Not that triple product

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?