  We study the zeta-regularized spectral determinant of the Friedrichs
Laplacians on the singular spheres obtained by cutting and glueing copies of
constant curvature (hyperbolic, spherical, or flat) double triangle. The
determinant is explicitly expressed in terms of the corresponding Belyi
functions and the determinant of the Friedrichs Laplacian on the double
triangle. The latter determinant was found in a closed explicit form in [V.
Kalvin, Calc. Var. 62 (2023), Paper 59, arXiv:2112.02771]. In examples we
consider the cyclic, dihedral, tetrahedral, octahedral, and icosahedral
triangulations, and find the determinant for the corresponding spherical,
Euclidean, and hyperbolic Platonic surfaces. These surfaces correspond to
stationary points of the determinant.
