Peter Stampfli

This image shows a regular tiling of hyperbolic space using pentagons. It uses multiple reflections at a central pentagon with circle arcs of equal length as sides and angles of 90 degrees at its corners. Inversion at the circles are used to do reflections. This generates a Poincaré representation of the tiled hyperbolic space. The central pentagon is decorated with an anamorphic image with 5-fold rotational symmetry and no mirror symmetry. Thus the entire image has 5-fold rotational symmetry. At each corner point four pentagons meet. Two of them are of opposite handedness as they are mirror images.

A kaleidoscope generates this image using reflections at the sides of a triangle. The triangle is filled with an input image. The angles at its corners are 180/7 degrees, 60 degrees and 90 degrees, resulting in 7, 3 and 2-fold rotational symmetries. Two sides of the triangle are straight lines with mirror symmetries. The third side of the triangle is a circle arc and I am using inversion at this circle. Multiple reflections give a Poincaré disc representation of a periodic wallpaper that covers the entire hyperbolic space. It is a decoration of a regular tiling with heptagons. Three of them meet at each corner.