  Using exact results, we determine the complex-temperature phase diagrams of
the 2D Ising model on three regular heteropolygonal lattices, $(3 \cdot 6 \cdot
3 \cdot 6)$ (kagom\'{e}), $(3 \cdot 12^2)$, and $(4 \cdot 8^2)$ (bathroom
tile), where the notation denotes the regular $n$-sided polygons adjacent to
each vertex. We also work out the exact complex-temperature singularities of
the spontaneous magnetisation. A comparison with the properties on the square,
triangular, and hexagonal lattices is given. In particular, we find the first
case where, even for isotropic spin-spin exchange couplings, the nontrivial
non-analyticities of the free energy of the Ising model lie in a
two-dimensional, rather than one-dimensional, algebraic variety in the
$z=e^{-2K}$ plane.
