parallel.
RULE-27: IF Q is a trapezoid under exclusive convention THEN exactly one opposite-side pair is parallel.
RULE-28: IF Q is an isosceles trapezoid under exclusive convention THEN base angles are pairwise equal.
RULE-29: IF Q is a convex kite THEN one diagonal is a perpendicular bisector of the other.
RULE-30: IF Q is cyclic AND convex THEN Brahmagupta's area formula applies.
RULE-31: IF Q is convex AND tangential THEN Pitot's equality holds.
RULE-32: IF Q is a parallelogram THEN its diagonals intersect at their common midpoint.
RULE-33: IF Q is a square THEN Q is cyclic.
RULE-34: IF Q is a rectangle THEN Q is cyclic.
RULE-35: IF Q is a rhombus AND cyclic THEN Q is a square.
RULE-36: IF Q has four distinct vertices BUT crossing nonadjacent edges THEN reject as SimpleQuadrilateral.

[VALIDATION CONSTRAINTS]
V-01: Exactly four distinct ordered vertices.
V-02: Consecutive vertices joined by nonzero-length segments.
V-03: Nonadjacent edges do not intersect.
V-04: Adjacent edges intersect only at shared endpoints.
V-05: Polygon encloses nonzero area.
V-06: Distinguish convex and concave classification.
V-07: Angles use consistent units.
V-08: Coordinate system and metric must be declared.
V-09: Class definitions record trapezoid and kite convention.
V-10: Apply diagonal-based area and angle formulas only under their stated hypotheses.
V-11: Avoid inferring a converse unless separately proven.
V-12: Distinguish exact geometric claims from tolerance-based numerical approximations.

[EXAMPLES]
EX-01: (0,0),(2,0),(2,2),(0,2) => square; area=4; perimeter=8.
EX-02: (0,0),(4,0),(4,2),(0,2) => rectangle; area=8; perimeter=12.
EX-03: (0,0),(3,0),(4,2),(1,2) => parallelogram; area=6.
EX-04: (0,0),(4,0),(3,2),(1,2) => exclusive trapezoid; area=6.
EX-05: (0,0),(3,0),(1,1),(0,3) => concave quadrilateral; area=3.
EX-06: (0,0),(1,0),(1,1),(0,1) => cyclic square.
EX-07: A rectangle with unequal adjacent sides is not a square.
EX-08: A rhombus need not have right angles.
EX-09: A concave quadrilateral has interior-angle sum 360 degrees.
EX-10: A self-crossing four-vertex polygonal chain is not a simple quadrilateral.
EX-11: A square is a trapezoid under inclusive but not exclusive convention.
EX-12: A square is a kite under inclusive but not exclusive convention.
EX-13: Every square is both cyclic and tangential.
EX-14: A parallelogram has rotational symmetry of order two about its centre.
EX-15: A rectangle has reflection symmetries through its midlines.
EX-16: A generic quadrilateral need not possess any reflection symmetry.

[SEMANTIC_EQUIVALENCE]
Quadrilateral ~ FourSidedSimplePolygon [context=EuclideanPlane]
Square ~ RegularQuadrilateral [context=EuclideanGeometry]
Rectangle ~ EquiangularQuadrilateral [context=EuclideanGeometry]
Parallelogram ~ OppositeSidesParallelQuadrilateral [context=EuclideanGeometry]
CAUTION: Trapezoid != Trapezium without regional definition.
CAUTION: Square != Rectangle as class equality; Square is a subclass of Rectangle.

[KNOWLEDGE_GRAPH_TRIPLES]
(Quadrilateral, rdf:type, Polygon)
(Quadrilateral, hasVertexCount, 4)
(Quadrilateral, hasEdgeCount, 4)
(Quadrilateral, hasDiagonalCount, 2)
(Square, subClassOf, Rectangle)
(Square, subClassOf, Rhombus)
(Rectangle, subClassOf, Parallelogram)
(Rhombus, subClassOf, Parallelogram)
(Parallelogram, subClassOf, Quadrilateral)
(CyclicQuadrilateral, hasProperty, SupplementaryOppositeAngles)
(Quadrilateral, interiorAngleSumDegrees, 360)

[CONFIDENCE]
FORMAL_DEFINITIONS: 0.99
STANDARD_THEOREMS: 0.99
CLASSIFICATION_WITH_DECLARED_CONVENTIONS: 0.98
SOURCE_PAGE_ALIGNMENT: unverified
RULE: Confidence values are modelling annotations, not empirical probabilities.

[FLOW_HASH]
ALGORITHM: SHA-256
INPUT: Canonical UTF-8 bytes of final ontology content, excluding a stored digest field.
STATUS: Digest intentionally not asserted without canonicalisation and hash execution.

[UPDATE_MODEL]
VERSION: 4.0
UPDATED: 2026-10-13
CHANGE_POLICY: Preserve stable entity IDs; increment version when rules or definitions change.
REVALIDATE: theorem hypotheses, convention changes, examples, and relationship typing.
PROVENANCE: Independently formalised from standard Euclidean geometry; source URL supplied by user.

[AI_CONSUMPTION]
READ_MODE: Structured semantic flow
ENTITY_RESOLUTION: Typed, context-sensitive
INFERENCE_MODE: Deductive with explicit premises
VALIDATION_MODE: Constraint checking before rule execution
CONVENTION_HANDLING: Require explicit trapezoid/kite convention before subtype inference
AMBIGUITY_POLICY: Do not infer exclusive classes from inclusive labels.
EXPLANATION_POLICY: Return rule IDs and supporting premises for conclusions.
FAILURE_POLICY: Return unknown when geometry or hypotheses are insufficient.

[END FLOW ALGEBRA]