# 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${\displaystyle 2\pi ^{2}Rr^{2}}$
Related polytopes
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryO(2)×A1
ConvexNo

A torus is the surface of revolution of a circle, visually forming the surface of a donut. It is a two-dimensional surface in 3D Euclidean space.

The 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.

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 defined as the Cartesian product of two circles. (It is not required for the circles to have the same radius.) It requires 4D Euclidean space, but it is topologically two-dimensional and is homeomorphic to the usual torus.

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 Clifford torus' symmetry group acts transitively on every point of the torus, so all points are indistinguishable (and the figure has zero curvature at all points). This is in contrast to the usual torus, whose symmetry group distinguishes points of negative and positive curvature.

In analogy to discretizing a circle into a convex regular polygon, the Clifford torus can be approximated by computing the comb product of any two convex regular polygons, forming the duocombs. The result is an isogonal and isotopic skew polyhedron in 4D space; it is in fact a regular skew polyhedron if the two convex regular polygons are congruent.