# torus (0-torus.org)

A torus is a doughnut-shaped surface. People picture a ring you could swim through, or the shape of a bagel or an inner tube.

## What a model may hear
- compact Lie group (algebraic topology and geometry): assume the user means the product of circles, often with dimension or rank implied by context
- network topology (distributed systems and parallel computing): route messages or place nodes on a wrap-around grid, expecting coordinates and neighbor rules
- algebraic torus (algebraic geometry and number theory): expect equations over fields, possibly involving character groups or Galois actions
- flat torus (Riemannian geometry): assume a quotient of Euclidean space by a lattice, with flat metric and periodic boundary conditions

## Where people and models part ways
- Says: "draw me a torus"
  Means: a 3D doughnut shape
  May be taken as: render a flat torus, a wireframe lattice, or a group diagram with no visual form
  Say instead: "draw a 3D doughnut-shaped surface"
- Says: "connect the nodes in a torus"
  Means: a ring-like network
  May be taken as: build a multi-dimensional wrap-around mesh with complex routing
  Say instead: "connect the nodes in a ring network with wrap-around links"
- Says: "the torus has a hole"
  Means: one visible hole in a doughnut
  May be taken as: discuss genus-one topology or fundamental groups
  Say instead: "the doughnut shape has one hole through the middle"

## Tips
- Say 'doughnut shape' if you want a picture, 'ring network' if you want connections, 'circle product' if you want the math object.
- Specify dimensions only if you mean them; '2-torus' means surface, '3-torus' means space, and models default differently.
- Mention 'flat' or 'curved' to clarify which geometry you intend.
- If you mean the network, say how many nodes and whether you want one ring or a grid with wrap-around.
- Avoid 'torus' alone for jewelry or anatomy; say 'ring shape' or 'tubular structure' instead.

## Often confused with
- toroid: any doughnut-like shape, not necessarily a surface of revolution
- annulus: a flat ring, not a 3D solid
- cylinder: no wrap-around, two open ends
- Klein bottle: one-sided, non-orientable cousin
- genus: a count of holes, not the shape itself
- mesh: any connected grid, not necessarily wrap-around