The torus may be defined parametrically by where A equation inside Cartesian coordinates for a torus azimuthally symmetrical just about a z-axis is A surface area and interior volume of this torus are given by

Based on data from a wide definition, the generator of the torus want non exist as a circle however can too exist as an ellipse or any other conic section.

Topology

Topologically, a torus occurs as closed surface defined as product of two circles: S¹ × S^Ace. A surface described above, given a relative topology from R³, is homeomorphic to a topologic torus when hanker when it doesn't intersect its have axis.

A torus can likewise exist as described as a quotient of the Euclidean plane under the identifications Or even, equivalently, when a quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon $ABA^$.

A fundamental group of the torus is just the direct product of the fundamental group of the circle by owning itself: Intuitively speaking, this means that the closed path that circles the torus' "hole" (say, a circle that traces retired the particular latitude) and so circles the torus' "body" (say, a circle that traces retired a particular longitude) may be deformed to the path that circles the person so the hole. And so, strictly 'latitudinal' & strictly 'longitudinal' paths commute.

A number 1 homology group of the torus is isomorphic to a fundamental class action (since the fundamental group is abelian).

The n-torus

1 might well generalize a torus to arbitrary dimensions. Anorth '''n-torus' is defined as a product of n circles: A torus discussed above is the Two-torus. A One-torus is just the circle. A Three-torus is like hard to visualize. Upright whenorth for a Two-torus, the n-torus may be described as a quotient of Rnorth under integral shifts in any co-ordinate. That is, a north-torus is Rnorth modulo a action of the integer lattice Zn (by having a action existence taken when vector addition). Equivalently, a n-torus is found from either a n-cube by gluing the paired faces together.

Anorth n-torus is anorth case of an n-dimensional compact manifold. These are likewise an lesson of the compact abelian Lie group. This follows from either a fact that a unit circle is a compact abelian Lie group (whilst identified by using a unit complex numbers with multiplication). Class action multiplication on the torus is so defined by coordinate-caring multiplication.

Toroidal groups play an significant a share in the theory of compact Lie groups. This flow from around section to the fact that in any compact Lie class action 1 could universally buy the maximal torus; that is, a closed subgroup which is a torus of the big imaginable dimension.

A fundamental group of an n-torus occurs as free abelian group of rank n. A k-th homology group of an n-torus occurs as loose abelianorth class action of rank n choose k. It follows that a Euler characteristic of the n-torus is Cipher for everthing n. A cohomology ring H^•(Tn,Z) may be identified by using a exterior algebra over the Z-module Zn whose generators come a duals of the n'' nontrivial rounds.