torus
Let an agent explain the torus, relay geometry, topology, higher-dimensional and algebraic forms and applications from mathematics and physics sources, describe the named forms and flag the loosely filed alias, and distinguish the torus from the sphere, annulus, solid torus, Klein bottle and toroidal inductors as objects.
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 the torus, relay geometry, topology, higher-dimensional and algebraic forms and applications from mathematics and physics sources, describe the named forms and flag the loosely filed alias, and distinguish the torus from the sphere, annulus, solid torus, Klein bottle and toroidal inductors as objects.
A surface of revolution generated by revolving a circle in three-dimensional space about an axis in its plane that does not intersect the circle, forming a doughnut shape, and in topology any space homeomorphic to the product of two circles, a closed orientable surface of genus one; related forms the registry aliases name include the Clifford torus, a flat torus embedded in four-dimensional space, maximal tori, abelian subgroups central to Lie group theory, and supertoroids, generalised torus shapes used in computer graphics; the alias tie ring may refer to a ring-shaped mechanical or fastening part and appears loosely filed. Tori appear in physics as the shape of tokamak plasma chambers.
What it is for: Describing ring-shaped surfaces and spaces.
It can be explain geometry and topology; relay higher-dimensional and algebraic forms; describe named forms; relay applications.
Distinguishing features
Genus one
Product of circles
Flat metrics possible
Many applications
What it looks like
A doughnut or ring shape.
Physical character
genus: 1 count
Euler characteristic: 0 count
surface area: 4 pi squared R r formula - ring torus
How it is recognised
Doughnut-shaped surface
Tie ring, maximal torus, Clifford torus, supertoroid, ring torus
Spheres have genus zero; annuli are flat rings; solid tori include the interior; Klein bottles are non-orientable; toroidal inductors are components
Related models
is a kind of - in registry terms
is the product of - two circles
includes -
is contrasted with -
In practice
Families and kinds
ring, horn and spindle tori
flat tori such as the Clifford torus
n-dimensional tori
maximal tori in Lie groups
supertoroids
Standards and regulation
No regulation
Failure modes and hazards
Confusing torus and solid torus
Loosely filed mechanical alias
Confusing embedded and abstract tori
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.torus
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 torus is.
Definition.
Definition
Definition.
Definition
Definition.
- What is a torus, and how does it differ from a sphere, annulus, solid torus, Klein bottle and toroidal components? definition
- Is the question about geometry, topology, Lie groups, graphics or physics? boundary
Forms
Named forms.
Forms
Forms.
- What are Clifford tori, maximal tori and supertoroids, and what might tie ring refer to? definition
- Which entry fits the specific form? action
Geometry Geometry and topology.
Science.
Surface
Surface of revolution.
Surface
Surface.
- How are area and volume of a ring torus calculated? measurement
- Which references are standard? provenance
Topology
Topology.
Topology
Topology.
- Why is the torus a genus-one surface, and how is it built from a square? provenance
- Which sources are cited? provenance
Advanced Higher forms.
Science.
Clifford torus
Clifford torus.
Clifford torus
Clifford torus.
- How does the Clifford torus sit flat in four dimensions? provenance
- Which entry fits Clifford torus? action
Lie groups
Maximal tori.
Lie groups
Lie groups.
- What role do maximal tori play in the theory of compact Lie groups? provenance
- Which entry fits maximal torus? action
Context Applications.
Context.
Tokamak
Fusion devices.
Tokamak
Tokamak.
- Why are tokamak plasma chambers toroidal? provenance
- Which entry fits tokamak? action
Games
Toroidal worlds.
Games
Games.
- Why do wrap-around video game maps form a torus? provenance
- Which entry fits wraparound (video games)? action
What the second pass must settle
- The registry alias tie ring should be reviewed and moved to mechanical entries if appropriate
- Should maximal torus be a separate entry?
- How should topology references be linked?