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A polygonal complex $X$ is Platonic if its automorphism group $G$ acts transitively on the flags (vertex, edge, face) in $X$. Compact examples include the boundaries of Platonic solids. Noncompact examples $X$ with nonpositive curvature (in an appropriate sense) and three polygons meeting at each edge were classified by \'Swi\c{a}tkowski, who also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete. For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete. A Fuchsian building is a two-dimensional also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete. For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete. A Fuchsian building is a two-dimensional hyperbolic building. We study lattices in automorphism groups of Platonic complexes and Fuchsian buildings. Using similar methods for both cases, we construct uniform and nonuniform lattices in $G=Aut(X)$. We also show that for some $X$ the set of covolumes of lattices in $G$ is nondiscrete, and that $G$ admits lattices which are not finitely generated. In fact our results apply to the larger class of Davis complexes, which includes examples in dimension > 2.