disk
Enable an agent to recognise a geometric disk, record its boundary and geometric assumptions, and determine which measurements, comparisons and transformations are valid.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an agent to recognise a geometric disk, record its boundary and geometric assumptions, and determine which measurements, comparisons and transformations are valid.
In plane geometry, a disk is the region consisting of all points whose distance from a fixed center is less than a given positive radius (open disk) or less than or equal to it (closed disk).
It can be Test whether a point belongs to an open or closed Euclidean disk.; Calculate Euclidean area and boundary length from a known radius.; Classify two Euclidean disks as disjoint, overlapping, tangent, nested or coincident.; Construct a disk from a centre, radius and boundary convention.; Apply a transformation and determine whether its image remains a Euclidean disk.; Assess a topological disk claim against a specified reference space and supporting argument..
Distinguishing features
A Euclidean disk contains all planar points whose distance from a fixed centre is less than, or at most, a specified positive radius.
A circle contains only points at the specified radius; a disk includes points closer to the centre.
An annulus with a positive inner radius excludes a central region that a Euclidean disk includes.
An open disk excludes its boundary; a closed disk includes it.
A topological disk is identified through a stated homeomorphism convention, which does not supply a Euclidean centre or radius.
Scope
+ Open and closed disks in the Euclidean plane
+ Centre, radius, interior and boundary relationships
+ Point membership and relationships between disks
+ Area, boundary length and transformations under stated geometric assumptions
+ Topological uses of disk, with their conventions explicitly recorded
- Computer storage disks, drives and their data
- Physical disk-shaped components and their material properties
- Circles considered solely as boundary curves
- Annuli and punctured disks as independent kinds of region
- Higher-dimensional balls and spheres
- General ellipses and arbitrary planar regions
Characteristics
- Interpretation
- Euclidean open disk | Euclidean closed disk | topological open disk | topological closed disk | unresolved Determines which recognition tests and geometric operations apply.
- Ambient space and metric
- Reference to the containing space and its distance function, where applicable Distance and boundary depend on the space and metric being used.
- Centre
- Point in the stated Euclidean plane; not intrinsic to a purely topological disk Anchors Euclidean membership, intersection and transformation calculations.
- Radius
- Positive real length in stated units; zero requires an explicit degeneracy convention Determines Euclidean extent, area and boundary length.
- Boundary inclusion
- Included | excluded | unresolved Changes membership at the edge and whether tangency creates an intersection.
- Area
- Squared length units; πr² for a Euclidean disk Supports size comparison under a specified measure.
- Boundary length
- Length units; 2πr for the Euclidean boundary circle Records edge extent even when the boundary is excluded from the disk.
- Membership determination
- Inside | on boundary | outside | indeterminate under available precision Separates mathematical classification from uncertainty in numerical inputs.
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.disk
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 9 layers · 15 findings · 23 questions.
Disk sense and conventions Establish which mathematical meaning of disk governs the instance.
The name alone does not resolve geometric versus topological meaning or boundary inclusion.
Geometric or topological
Separate distance-defined disks from spaces identified by topological equivalence.
Governing disk definition
Record the adopted definition before assigning centre, radius or metric properties.
- Does disk mean a Euclidean distance-defined region or a space homeomorphic to a specified standard disk? definition
- Which registry context or mathematical reference supports this interpretation? provenance
Boundary and degeneracy
Make edge inclusion and exceptional radius conventions explicit.
Admissible radius and edge
Distinguish strict and non-strict distance inequalities and identify any admitted degenerate cases.
- Is membership defined by distance less than the radius or less than or equal to it? boundary
- Must the radius be positive, or does the adopted convention admit a zero-radius case? definition
Euclidean extent and measure Record the information needed to locate and measure a Euclidean disk.
A centre and radius support calculations only within an identified plane, metric and unit system.
Centre, radius and plane
Establish the geometric inputs that determine the disk.
Distance-based construction
Record a centre point, radius and containing plane without confusing a planar disk embedded in three dimensions with a solid ball.
- What centre, radius, units and Euclidean plane determine this disk? measurement
- If coordinates have three components, what restricts the region to a plane rather than a three-dimensional ball? boundary
Area and boundary length
Derive measures while preserving their geometric assumptions.
Valid Euclidean measures
For a Euclidean disk of radius r, record area πr² and boundary-circle length 2πr; edge exclusion does not change its area.
