Torus

Lore & Background

The torus is defined as a surface of revolution created by rotating a circle around a coplanar axis. When the axis does not touch the circle, the result is a ring torus; when tangent, a horn torus; when it passes twice through the circle, a spindle torus. If the axis passes through the circle's center, the surface degenerates to a sphere. Real-world approximations include swim rings, inner tubes, and ringette rings. A solid torus, formed by rotating a disk rather than a circle, includes the interior volume and is approximated by O-rings, lifebuoys, ring doughnuts, and bagels.

Reader's Guide

The torus is a fundamental shape in geometry and topology, bridging intuitive real-world objects with abstract mathematical concepts. Its parametric and implicit equations allow precise description in three-dimensional space, while its topological characterization as the product of two circles (S¹ × S¹) makes it a key example of a compact 2-manifold of genus 1. The distinction between a torus (surface) and a solid torus (volume) clarifies applications from engineering (O-rings) to food (bagels). The aspect ratio R/r determines the torus's form—ring, horn, or spindle—and the degenerate cases (R=0 gives a sphere; r=0 gives a circle) illustrate continuity in geometric families. In topology, the torus's homeomorphism to a coffee cup surface demonstrates the field's focus on intrinsic properties rather than specific shape. The construction by joining opposite edges of a rectangle without half-twists provides a simple model for understanding its topology.

Did You Know?

• A ring torus is sometimes colloquially referred to as a doughnut.
• If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, specifically a sphere.
• The Clifford torus is a geometric object in 4-space produced by the Cartesian product of the embedding of S¹ in the plane with itself.
• The surface of a coffee cup and a doughnut are both topological tori with genus one.