  We study the thermodynamics of Ising spins on the triangular kagome lattice
(TKL) using exact analytic methods as well as Monte Carlo simulations. We
present the free energy, internal energy, specific heat, entropy, sublattice
magnetizations, and susceptibility. We describe the rich phase diagram of the
model as a function of coupling constants, temperature, and applied magnetic
field. For frustrated interactions in the absence of applied field, the ground
state is a spin liquid phase with integer residual entropy per spin
$s_0/k_B={1/9} \ln 72\approx 0.4752...$. In weak applied field, the system maps
to the dimer model on a honeycomb lattice, with irrational residual entropy
0.0359 per spin and quasi-long-range order with power-law spin-spin
correlations that should be detectable by neutron scattering. The power-law
correlations become exponential at finite temperatures, but the correlation
length may still be long.
