  We study renormalization group (RG) fixed points of scalar field theories
endowed with the discrete symmetry groups of regular polytopes. We employ the
functional perturbative renormalization group (FPRG) approach and the
$\epsilon$-expansion in $d=d_c-\epsilon$. The upper critical dimensions
relevant to our analysis are $d_c =
6,4,\frac{10}{3},3,\frac{14}{5},\frac{8}{3},\frac{5}{2},\frac{12}{5}$; in order
to get access to the corresponding RG beta functions, we derive general
multicomponent beta functionals $\beta_V$ and $\beta_Z$ in the aforementioned
upper critical dimensions, most of which are novel. The field theories we
analyze have $N=2$ (polygons), $N=3$ (Platonic solids) and $N=4$
(hyper-Platonic solids) field components. The main results of this analysis
include a new candidate universality class in three physical dimensions based
on the symmetry group $\mathbb{D}_5$ of the Pentagon. Moreover we find new
Icosahedron fixed points in $d<3$, the fixed points of the $24$-Cell,
multi-critical $O(N)$ and $\phi^n$-Cubic universality classes.
