We analyse a certain family of cellular integrals, which are period integrals
on the moduli space $\mathcal{M}_{0,8}$ of curves of genus zero with eight
marked points, which give rise to simultaneous rational approximations to
$\zeta(3)$ and $\zeta(5)$. By exploiting the action of a large symmetry group
on these integrals, we construct infinitely many effective rational
approximations $p/q$ to $\zeta(5)$ satisfying \[ 0<\bigg|\zeta(5)-\frac
pq\bigg|<\frac1{q^{0.86}}. \]
