Not that triple product

Geometry Level 2

We call the regular circle \(S^{1}\), and the regular sphere \(S^{2}\). It is known the torus is the product of two circles \(S^{1} \times S^{1} \neq S^{2}\), and it is embedded in three dimensions to avoid self-intersection.

What is the minimum number of dimensions \(S^{1} \times S^{1} \times S^{1}\) must be embedded in to prevent self-intersection?

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