Bending Circle Limits

Vladimir Bulatov
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 167–174


M.C.Escher's hyperbolic tessellations “Circle Limit I-IV” are based on a tiling of hyperbolic plane by identical triangles. These tilings are rigid because hyperbolic triangles are unambiguously defined by their vertex angles. However, if one reduce the symmetry of the tiling by joining several triangles into a single polygonal tile, such tiling can be deformed. Hyperbolic tilings allow a deformation which is called bending. One can extend tiling of the hyperbolic plane by identical polygons into tiling of the hyperbolic space by identical infinite prisms (chimneys). The original polygon being the chimney's cross section. The shape of these 3D prisms can be carefully changed by rotating some of its sides in space and preserving all dihedral angles.

The resulting tiling of 3D hyperbolic space creates 2D tiling at the infinity of hyperbolic space, which can be thought of as the sphere at infinity. This sphere can be projected back into the plane using stereographic projection. After small bending the original circle at infinity of the 2D tiling becomes fractal curve. Further bending results in thinning the fractal features which eventually form a fractal set of circular holes which in the end disappear.