Sphere Ontology

FLOW ALGEBRA
ONTOLOGY: Sphere Ontology
VERSION: 4.0
DOMAIN: Mathematics / Geometric Entities / Euclidean Geometry
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/sphere-ontology-2.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_ALIGNMENT: Independent mathematical formalisation; source page content not verified.

1. PURPOSE
Represent spheres as mathematically typed geometric entities, with precise distinctions between sphere surfaces and solid balls; support geometric reasoning, validation, retrieval, and knowledge graph integration.

2. TYPED ENTITIES
Sphere: {centre: Point3, radius: RealPositive, ambient_space: Euclidean3, boundary: Surface2}
Ball: {centre: Point3, radius: RealPositive, boundary: Sphere, interior: SetPoint3}
Point3: (x: Real, y: Real, z: Real)
Radius: r in Real, r > 0
Diameter: d in Real, d = 2r
GreatCircle: intersection(Sphere, PlaneThroughCentre)
SmallCircle: intersection(Sphere, PlaneNotThroughCentre) where intersection nonempty and non-tangent
TangentPlane: plane meeting sphere at exactly one point
SecantPlane: plane intersecting sphere in a circle
Chord: segment joining two distinct points on sphere
AntipodalPair: (p,q) where q = 2c - p
SphericalCap: portion of sphere cut by a plane
Hemisphere: one of two halves separated by a plane through centre

3. FORMAL DEFINITIONS
SPHERE(c,r) := {p in R^3 : ||p-c||_2 = r}, r>0
CLOSED_BALL(c,r) := {p in R^3 : ||p-c||_2 <= r}
OPEN_BALL(c,r) := {p in R^3 : ||p-c||_2 < r}
SPHERE(c,r) = BOUNDARY(CLOSED_BALL(c,r))
INTERIOR(SPHERE(c,r)) = empty_set (in ambient R^3)
DIMENSION_TOPOLOGICAL(SPHERE(c,r)) = 2

4. COORDINATE AND PARAMETRIC MODELS
Cartesian: (x-a)^2+(y-b)^2+(z-c)^2=r^2
Parametric: p(theta,phi) = centre + r(sin(phi)cos(theta), sin(phi)sin(theta), cos(phi))
Domain: 0 <= theta < 2*pi; 0 <= phi <= pi
Implicit function: F(x,y,z) = (x-a)^2+(y-b)^2+(z-c)^2-r^2
Normal at p: n(p) = (p-centre)/r
Tangent plane at p: (p-centre) dot (x-p) = 0

5. MEASURES
Sphere surface area: A=4*pi*r^2
Ball volume: V=4*pi*r^3/3
Great-circle circumference: C=2*pi*r
Diameter: d=2*r
Surface area as diameter: A=pi*d^2
Ball volume as diameter: V=pi*d^3/6
Area-to-volume ratio (ball): A/V=3/r
Spherical cap surface area (height h): A_cap=2*pi*r*h, 0<=h<=2r
Spherical cap solid volume: V_cap=pi*h^2*(r-h/3), 0<=h<=2r

6. RELATIONSHIPS
Sphere IS_A GeometricEntity
Sphere IS_A Surface
Sphere BOUNDS Ball
Sphere HAS_CENTRE Point3
Sphere HAS_RADIUS Radius
Sphere HAS_DIAMETER Diameter
Sphere INTERSECTS Plane
Sphere HAS_GREAT_CIRCLE GreatCircle
Sphere HAS_TANGENT_PLANE TangentPlane
Sphere IS_INVARIANT_UNDER RotationAboutCentre
Sphere IS_HOMEOMORPHIC_TO StandardTwoSphere
Ball HAS_BOUNDARY Sphere
GreatCircle IS_INTERSECTION_OF Sphere AND PlaneThroughCentre

7. GEOMETRIC CONSTRAINTS
C01: radius > 0 for non-degenerate sphere
C02: all sphere points have distance radius from centre
C03: tangent plane normal parallel to radius at tangency point
C04: great circle plane passes through centre
C05: great circle radius equals sphere radius
C06: antipodal points lie at Euclidean distance 2r
C07: all diameters pass through centre
C08: a plane at distance h from centre intersects sphere iff h<=r
C09: plane section radius sqrt(r^2-h^2) for 0<=h<r
C10: h=r gives tangent singleton, not a positive-radius circle
C11: surface area has squared-length units
C12: enclosed ball volume has cubed-length units
C13: sphere surface has zero three-dimensional Lebesgue volume
C14: scaling by factor k>0 changes area by k^2 and volume by k^3
C15: not every closed curved surface is a sphere
C16: spherical surface and solid ball are distinct entity types

