  The fractal structure and scaling properties of a 2d slice of the 3d Ising
model is studied using Monte Carlo techniques. The percolation transition of
geometric spin (GS) clusters is found to occur at the Curie point, reflecting
the critical behavior of the 3d model. The fractal dimension and the winding
angle statistics of the perimeter and external perimeter of the geometric spin
clusters at the critical point suggest that, if conformally invariant in the
scaling limit, they can be described by the theory of Schramm-Loewner evolution
(SLE_\kappa) with diffusivity of \kappa=5 and 16/5, respectively, putting them
in the same universality class as the interfaces in 2d tricritical Ising model.
It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the
cross sections undergo a nontrivial percolation transition, in the same
universality class as the ordinary 2d critical percolation.
