Research Article
On a Diophantine Equation That Generates All Integral Apollonian Gaskets
Table 1
Bend quintets for Apollonian integer disk packings for principal curvatures 1 through 15, including four indices and shifts.
| | | Symmetry | Curvatures | Four indices | | | | type | | | | | | | | | | Shift |
| 1 | 1 | Odd | 1 | 2 | 2 | 3 | 3 | 1 | 1 | 1 | 0 | 0 |
| 2 | 1 | Odd | 2 | 3 | 6 | 7 | 7 | 2 | 1 | 4 | 0 | 0 | 3 | 1 | Odd | 3 | 4 | 12 | 13 | 13 | 3 | 1 | 9 | 0 | 0 | 4 | 2 | Even | 3 | 5 | 8 | 8 | 12 | 3 | 2 | 5 | 1 | 1 |
| 5 | 1 | Odd | 4 | 5 | 20 | 21 | 21 | 4 | 1 | 16 | 0 | 0 | 6 | 2 | Even | 4 | 8 | 9 | 9 | 17 | 4 | 4 | 5 | 2 | 1 |
| 7 | 1 | Odd | 5 | 6 | 30 | 31 | 31 | 5 | 1 | 25 | 0 | 0 | 8 | 2 | Even | 5 | 7 | 18 | 18 | 22 | 5 | 2 | 13 | 1 | 1 |
| 9 | 1 | Odd | 6 | 7 | 42 | 43 | 43 | 6 | 1 | 36 | 0 | 0 | 10 | 2 | Odd | 6 | 10 | 15 | 19 | 19 | 6 | 4 | 9 | 0 | 0 | 11 | 3 | Skew | 6 | 11 | 14 | 15 | 23 | 6 | 5 | 8 | 2 | 4/5 |
| 12 | 1 | Odd | 7 | 8 | 56 | 57 | 57 | 7 | 1 | 49 | 0 | 0 | 13 | 2 | Even | 7 | 9 | 32 | 32 | 36 | 7 | 2 | 25 | 1 | 1 | 14 | 3 | Skew | 7 | 12 | 17 | 20 | 24 | 7 | 5 | 10 | 1 | 2/5 |
| 15 | 1 | Odd | 8 | 9 | 72 | 73 | 73 | 8 | 1 | 64 | 0 | 0 | 16 | 2 | Skew | 8 | 13 | 21 | 24 | 28 | 8 | 5 | 13 | 1 | 2/5 | 17 | 3 | Even | 8 | 12 | 25 | 25 | 33 | 8 | 4 | 17 | 2 | 1 |
| 18 | 1 | Odd | 9 | 10 | 90 | 91 | 91 | 9 | 1 | 81 | 0 | 0 | 19 | 2 | Even | 9 | 11 | 50 | 50 | 54 | 9 | 2 | 41 | 1 | 1 | 20 | 3 | Skew | 9 | 14 | 26 | 27 | 35 | 9 | 5 | 17 | 2 | 4/5 | 21 | 4 | Skew | 9 | 18 | 19 | 22 | 34 | 9 | 9 | 10 | 3 | 2/3 |
| 22 | 1 | Odd | 10 | 11 | 110 | 111 | 111 | 10 | 1 | 100 | 0 | 0 | 23 | 2 | Odd | 10 | 14 | 35 | 39 | 39 | 10 | 4 | 25 | 0 | 0 | 24 | 3 | Skew | 10 | 18 | 23 | 27 | 35 | 10 | 8 | 13 | 2 | 1/2 |
| ⋯ | | | | | | | | | | | | |
| 34 | 6 | Even* | 12 | 25 | 25 | 28 | 48 | 12 | 13 | 13 | 5 | 10/13 |
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