  The antiferromagnetic Heisenberg Hamiltonian is investigated on a truncated
tetrahedron, which is a closed 12 site system. We find that the ground state
has many similarities to that of $C_{60}$. We study 2- and 4-spin correlations
in the classical ground state of the truncated tetrahedron and calculate the
same correlations in the exact S=1/2 ground state. We find that the classical
correlations survive for a range of bond strengths in the Heisenberg
Hamiltonian and that one can construct a good trial wavefunction based on the
classical ground state. This suggests that the correlations present in the
classical ground state of $C_{60}$ also survive in the exact ground state of
that system, for a range of bond strengths about the physically relevant $J_2
\approx J_1$. We calculate the momentum-space correlation function $S ( q )$,
which is measurable by neutron scattering, for both $C_{12}$ and $C_{60}$. We
also calculate correlations at finite temperature.
