Reading the fascinating book « Indra`s Pearls », written by David Mumford, Caroline Series and David Wright, you discover that the Apollonian gasket can be created by multiple inversions at four touching circles. Three of the circles are of equal size. The points lying outside these circles belong to two different regions. One lies in between the circles. It is a hyperbolic triangle with vanishing angles. The second region surrounds the circles and is an inverted image of the first region. It includes the point at infinity. We get a hyperbolic tiling of the plane by multiple inversion in these three circles. For each pixel we take its position and invert it at any circle if it lies inside. We repeat this until the position lies outside of all three circles. Then we colour the pixel depending on the number of reflections and its final position. We get this :
Here pixels that are mapped into the inner triangle get a light blue colour for an even number of reflections and a dark blue colour for an odd number. Pixels going to the outer inverted triangle are similarly coloured in light and dark yellow. In blue colours we see a Poincaré disc representation of tiled hyperbolic space inside an inverted disc of the same geometry in yellow colours.
We get a generator for the Apollonian gasket by adding a fourth circle inside the inner triangle. The fourth circle touches the three outer circles. Note that this cuts the inner triangle into three smaller triangles. Each of them has this additional circle as one of its sides and two of the three larger circles as other sides. Repeated inversion on the sides of one of the small triangles creates a Poincaré disc representation of tiled hyperbolic space similarly as for the figure above. Inversion in the three larger circles makes an inverted Poincaré disc, which is shown in blue in the above figure. By the way, this inverted disc is also neeeded for the tetrahedral symmetry of the Apollonian gasket. Thus we get four basic discs as building blocks. Inversion in all four circles generates infinitely many images of these discs. They cover the yellow disc of the above figure. It is important that the number of inversions becomes infinite at the border of each of these discs. Thus, we can approximate the Apollonian gasket by colouring those pixels that require many inversions. Superimposing on the image above we get :
We see that the Apollonian gasket nicely fits the tiled space as each hyperbolic triangle is decorated by the same fractal triangle. By the way, these triangles have a close similarity to the Sierpinsky triangle. Note that the projection of hyperbolic space into the Euclidean drawing plane distorts the tiles and their decorations in the same way.
To see how the Apollonian gasket is composed of discs representing hyperbolic spaces, we can colour the pixels depending on the triangle its position gets mapped to and on the number of inversions used. Using red, green and blue for the inner triangles and yellow for the surrounding inverted one we get :
Note that discs touching each other never get the same colour. Discs of the same colour are inverted images of each other. Their tilings are all the same and they only appear to be different. That’s because of the inversion mapping.
The Cayley transform changes the Poincaré disc into a Poincaré plane representation of hyperbolic space. Actually, an inversion in a circle that has its center on the border of the gasket gives an equivalent result. Applying it to the figure that superimposes the gasket and the embedding hyperbolic space, we get :
The triangles of the tiling again match the Apollonian gasket. In comparison, their sizes vary much stronger than in the figure above. Thus, at first sight the decorations of the tiles seem to be different. But this is only an effect of their different sizes. Because of the fractal nature of the gasket we see more or less details resulting in a different appearance. Again, we can show the discs of the gasket in more detail :
Note that in this image we see the gasket as a periodic frieze resulting from mirror images in two vertical lines and inversions in a string of touching circles.
I conclude that the Apollonian gasket is a fractal covering of a tiled hyperbolic space by Poincaré disc representations of hyperbolic space. But this is only one of its many faces.