Platonic self-dual solitons in a (1+1)-dimensional ${\mathbb{C}\mathbb{P}}^1$ model

We construct a family of self-dual $(1+1)$-dimensional ${\mathbb{C}\mathbb{P}}^1$ sigma models whose potentials $V$ are determined from pre-potentials $U$ by Platonic geometry. Using the round metric on ${\mathbb{S}}^2\simeq {\mathbb{C}\mathbb{P}}^1$ and the generalized self-duality formalism, we build symmetry-invariant pre-potentials $U$ from the unit vertices of each Platonic solid intersecting the unit circumsphere. The regular-lattice sites associated with vertex, face-center, and edge-midpoint directions are critical points of $U$ and degenerate vacua of the scalar potential $V$. Dual solids share the same regular lattice but generate different pre-potentials, charges, potentials, and metric-gradient flows. Numerical integration reveals distinct self-dual trajectories and energy-density profiles, including multi-peak structures for paths passing near saddle points, while solutions with the same endpoint difference $\mathrm{\Delta }U$ have the same total energy. The positive maxima of $V$ are also organized into symmetry orbits. This construction provides a geometric prescription for controlling the vacuum structure and self-dual sectors of ${\mathbb{C}\mathbb{P}}^1$ models with a potential.