8. INFERENCE RULES
R01 IF p belongs_to Sphere(c,r) THEN norm(p-c)=r
R02 IF norm(p-c)=r AND r>0 THEN p belongs_to Sphere(c,r)
R03 IF radius=r THEN diameter=2r
R04 IF radius=r THEN surface_area=4*pi*r^2
R05 IF ball_radius=r THEN ball_volume=4*pi*r^3/3
R06 IF surface_area=A>0 THEN radius=sqrt(A/(4*pi))
R07 IF ball_volume=V>0 THEN radius=cuberoot(3V/(4*pi))
R08 IF diameter=d>0 THEN radius=d/2
R09 IF plane_distance_from_centre>r THEN intersection=empty_set
R10 IF plane_distance_from_centre=r THEN intersection=tangent_point
R11 IF plane_distance_from_centre<r THEN intersection=circle
R12 IF plane_distance_from_centre=0 THEN section=great_circle
R13 IF section=great_circle THEN section_radius=r
R14 IF section_plane_distance=h AND 0<h<r THEN section_radius=sqrt(r^2-h^2)
R15 IF p belongs_to Sphere(c,r) THEN tangent_plane_normal=p-c
R16 IF p belongs_to Sphere(c,r) THEN antipode=2c-p
R17 IF points p,q are antipodal THEN distance(p,q)=2r
R18 IF scale_factor=k>0 THEN new_radius=k*r
R19 IF scale_factor=k>0 THEN new_surface_area=k^2*A
R20 IF scale_factor=k>0 THEN new_ball_volume=k^3*V
R21 IF sphere_radius=r THEN mean_curvature_magnitude=1/r (average-principal-curvature convention)
R22 IF sphere_radius=r THEN Gaussian_curvature=1/r^2
R23 IF sphere_radius=r THEN principal_curvatures_have_magnitude=1/r
R24 IF surface_is_sphere THEN surface_is_closed_and_bounded
R25 IF surface_is_sphere THEN surface_is_connected
R26 IF surface_is_sphere THEN surface_is_orientable
R27 IF surface_is_sphere THEN Euler_characteristic=2
R28 IF surface_is_sphere THEN genus=0
R29 IF sphere_centre=c THEN rotations_about_c preserve_sphere
R30 IF point p lies_inside_open_ball THEN norm(p-c)<r
R31 IF point p lies_outside_closed_ball THEN norm(p-c)>r
R32 IF cap_height=h AND 0<=h<=2r THEN cap_area=2*pi*r*h
R33 IF cap_height=h AND 0<=h<=2r THEN cap_ball_volume=pi*h^2*(r-h/3)
R34 IF r>0 THEN sphere_is_boundary_of_ball
R35 IF r>0 THEN surface_dimension=2 AND ambient_dimension=3
R36 IF metric_is_Euclidean AND centre_and_radius_known THEN sphere_is_uniquely_determined

9. SEMANTIC_EQUIVALENCE
SphereSurface ~ 2Sphere ~ S2 (when standard unit sphere is intended)
UnitSphere ~ Sphere(origin,1) (relative to stated Euclidean coordinate space)
SphericalSurface ~ Sphere (geometry context)
SolidSphere ~ ClosedBall (only where explicitly defined as solid region)
GreatCircle ~ CentralPlaneSectionOfSphere
CAUTION: Sphere is NOT unconditionally equivalent to Ball.
CAUTION: Circle is a one-dimensional boundary in a plane; sphere is a two-dimensional surface in 3D.

