  In this article, we study the geometry of log Calabi-Yau pairs $(X,B)$ of
index one and birational complexity zero. Firstly, we propose a conjecture that
characterizes such pairs $(X,B)$ in terms of their dual complex and the
rationality of their log canonical places. Secondly, we show that for these
pairs the open set $X\setminus B$ is divisorially covered by open affine
subvarieties which are isomorphic to open subvarieties of algebraic tori. We
introduce and study invariants that measure the geometry and the number of
these open subvarieties of algebraic tori. Thirdly, we study boundedness
properties of log Calabi-Yau pairs of index one and birational complexity zero.
For instance, in dimension $2$ we prove that such pairs are affinely bounded.
