ball
Let an agent explain balls in mathematics and their kinds, relay definitions, properties and uses in topology, analysis and coding theory from mathematics references, describe unit and Hamming balls, and distinguish balls from spheres, discs, neighbourhoods and physical balls.
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.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain balls in mathematics and their kinds, relay definitions, properties and uses in topology, analysis and coding theory from mathematics references, describe unit and Hamming balls, and distinguish balls from spheres, discs, neighbourhoods and physical balls.
In mathematics, the set of all points within a given distance of a centre point, a solid figure generalising the disc and the solid sphere to any metric space, either open, excluding the boundary, or closed, including it, with the unit ball of radius one in a normed space and the Hamming ball of strings within a given Hamming distance in coding theory; balls are convex in normed spaces and are the basic neighbourhoods of metric topology.
What it is for: Not applicable; a mathematical object.
It can be explain the concept; relay properties; describe kinds; distinguish related concepts.
Distinguishing features
Metric definition
Open and closed
Convex in normed spaces
Basis of topology
What it looks like
Not a visible object; a solid sphere in three dimensions.
Physical character
volume, 3D radius r: 4/3 pi r cubed note
unit ball volume, dimension n: tends to zero as n grows note
Hamming ball size: sum of binomial coefficients note
How it is recognised
Set of points within a distance of a centre
Open ball, closed ball, unit ball, closed unit ball, Hamming ball
A sphere is the boundary; a disc is a 2D ball; neighbourhoods are more general; physical balls are objects
Related models
is a kind of - in normed spaces, in registry terms
is a kind of - in registry terms
is bounded by - its boundary
is used in - as a neighbourhood
In practice
Families and kinds
open and closed balls in metric spaces
unit balls in normed spaces
balls in Euclidean space
Hamming balls in coding theory
p-adic and ultrametric balls
Standards and regulation
No regulation
Failure modes and hazards
Confusing ball with sphere
Assuming Euclidean intuition in other metrics
Registry aliases mixing analysis and coding theory
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.ball-act
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What a ball is.
Science.
Definition
Definition.
Definition
Definition.
- What is a ball in mathematics, and how does it differ from a sphere, disc, neighbourhood and physical ball? definition
- Is the question about topology, analysis, geometry or coding theory? boundary
Kinds
Kinds.
Kinds
Kinds.
- What are open, closed, unit and Hamming balls? definition
- Which entry fits the specific kind? action
Properties Properties.
Science.
Geometry
Geometry.
Geometry
Geometry.
- What are the volume and surface formulas, and how do they behave in high dimensions? provenance
- Which references are standard? provenance
Topology
Topology.
Topology
Topology.
- How do balls define open sets and neighbourhoods in metric spaces? provenance
- Which sources are cited? provenance
Applications Applications.
Application.
Analysis
Analysis.
Analysis
Analysis.
- How are unit balls used in functional analysis and convexity? provenance
- Which entry fits normed vector space? action
Coding
Coding theory.
Coding
Coding.
- How are Hamming balls used in error-correcting code bounds? provenance
- Which entry fits Hamming bound? action
Context Beyond Euclid.
Context.
Metrics
Other metrics.
Metrics
Metrics.
- How do balls look in taxicab, ultrametric and p-adic spaces? provenance
- Which entry fits ultrametric space? action
History
History.
History
History.
- How did the abstract ball arise with metric spaces? provenance
- Which entry fits the history of topology? action
What the second pass must settle
- Should unit ball and Hamming ball be separate primary entries?
- How should mathematics references be linked?
- The registry entry has merged aliases from analysis and coding theory; should they be split off?