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We obtain full moduli parameters for generic non-planar BPS networks ofdomain walls in an extended Abelian-Higgs model with $N$ complex scalar fields,and exhaust all exact solutions in the corresponding $\mathbb{C}P^{N -1}$model. We develop a convenient description by grid diagrams which are polytopesdetermined by mass parameters of the model. To illustrate the validity of ourmethod, we work out non-planar domain wall networks for lower $N$ in $3+1$dimensions. In general, the networks can have compact vacuum bubbles, which arefinite vacuum regions surrounded by domain walls, when the polytopes of thegrid diagrams have inner vertices, and the size of bubbles can be controlled bymoduli parameters. We also construct domain wall networks with bubbles in theshapes of the Platonic, Archimedean, Catalan, and Kepler-Poinsot solids.

Exhausting all exact solutions of BPS domain wall networks in arbitrary dimensions