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.
|
|
Read more about this topic: Apollonian Gasket
Famous quotes containing the words integral, apollonian and/or circle:
“Make the most of your regrets; never smother your sorrow, but tend and cherish it till it come to have a separate and integral interest. To regret deeply is to live afresh.”
—Henry David Thoreau (1817–1862)
“Cinema is the culmination of the obsessive, mechanistic male drive in western culture. The movie projector is an Apollonian straightshooter, demonstrating the link between aggression and art. Every pictorial framing is a ritual limitation, a barred precinct.”
—Camille Paglia (b. 1947)
“It is a good lesson—though it may often be a hard one—for a man who has dreamed of literary fame, and of making for himself a rank among the world’s dignitaries by such means, to step aside out of the narrow circle in which his claims are recognized, and to find how utterly devoid of all significance, beyond that circle, is all that he achieves, and all he aims at.”
—Nathaniel Hawthorne (1804–1864)