# torus (1-torus.org)

A torus is a doughnut-shaped surface. People say it to describe anything with that rounded ring shape, from a life preserver to a fancy light fixture.

## What a model may hear
- compact Lie group (algebraic topology and Lie theory): assumes the user wants group operations, representations, or cohomology calculations on the circle group or its products
- algebraic torus (algebraic geometry): expects discussion of schemes isomorphic to products of multiplicative groups, often over arbitrary fields
- periodic domain (scientific computing and PDEs): sets boundary conditions to wrap around in one or more dimensions, treating space as a flat torus
- surface of revolution (differential geometry): computes Gaussian curvature, geodesics, or embeddings in three-space
- network topology (distributed systems): interprets as a ring with redundant cross-connections, a fault-tolerant mesh

## Where people and models part ways
- Says: "Draw me a torus"
  Means: a simple 3D doughnut picture
  May be taken as: generates a mathematical diagram with parameterization formulas, UV coordinates, and grid lines
  Say instead: "Draw a smooth doughnut-shaped 3D object, no formulas or grid lines"
- Says: "What is the volume of a torus"
  Means: the space inside a doughnut shape
  May be taken as: gives the topological invariant or assumes a flat torus with no interior volume
  Say instead: "What is the volume inside a doughnut-shaped solid with major radius R and minor radius r"
- Says: "Torus in my data"
  Means: a circular pattern or loop in visualization
  May be taken as: assumes the data lies on a manifold requiring dimensionality reduction with torus topology
  Say instead: "My data points form a circular loop when plotted, like a ring"
- Says: "Living on a torus"
  Means: a science fiction setting or thought experiment
  May be taken as: models physics on a flat torus with periodic boundary conditions, no gravity well
  Say instead: "A story set on a doughnut-shaped planet with normal gravity pointing toward the surface"

## Tips
- Say 'doughnut shape' if you want the everyday geometric solid.
- Specify 'flat torus' or 'periodic boundary conditions' for the wrapped-space computing sense.
- Add 'Lie group' or 'algebraic' to get the advanced math structures.
- Say 'network torus' or 'torus topology' for the distributed systems mesh.
- Mention 'embedded in 3D' to avoid the abstract flat version.

## Often confused with
- toroid: the solid body, not just the surface
- annulus: flat ring, not a 3D doughnut
- circle: one dimension lower, no hole through the middle
- manifold: general term, any smooth shape
- klein bottle: one-sided, non-orientable cousin
- genus: count of holes, not the shape itself