  We introduce a model of interacting bosons exhibiting an infinite collection
of fractal symmetries -- termed "Pascal's triangle symmetries" -- which
provides a natural $U(1)$ generalization of a spin-(1/2) system with Sierpinski
triangle fractal symmetries. The Pascal's triangle symmetry gives rise to exact
degeneracies, as well as a manifold of low-energy states which are absent in
the Sierpinski triangle model. Breaking the $U(1)$ symmetry of this model to
$Z_p$, with prime integer $p$, yields a lattice model with a unique fractal
symmetry which is generated by an operator supported on a fractal subsystem
with Hausdorff dimension $d_H = \ln (p(p+1)/2)/\ln p$. The Hausdorff dimension
of the fractal can be probed through correlation functions at finite
temperature. The phase diagram of these models at zero temperature in the
presence of quantum fluctuations, as well as the potential physical
construction of the $U(1)$ model are discussed.
