Peter Stampfli
I have fun playing around with symmetries and using them to create images no one has ever seen before. Then the real work begins trying to figure out what it means, explaining details and getting better images.
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.