The self-assembly of lanthanide ions with ditopic organic spacers results in the formation of complex tiling patterns that mimic the structural motifs of quasi-periodic 2D materials. The linking of trans-{LnI2}+ nodes (Ln = Gd, Dy) by both closed-shell and anion radicals of 4,4'-bipyridine affords rare examples of Archimedean tessellations in a metal-organic framework. We furthermore demonstrate the occurrence of sizable magnetic exchange interactions and slow relaxation of magnetization behavior in a complex tessellation pattern. The implementation of Archimedean tessellations in lanthanide(III) coordination solids couriers a strategy to design elusive quasi-periodic metal-organic frameworks with inimitable magnetic properties.