  We study the antiferromagnetic classical Ising (AFI) model on the sorrel net,
a 1/9th site depleted and 1/7th bond depleted triangular lattice. Our classical
Monte Carlo simulations, verified by exact results for small system sizes, show
that the AFI model on this corner-shared triangle net (with coupling constant
$J_1$) is highly frustrated, with a residual entropy of $\frac{S}{N}$=
0.48185$\pm$0.00008. Anticipating that it may be difficult to achieve perfect
bond depletion, we investigate the physics originating from turning back on the
deleted bonds ($J_2$) to create a lattice of edge-sharing triangles. Below a
critical temperature which grows linearly with $J_2$ for small $J_2$, we
identify the nature of the unusual magnetic ordering and present analytic
expressions for the low temperature residual entropy. We compute the static
structure factor and find evidence for long range partial order for
antiferromagnetic $J_2$, and short range magnetic order otherwise. The magnetic
susceptibility crosses over from following a Curie-Weiss law at high
temperatures to a low temperature Curie law whose slope clearly distinguishes
ferromagnetic $J_2$ from the $J_2 = 0$ case. We briefly comment on a recent
report of the creation of a 1/9th site depleted triangular lattice cobalt
hydroxide oxalate.