- What area and boundary length follow from the radius in the declared units? measurement
- Is a Euclidean metric available, or would assigning these measurements to a topological disk require additional structure? boundary
Membership and disk relations Support point classification and spatial comparisons between Euclidean disks.
Boundary conventions affect containment and intersection, especially at exact contact.
Point membership
Evaluate distance against the radius and distinguish uncertainty from edge inclusion.
Interior, edge and exterior test
Compare point-to-centre distance with the radius, then apply the disk's boundary convention to decide membership.
- Is the point-to-centre distance smaller than, equal to or greater than the radius? measurement
- If finite precision prevents a reliable comparison, should the result remain indeterminate or follow a declared approximation rule? action
Pairwise position
Compare centre separation and radii for disks in the same Euclidean plane.
Overlap, containment and contact
Use centre separation, radii and edge inclusion together to distinguish boundary-circle contact from actual disk intersection.
- How does centre separation compare with the sum and absolute difference of the two radii? measurement
- At a contact point or coincident boundary, do the inclusion conventions permit intersection or containment? boundary
Transformations and topological equivalence Determine what survives when a disk is moved, scaled or continuously deformed.
Preserving disk topology does not necessarily preserve circular geometry or its measurements.
Euclidean disk transformations
Track the effects of transformations on centre, radius and circular shape.
Disk preservation test
Euclidean similarities with positive scale preserve disks; unequal scaling along perpendicular axes produces an ellipse instead.
- Does the proposed transformation preserve distances up to one positive scale factor? action
- What are the transformed centre, radius, area and boundary length if it does? measurement
Topological disk claims
Evaluate equivalence to an open or closed standard disk without importing unsupported metric properties.
Homeomorphism and boundary context
Record the reference disk and evidence of homeomorphism, distinguishing a manifold boundary from a boundary relative to an ambient space.
- What homeomorphism or applicable theorem supports equivalence to the chosen open or closed standard disk? provenance
- Does boundary refer to the space's manifold boundary or its boundary in a specified ambient space? definition
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- The geometric sense is inferred from the domain context; the registry supplies no explicit sense.
- Unqualified 'disk' can mean open or closed depending on the author's convention.
- A topological disk is defined by homeomorphism to a standard disk, rather than by a center and radius; boundary conventions should be checked.
- Which of these check these first hold for the sense of disk this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Open disk
- Closed disk
- Unit disk
- Topological disk
- Which of these kinds and varieties hold for the sense of disk this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Representing circular regions in geometry and spatial modelling
- Defining neighborhoods in mathematical analysis
- Providing domains for complex functions and boundary-value problems
- Modelling idealized circular cross-sections
- Which of these real-world use hold for the sense of disk this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Radius - Any positive real value; no universal typical range - Length units, or dimensionless in abstract mathematics
- Area - πr² for a Euclidean disk of radius r - Square length units, or dimensionless
- Boundary circumference - 2πr for a Euclidean disk of radius r - Length units, or dimensionless
- Which of these typical measurements hold for the sense of disk this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Confusing the disk with its boundary circle
- Leaving boundary inclusion unspecified when it affects membership, compactness, or existence of extrema
- Applying Euclidean area and circumference formulas to a topological disk without specifying its geometry
- Which of these failure modes and hazards hold for the sense of disk this model covers, and on what evidence? provenance
Regional variation
Recalled without web access and unsourced; every item is a lead to verify.
- Both 'disk' and 'disc' occur in mathematical English; spelling does not reliably distinguish mathematical senses.
- Which of these regional variation hold for the sense of disk this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- circle - A circle consists only of points exactly one radius from the center; a disk includes the interior.
- ball - A Euclidean disk is a two-dimensional ball; balls also occur in other dimensions and metric spaces.
- annulus - An annulus has a central hole bounded by an inner circle; a disk has no such hole.
- physical disk - A physical disk is a material object with thickness; a geometric disk is an abstract two-dimensional region.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of disk this model covers, and on what evidence? provenance
What the second pass must settle
- Does the registry's originating record confirm the geometric sense of disk rather than another abstract or physical sense?
- Does an existing Vercy world model already own this concept and require the registry entry to link to it?
- Should an unqualified disk default to a closed Euclidean disk, or must every instance declare its convention?
- Should topological disks and disks defined using non-Euclidean metrics remain within this entry, and what limits should govern those extensions?
- Which source-backed convention should govern zero-radius cases and their recognition as disks?