Apollonian Gasket - Integral Apollonian Circle Packings

Integral Apollonian Circle Packings

  • Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3)

  • Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8)

  • Integral Apollonian circle packing defined by circle curvatures of (−12, 25, 25, 28)

  • Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19)

  • Integral Apollonian circle packing defined by circle curvatures of (−10, 18, 23, 27)

If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature. The first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket – all other curvatures can be derived from these three.

Integral Apollonian gaskets
Beginning curvatures Symmetry
−1, 2, 2, 3, 3 D2
−2, 3, 6, 7, 7 D1
−3, 4, 12, 13, 13 D1
−3, 5, 8, 8, 12 D1
−4, 5, 20, 21, 21 D1
−4, 8, 9, 9, 17 D1
−5, 6, 30, 31, 31 D1
−5, 7, 18, 18, 22 D1
−6, 7, 42, 43, 43 D1
−6, 10, 15, 19, 19 D1
−6, 11, 14, 15, 23 C1
−7, 8, 56, 57, 57 D1
−7, 9, 32, 32, 36 D1
−7, 12, 17, 20, 24 C1
−8, 9, 72, 73, 73 D1
−8, 12, 25, 25, 33 D1
−8, 13, 21, 24, 28 C1
−9, 10, 90, 91, 91 D1
−9, 11, 50, 50, 54 D1
−9, 14, 26, 27, 35 C1
−9, 18, 19, 22, 34 C1
−10, 11, 110, 111, 111 D1
−10, 14, 35, 39, 39 D1
−10, 18, 23, 27, 35 C1
Integral Apollonian gaskets
Beginning curvatures Symmetry
−11, 12, 132, 133, 133 D1
−11, 13, 72, 72, 76 D1
−11, 16, 36, 37, 45 C1
−11, 21, 24, 28, 40 C1
−12, 13, 156, 157, 157 D1
−12, 16, 49, 49, 57 D1
−12, 17, 41, 44, 48 C1
−12, 21, 28, 37, 37 D1
−12, 21, 29, 32, 44 C1
−12, 25, 25, 28, 48 D1
−13, 14, 182, 183, 183 D1
−13, 15, 98, 98, 102 D1
−13, 18, 47, 50, 54 C1
−13, 23, 30, 38, 42 C1
−14, 15, 210, 211, 211 D1
−14, 18, 63, 67, 67 D1
−14, 19, 54, 55, 63 C1
−14, 22, 39, 43, 51 C1
−14, 27, 31, 34, 54 C1
−15, 16, 240, 241, 241 D1
−15, 17, 128, 128, 132 D1
−15, 24, 40, 49, 49 D1
−15, 24, 41, 44, 56 C1
−15, 28, 33, 40, 52 C1
−15, 32, 32, 33, 65 D1

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