Before discussing the relation between the Apollonian gasket and tilings of the sphere, I want to present briefly the spherical kaleidoscope with tetrahedral symmetry.

A tetrahedron has three different kinds of points with rotational symmetry. Four equilateral triangles make up the tetrahedron. It has three-fold rotational symmetry with respect to the points lying at the center of triangles. Another set of points with the same rotational symmetry is made of the points at the corners of the triangles, where three triangles meet. Points of two-fold rotational symmetry lie in the middle of the sides of the triangles.

Note that reflections in two mirrors that intersect at an angle of 180/n degrees result in an n-fold rotational symmetry. Thus, a kaleidoscope with three planar mirrors makes an image with tetrahedral symmetry if its angles between the mirrors are 90 degrees and twice 60 degrees. These mirrors are not parallel, instead they make up a pyramid. To create a two-dimensional image with these symmetries, we replace the mirrors by mirror lines and the kaleidoscope becomes a triangle. However, the sum of its angles is 210 degrees. This is larger than 180 degrees and thus it is a spherical triangle. At least one of its sides has then to be a circle arc instead of a straight line.

To get the kaleidoscopic image we repeatedly mirror the position of a pixel at straight lines and invert it at circle arcs until it lies inside the kaleidoscopic triangle. The colour of the pixel can depend on the number of reflections and on its final position. A typical result for the tiling generated by the kaleidoscope looks like this :

Stereoscopic projection of the spherical tiling with tetrahedral symmetry.

The solid black lines mark the kaleidoscopic triangle. To show the structure of the tiling I coloured pixels already lying inside the triangle in light yellow. Pixels needing an even number of reflections to get into the triangle are shown in primary yellow and dark yellow shows those with an odd number. Here you can see how images of the kaleidoscopic triangle are put together to cover the entire plane. Isn’t it surprising that the triangles have such different shapes and sizes ? Shouldn’t they all be equal ?

What you actually see is a stereographic projection of the tiled sphere. This projection causes strong distortions. The tiled sphere can better be recognized in a normal projection. To show this, we first make an inverse stereographic projection from the plane to the sphere. The dashed line in the figure above shows the equator of the sphere. Its inside gets projected to the lower hemisphere with the south pole at its center. The south pole corresponds to the intersection point of the straight lines. The outside of this circle goes to the upper hemisphere having the north pole at its center. Stereographic projection maps the north pole to infinity in the plane. We now see that all triangles of the tiling have really the same shape :

Normal projection of the tiled sphere. Lower hemisphere at left, upper at the right. Both as seen from above.

These images are somewhat underwhelming and we do not even see any difference between the points of three-fold rotational symmetry! We get more interesting kaleidoscopic images from patching a part of a photography on the triangles. For each output pixel we simply read the colour of the input photography at the mapped position of the pixel position inside the kaleidoscopic triangle. This is an example :

Stereographic projection of a sphere covered by a kaleidoscopic image with tetrahedral symmetry. Solid white lines show the kaleidoscopic triangle. The dashed line is the projected equator.

We now see a difference between the two kinds of points with three-fold rotational symmetry. Some points lie inside a grey shape and the other points lie inside a dark brown shape. The distortion of the stereographic projection appears to be even more pronounced in this image. Again, the normal projections of the decorated sphere show the symmetries and equivalencies much better :

Normal projection of the sphere with a kaleidoscopic image. Lower hemisphere at left, upper at the right. Both as seen from above.

The Apollonian gasket too is a decoration of the tiled sphere. It results from multiple reflection similar to the tetrahedral tiling. Adding a circle to the kaleidoscopic triangle we get :

Stereographic projection of the tiled sphere together with an Apollonian gasket. Solid green lines show the generating elements. The dotted line is a projection of the equator.

Now, reflections at the two lines and two circles are repeated until a point gets mapped into the small triangle made of two circles and one straight line. This triangle has two angles of 60 degrees and an angle of 0 degrees. It is a hyperbolic triangle, because the sum of its angles is less than 180 degrees. On itself, it would only create a Poincaré disc representation of a tiled hyperbolic space. All four reflecting elements together make a fractal covering of the entire plane by non-overlapping images of this disc. The image above shows essentially the borders of those discs superimposed on the tetrahedral tiling. I am using that the number of reflections required to map a point into the kaleidoscopic triangle becomes infinite at the border of the discs. Thus pixels are shown in black if they require more than a certain number of reflections. This indicates the borders.

In normal projection we see the symmetry of the gasket much better:

Normal projection of the Apollonian gasket on a sphere. Lower hemisphere at left, upper at the right.

This view shows that each triangle really bears the same decoration.

We get more from these images if we relate them to the standard way of drawing the Apollonian gasket. It starts with a single circle. Then, three circles of equal radius are drawn inside this circle, touching each other and also the first circle. You can easily identify these four circles in the two figures above. Note that they all have the same size in the normal projection. Their centers lie on points of three-fold rotational symmetry. All these points lie in the grey shapes of the other kaleidoscopic image. One of these points is the north pole of the sphere and does not appear in the stereoscopic image. Note that there are four triangular gaps between the four circles. The all have the same size and shape in the normal projection. In a second step, a circle is drawn inside each of these gaps such that it touches its borders. The centers of these circles lie at other points of three-fold rotational symmetry. In the other kaleidoscopic image, they are in the center of the dark brown shapes. One of these points is the south pole of the lower hemisphere. These additional circles leave more gaps, which are filled again in the same manner. Repeating this procedure gives the same fractal decoration for all triangles of the tetrahedral tiling, as you can see in the images above.

We get a nice and instructive image if we draw the sphere as a black shadow with the Apollonian gasket in light colour. It shows how the two hemispheres fit together :

Normal projection of the Apollonian gasket. Lower hemisphere in pale blue and upper hemisphere in pale yellow.

To get another view of the Apollonian gasket we rotate the sphere, such that a point of two-fold rotational symmetry lies at the north pole and another one at the south pole. This results in an Apollonian gasket, that looks like a fractal decoration of an Euclidean frieze :

Stereographic projection of the rotated tiled sphere together with an Apollonian gasket.

This periodic repetition is an effect of the distorting stereographic projection. Note that the parallel lines and the periodicity do not match the tiling of the sphere. In the normal projection we see that the sphere is really only rotated :

Normal projection of the Apollonian gasket on a sphere. Lower hemisphere at left and upper at the right.

The combined view shows that now the upper and lower hemispheres have the same decoration up to a rotation by 90 degrees :

Normal projection of the Apollonian gasket. Lower hemisphere in pale blue and upper hemisphere in pale yellow.

I have made all these images using my public browser app at http://geometricolor.ch/circleInTriangle.html and with a little help of GIMP. Try out this browser app. It allows you to zoom into the gasket without limits, except for computer time. In this blog and on http://geometricolor.ch/home.html you find more information on kaleidoscopes.