  Archimedes' hat-box theorem states that uniform measure on a sphere projects
to uniform measure on an interval. This fact can be used to derive Simpson's
rule. We present various constructions of, and lower bounds for, numerical
cubature formulas using moment maps as a generalization of Archimedes' theorem.
We realize some well-known cubature formulas on simplices as projections of
spherical designs. We combine cubature formulas on simplices and tori to make
new formulas on spheres. In particular $S^n$ admits a 7-cubature formula
(sometimes a 7-design) with $O(n^4)$ points. We establish a local lower bound
on the density of a PI cubature formula on a simplex using the moment map.
  Along the way we establish other quadrature and cubature results of
independent interest. For each $t$, we construct a lattice trigonometric
$(2t+1)$-cubature formula in $n$ dimensions with $O(n^t)$ points. We derive a
variant of the M\"oller lower bound using vector bundles. And we show that
Gaussian quadrature is very sharply locally optimal among positive quadrature
formulas.