10. KNOWLEDGE GRAPH TRIPLES
(Sphere, rdf:type, GeometricEntity)
(Sphere, subClassOf, Surface)
(Sphere, hasParameter, Radius)
(Sphere, hasParameter, Centre)
(Sphere, bounds, Ball)
(Sphere, hasSection, GreatCircle)
(Sphere, admitsOperation, Rotation)
(Sphere, hasInvariant, ConstantGaussianCurvature)
(Sphere, relatedTo, Circle)
(Sphere, relatedTo, SphericalGeometry)

11. WORKED EXAMPLES
E01: c=(0,0,0), r=3 => x^2+y^2+z^2=9.
E02: r=3 => surface area=36*pi.
E03: r=3 => enclosed ball volume=36*pi.
E04: r=5 => diameter=10.
E05: c=(1,2,3), r=2 => (x-1)^2+(y-2)^2+(z-3)^2=4.
E06: c=(0,0,0), r=5, plane z=3 => section radius=4.
E07: c=(0,0,0), r=5, plane z=5 => tangent at (0,0,5).
E08: c=(0,0,0), r=5, plane z=6 => no intersection.
E09: p=(3,0,0), c=origin, r=3 => antipode=(-3,0,0).
E10: r=2, h=1 => spherical cap area=4*pi.
E11: r=2, h=1 => cap ball volume=5*pi/3.
E12: r=2, scale=3 => new r=6, area factor=9, volume factor=27.
E13: r=1 => Gaussian curvature=1.
E14: r=4 => great circle circumference=8*pi.
E15: c=(1,1,1), p=(1,1,4), r=3 => tangent plane z=4.
E16: r=1 => sphere surface area=4*pi, ball volume=4*pi/3.

12. VALIDATION MODEL
V01 REQUIRE declared geometry: Euclidean R^3 unless explicitly overridden
V02 REQUIRE radius is numeric, finite, strictly positive
V03 REQUIRE centre has exactly three finite real coordinates
V04 REQUIRE unit consistency for radius, area and volume
V05 REQUIRE distinguish boundary area from enclosed volume
V06 REQUIRE domain guards before applying cap or section formulas
V07 REQUIRE point membership validated using metric and tolerance when numeric
V08 REQUIRE floating-point comparisons use explicit tolerance, not exact equality
V09 REQUIRE topological claims apply to non-degenerate full sphere
V10 REQUIRE geodesic distance and Euclidean chord distance not conflated
V11 REQUIRE sphere surface not confused with ellipsoid
V12 REQUIRE inference provenance and preconditions retained

13. CONFIDENCE
Mathematical definition and standard formulas: HIGH, conditional on stated Euclidean assumptions.
Derived inference: HIGH when preconditions are satisfied.
Numerical point membership: CONDITIONAL on measurement precision and tolerance.
Website alignment: UNVERIFIED; source page content not retrieved.
Confidence values describe evidence and validation, not a substitute for proof.

14. FLOW_HASH
Algorithm: SHA-256
Canonicalisation: UTF-8, LF newlines, normalised key ordering, exclude FLOW_HASH output field.
FLOW_HASH_VALUE: TO_BE_COMPUTED_BY_DEPLOYMENT_PIPELINE
Purpose: detect changes, support deduplication and integrity verification.

15. UPDATE_MODEL
Trigger: revised source ontology, changed mathematical scope, correction to axioms, new relationships, or revised inference rule.
On update: validate formulas, compare entity definitions, increment semantic revision, regenerate canonical hash, retain provenance and changelog.
Conflict policy: prefer explicitly scoped mathematical definitions; flag unresolved ambiguity rather than silently merge sphere and ball.

16. AI_CONSUMPTION
Parse typed entities and relations before applying inference.
Resolve 'sphere' versus 'solid sphere' from context; request disambiguation if necessary.
Check ambient dimension, metric, coordinate frame and units.
Apply only rules with satisfied guards; record inferred facts separately from asserted facts.
Use formula provenance and confidence metadata in retrieval and reasoning.
Expose validation errors for nonpositive radius, invalid plane-section assumptions and incompatible units.
Do not treat descriptive text as a proof without checking mathematical conditions.

17. CROSS-ONTOLOGY LINKS
Geometric Entity Ontology
Point Ontology
Circle Ontology
Surface Ontology
Plane Ontology
Curve Ontology
Polyhedron Ontology
Dimension Ontology
Vector Ontology
Distance and Metric Ontologies
Topology and Manifold Ontologies

END FLOW ALGEBRA