I will describe interactions between the number-theoretic properties of a closed, orientable hyperbolic 3-manifold $M$ and its quantitative geometric properties. The number-theoretic properties in question are captured by the trace field $K$ of $M$, while the geometric properties in question are expressed through such quantities as the Margulis number and estimates for tube radii about closed geodesics. The link between the two involves the algebra of finite simple groups, deep number-theoretic results due to Mahler and Siegel, and joint work of mine with Wagreich and with Anderson, Canary and Culler.