Does birational equivalence between varieties preserves dimensions?

Question

I assume that birational equivalence may not preserve dimensions, i.e. if there are two birationally equivalent varieties $X$ and $Y$ (birational maps $\varphi:X\to Y, \langle U,\varphi \rangle$ and $\phi:Y\to X, \langle V,\phi \rangle$ such that $\phi\circ\varphi=id_{\varphi^{-1}(V)},\varphi\circ\phi=id_{\phi^{-1}(U)}$), then $\dim(X)$ and $\dim(Y)$ are not necessarily to be same. However, I could not figure out an appropriate example. I would be really appreciated if someone could figure it out.