# Torus

A torus is the surface of revolution of a circle. That circle's radius is called the minor radius while the distance from the center of the circle to the center of the torus is called the major radius of the torus. If the major radius is greater than the minor radius, it is called a ring torus. If the major radius is equal to the minor radius, it is called a horn torus. If the minor radius is greater, causing self-intersection, it is a spindle torus.

Torus
Dimensions2
ConnectedYes
CompactYes
Euler characteristic0
OrientableYes
Genus1
SymmetryO(2)×O(2) (4-space)
O(2)×A1 (3-space)
Torus (solid)
Rank3
Notation
Toratopic notation((II)I)
Elements
Faces1 torus (surface)
Measures (edge length 1)
Volume${\displaystyle 2\pi^2Rr^2}$
Related polytopes
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryO(2)×A1
ConvexNo

It can also be formed by inflating a circle in one dimension.

It is represented as ((II)I) in toratopic notation.

Its expanded rotatope is the duocylinder.

## Coordinates

Where r is the minor radius and R is the major radius:

Points on the surface of a torus are all points (x,y,z) such that

• ${\displaystyle \left(\sqrt{x^2+y^2}-R\right)^2+z^2=r^2.}$

Points in the interior of a torus are all points (x,y,z) such that

• ${\displaystyle \left(\sqrt{x^2+y^2}-R\right)^2+z^2

## Clifford torus

The Clifford torus or flat torus is a shape in 4D space which is homeomorphic to the usual torus in 3D space. The Clifford torus is the Cartesian product of two circles. Since circles live in 2D space their Cartesian product is 4D. The Clifford torus has a higher degree of symmetry than the usual torus (${\displaystyle O(2)\times O(2)}$  versus ${\displaystyle O(2)\times A_1}$ ). The space on the inner parts of the usual torus is different from the space on the outer parts (e.g. there inside has negative curvature and the outside has positive curvature). In the Clifford torus the symmetry acts transitively on every point in the torus, and thus they are all indistinguishable (and have zero curvature).

Since the comb product corresponds to taking the Cartesian product of a polytope's surface, the Clifford torus appears as the surface of n-gonal duocombs. While polyhedra with surfaces corresponding to the usual torus can never be regular, because of the high degree of symmetry the Clifford torus, skew polyhedra with the surface of the Clifford torus, such as the n-gonal duocombs, can be regular.