  Recently a ``Pascal's triangle model" constructed with $\text{U}(1)$ rotor
degrees of freedom was introduced, and it was shown that ($\textit{i}$.) this
model possesses an infinite series of fractal symmetries; and ($\textit{ii}$.)
it is the parent model of a series of $Z_p$ fractal models each with its own
distinct fractal symmetry. In this work we discuss a multi-critical point of
the Pascal's triangle model that is analogous to the Rokhsar-Kivelson (RK)
point of the better known quantum dimer model. We demonstrate that the
expectation value of the characteristic operator of each fractal symmetry at
this multi-critical point decays as a power-law of space, and this
multi-critical point is shared by the family of descendent $Z_p$ fractal
models. Afterwards, we generalize our discussion to a $(3+1)d$ model termed the
``Pascal's tetrahedron model" that has both planar and fractal subsystem
symmetries. We also establish a connection between the Pascal's tetrahedron
model and the $\text{U}(1)$ Haah's code.
