# 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-dimensional torus, or triple torus, is defined as the Cartesian product of three circles,

${\displaystyle \mathbb {T} ^{3}=S^{1}\times S^{1}\times S^{1}.}$

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 three pairs of opposite faces of a cube. (After gluing the first pair of opposite faces the cube looks like a thick washer, after gluing the second pair — the flat faces of the washer — it looks like a hollow torus, the last gluing — the inner surface of the hollow torus to the outer surface — is physically impossible in three-dimensional space so it has to happen in four dimensions.)

## 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.