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, and give rise to simultaneous rational approximations to $ζ(3)$ and $ζ(5)$. By exploiting the action of a large symmetry group on these integrals, we construct an infinite $effective$ sequence of rational approximations $p/q$ to $ζ(5)$ satisfying \[ 0<\bigg|ζ(5)-\frac pq\bigg|<\frac1{q^{0.86}}. \]