Archimedean tilings are periodic polygonal tessellations that are created by placing regular polygons edge-to-edge around a vertex to fill the plane. Here we show that three- and four-arm DNA junction tiles with specifically designed arm lengths and intertile sticky-end interactions can be used to form sophisticated two-dimensional (2D) and three-dimensional (3D) tessellation patterns. We demonstrate two different complex Archimedean patterns, (3(3).4(2)) and (3(2).4.3.4), and the formation of 2D lattices, 3D tubes, and sealed polygon-shaped pockets from the tessellations. The successful growth of hybrid DNA tile motif arrays suggests that it maybe possible to generate 2D quasi-crystals from DNA building blocks.