For each jump vector, there are exactly two positions of the flea for which the jump is defined by this vector. Therefore, the sequence of jumps is periodic if and only if there is only a finite number of different jump vectors. Let $\vec{a}_{1}$ be the jump vector of the flea from line $l_{2}$ to line $l_{1}; \vec{a}_{2}, \vec{a}_{3}, \vec{a}_{4}, \ldots$ be the vectors of subsequent jumps. Then $\vec{a}_{2}=S_{\mathrm{l}_{2}}\left(\vec{a}_{1}\right), \vec{a}_{3}=S_{\mathrm{l}_{1}}\left(\vec{a}_{2}\right), \vec{a}_{4}=S_{\mathrm{l}_{2}}\left(\vec{a}_{3}\right), \ldots$ Since the composition $S_{\mathrm{l}_{1}} \mathrm{O}_{\mathrm{l}_{2}}$ is a rotation by an angle $2 \alpha$ (or by an angle $2 \pi-2 \alpha$), the vectors $\vec{a}_{3}, \vec{a}_{5}, \vec{a}_{7}, \ldots$ are obtained from the vector $\vec{a}_{1}$ by rotations of $2 \alpha, 4 \alpha, 6 \alpha, \ldots$ (or by $2(\pi-\alpha), 4(\pi-\alpha), 6(\pi-\alpha), \ldots$). Therefore, the set $\vec{a}_{1}, \vec{a}_{3}, \vec{a}_{5}, \ldots$ contains a finite number of different vectors if and only if $\alpha / \pi$ is a rational number. The set $\vec{a}_{2}, \vec{a}_{4}, \vec{a}_{6}$,