By connecting the midpoints of the edges of any hyperbolic (or Euclidean) tessellation by regular polygons, you can obtain a [b]quasi-regular[/b] tessellation. Pictured is the Quasi(5,4) regular tessellation in red over the (5,4) regular tessellation that generated it.

M.C. Escher used such tessellations to produce his Circle Limit series.

[img]https://upload.wikimedia.org/wikipedia/en/5/55/Escher_Circle_Limit_III.jpg[/img]