Let $\rho:{{\bf Z}}^{n}\to(K^{\times})^{n}$ be a group homomorphism such that the image of the composition $val\circ\rho:{{\bf Z}}^{n}\to{{\bf R}}^{n}$ is a rank $n$ lattice in ${{\bf R}}^{n}$. Group $(K^{\times})^{n}$ acts by translations on the analytic space $({\bf G}_{m}^{an})^{n}$. Restriction of this action to ${{\bf Z}}^{n}$ (via $\rho$) is discrete and cocompact. The quotient is a $K$-analytic space $X$ called Tate torus. There is an obvious map $\pi:X\to B:={{\bf R}}^{n}/(val\circ\rho)({{\bf Z}}^{n})$. All points of $B$ are smooth. The space $X$ depends on $n^{2}$ parameters taking values in $K^{\times}$(cf. with the flat tori example in Section 3.2.1).