I want to show you a fractal tiling which can be seen as a decoration of a sphere with octahedral symmetry and at the same time as another decoration of a sphere with icosahedral symmetry. It arises as the limit set of a dihedral group and inversion in two circles. You can reproduce these results with my public browser app at http://geometricolor.ch/circleInTriangle.html. The basic setup is shown in figure 1.
In both parts of figure 1 two straight lines meet at an angle of 60 degrees. Reflections at these pairs of lines generate a dihedral group with three fold rotational symmetry. In the upper part, a circle arc of large radius intersects the lines at angles of 90 and 45 degrees. This makes an elliptic triangle with a sum of angles of 195 degrees. Inversions at the circle arc together with the dihedral group create a tiling with octahedral symmetry. The additional smaller circle touches the circle arc and intersects the lines at angles of 90 and 36 degrees. This creates a very small elliptic triangle at the intersection of the straight lines, which is a reduced copy of the large triangle in the lower part of figure 1. On itself, it would create a tiling with icosahedral symmetry. The two circles and the horizontal straight line define a hyperbolic triangle. It is coloured yellow and has angles of 45 and 36 degrees and a corner with a vanishing angle. Reflections at its sides make a Poincaré disc representation of hyperbolic space tiled by images of this triangle. Note that the lower part of figure 1 is simply an inverted image of the upper part. The center of the inversion lies at the intersection of the two straight lines. This exchanges the angles of 36 and 45 degrees as well as the octahedral and icosahedral symmetries.
We use repeated inversion in the two circles and the elements of the dihedral group of the two straight lines to map any point into the hyperbolic triangle. Thus we get a fractal tiling of the plane by images of the Poincaré disc mentioned above. We can best see the structure of the tiling if we draw only the borders of the discs. To do this exactly would be very difficult, but it is easy to find an approximation. We simply highlight all pixels that need a large number of inversions to go into the hyperbolic triangle. These pixels lie near the border of Poincaré discs as their hyperbolic distance to the center of the disc diverges at the border and the number of inversions is related to this distance. Figures 2 and 3 show the result at different scales. We see an interesting recursive structure made of two different steps. In each step there are triangular gaps with vanishing corner angles. They are filled by several discs, which leaves smaller gaps to be filled in the next step. This is similar to the Apollonian gasket.
In one step, three discs are put into a gap. Together with the surrounding three discs they make a stereographic projection of circles inscribed on the sides of a cube. They touch each other:
This step is followed by putting nine discs in each gap. Together with the surrounding three discs they are a stereographic projection of circles drawn on the sides of a dodecahedron. Again, they are touching:
We can now make inverse stereographic projections to spheres. From Figure 2 we get a decoration with octahedral symmetry:
Here I have used different colours to distinguish the upper and lower hemispheres. Figure 3 gives a decoration with icosahedral symmetry:
Although the two spheres have different decorations they show the same fractal covering of the plane.