# Embedding torus in Euclidean space

For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?

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It seems like you should be able to prove this more or less by induction. –  Aaron Mazel-Gee Nov 30 '10 at 6:41
Let $e_0, \ldots, e_n$ be the standard basis of $\mathbb {R}^{n+1}$. Take $\epsilon$ small. Consider the vector $v_1$ of length 1 in the span of $e_0, e_1$. Then the vector $v_2$ of length $\epsilon$ in the span of $v_1, e_2$, and in general the vector $v_i$ of length $\epsilon^{i-1}$ in the span of $v_{i-1}, e_i$. Now consider the vector $w=v_1+\ldots v_n$ For small $\epsilon$ the set of $w$'s is a torus embedded $\mathbb {R}^{n+1}$ (any $\epsilon < 1$ will do, actually).