Triple torus

Triple torus

Triple torus or three-torus can refer to one of the two following concepts, both related to a torus.

Three-dimensional torus

The three-dimensonal torus, or triple torus, is defined as the Cartesian product of three circles,

:mathbb{T}^3 = S^1 imes S^1 imes S^1.

In contrast, the usual torus is the Cartesian product of two circles only.

The triple torus is a three-dimensional compact manifold with no boundary. It can be obtained by gluing the opposite faces of a cube.

Torus-like surface with three holes

In the theory of surfaces, a triple torus refers to a smooth closed surface with three holes, or, in other words, a surface of genus three. It can be obtained by attaching three handles to a sphere or by gluing (taking the connected sum) of three tori.



handles
connected sum of three tori
Pretzel-style triple torus

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