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A lattice $\Gamma \subset \mathbb{R}^d$ is said to be rational if for any two vectors $\gamma_1, \gamma_2 \in \Gamma$, their inner product satisfies the relation

$$\displaystyle \langle\gamma_1, \gamma_2 \rangle = \beta_{\gamma}r_{12},$$

where $\beta_\Gamma \neq 0$ is a real-valued constant independent of both $\gamma_1$ and $\gamma_2$, and $r_{12} = r_{21}$ is an integer. Otherwise, we say that the lattice is irrational.

I'd like to talk about the quotient $\mathbb{R}^d / \Gamma$. If $\Gamma$ is a rational lattice, say, $\Gamma = \mathbb{Z}^d$, then we have $\mathbb{R}^d / \Gamma = (0,1)^d,$ a torus.

My question is, what happens if $\Gamma$ is irrational? Do we still obtain a torus? Could someone give me an example of an irrational lattice and what the resulting quotient looks like?

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The quotient of $\mathbb R^d$ by any full-rank lattice $\Gamma$ is a torus, no matter what are the properties of $\Gamma$. This is because any two lattices are related by an invertible matrix, the action of this matrix on the quotient space will give an homeomorphism between the tori.

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