![]() ![]() ![]() Three of the seed disks appear as disks, and the fourth disk’s boundary is indicated by the red line. As the traces Ta, Tb reduce 3 through quasi-Fuchsian 2.2 (right) to 2 (left), the fractal circle becomes a Kleinian Apollonian Gasket. In the following program, set nbrLevels to 0 and model to ‘disks’. 2 Not only are the four seed disks tangent in a chain, but paired disks are also tangent: to, and to. Remarkably, the Apollonian gasket also arises under Schottky dynamics. In the following figure, the Apollonian gasket is approximated by the black curve. It’s what the red ideal triangles ultimately shrink to. The following figure follows the process for two more steps, yielding the green circles and then the purple circles.Īs the process continues, we approach the limit set known as the Apollonian gasket, the set of ‘red’ points that remain when all inscribed circles have been removed. The process proceeds indefinitely: we inscribe a disk in each ideal triangle, thereby partitioning it into three smaller ideal triangles. When we remove each yellow disk, three smaller red ideal triangles remain. Next, inscribe a yellow disk in each of the two red triangles. Next, we remove the three blue disks, which leaves behind two red ideal triangles whose sides meet at zero degrees at the vertices. We can think of these three disks as lying in a common sphere. 1 The two ‘smaller’ blue disks render as conventional disks whereas the third appears as a blue ring: This disk contains the point at infinity. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators. These are the three blue disks in the following figure. Suppose we start with three non-overlapping disks, each tangent to the other two. Inch your way through dead dreams to another land ![]()
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