We study chiral fields [${U}_{i}$ in the group $\mathrm{U}(N)$] on a periodic lattice (${U}_{i}={U}_{i+L}$), with action $S=(\frac{1}{{g}^{2}})\ensuremath{\Sigma}{l=1}^{L}\mathrm{Tr}({U}_{l}{U}_{l+1}^{\ifmmode\dagger\else\textdagger\fi{}}+{U}_{l}^{\ifmmode\dagger\else\textdagger\fi{}}{U}_{l+1})$, as prototypes for lattice gauge theories [quantum chromodynamics (QCD)] at ${N}_{c}=\ensuremath{\infty}$. Indeed, these chiral chains are equivalent to gauge theories on the surface of an $L$-faced polyhedron (e.g., $L=4$ is a tetrahedron, $L=6$ is the cube, and $L=\ensuremath{\infty}$ is two-dimensional QCD). The one-link Schwinger-Dyson equation of Brower and Nauenberg, which gives the square of the transfer matrix, is solved exactly for all $N$. From the large-$N$ solution, we solve exactly the finite chains for $L=2, 3, 4, \mathrm{and} \ensuremath{\infty}$, on the weak-coupling side of the Gross-Witten singularity, which occurs at $\ensuremath{\beta}={({g}^{2}N)}^{\ensuremath{-}1}=\frac{1}{4}, \frac{1}{3}, \frac{\ensuremath{\pi}}{8}, \mathrm{and} \frac{1}{2}$, respectively. We carry out weak and strong perturbation expansions at ${N}_{c}=\ensuremath{\infty}$ to estimate the singular part for all $L$, and to show confinement (as ${g}^{2}N\ensuremath{\rightarrow}\ensuremath{\infty}$) and asymptotic freedom (${g}^{2}N\ensuremath{\rightarrow}0$) in the Migdal $\ensuremath{\beta}$ function for QCD. The stability of the location of the Gross-Witten singularity for different-size lattices ($L$) suggests that QCD at ${N}_{c}=\ensuremath{\infty}$ enjoys this singularity in the transition region from strong to weak coupling.