time.

Apollonian Gaskets. An interesting new kaleidoscope is enough to keep mathematicians happy for years.Kontorovich learned about the Apollonian kaleidoscope from his mentor, Peter Sarnak of Princeton University, who learned about it from Lagarias, who learned about it from Wilks and Mallows. using

Here are a couple examples: means creating a files in some "standard" format with

I have to confess that I spent a few idle minutes watching the fractals instead of writing.Before I became a full-time writer, I used to be a mathematician. triple of mutually tangent circles, and successively filling However, Apollonius’s surviving book No one knows, of course, what Apollonius’ solution was, or whether it was correct. At this point, though, he is far from proving this conjecture—the necessary math just doesn’t exist yet.An Apollonian gasket is built up through successive “generations.” For instance, in generation 1 And even if all the problems concerning the classic Apollonian gaskets are solved, there are still gaskets galore for mathematicians to work on.

“For me, what’s attractive about Apollonian gaskets is that even my 14-year-old daughter finds them interesting,” says Sarnak.

The empirically observed dimension is 1.56. Now you have four roughly triangular spaces between the circles.

Otherwise I might not have gotten any work done.Numbers in an Apollonian gasket correspond to the curvatures or “bends” of the circles, with larger bends corresponding to smaller circles. ;) (circle/sphere centers and radii). These numbers must satisfy certain “congruence restrictions.” For example, in the bugeye gasket, the only legal bends have a remainder of 2, 3, 6 or 11 when divided by 12. More recently, Stefan Hutzler of Trinity College Dublin, along with Gary Delaney and Tomaso Aste of the University of Canberra, studied the effect of bubbles with different shapes in a random Apollonian packing. Amongst all the possible variations I'm interested on the basic Apollonian gasket generated by the outer circle (curvature -1) and 2 congruent inner circles with curvature 2 (model -1, 2, 2 as in the image above). renderings. If there were too many, then previously known methods, such as the ones Wiles used, would already answer all your questions. possible 3-tuple of circles. In these simulations, new bubbles or grains nucleate in a random place and grow, either with rotation or without, until they encounter another bubble or grain. The following year, Thorold Gosset published an (The authors note that the poem is to be pronounced in the Queen’s English. Circles Kontorovich and Oh exploited this symmetry in an extraordinary and amusing way to prove their estimate of the function This property makes the “lightbulb counting function” a very special kind of function, one which is invariant under the same symmetries as the Apollonian gasket itself. For other Apollonian gaskets, such as the “coins” gasket in the fifth figure, there are some absentees—numbers that obey the congruence restrictions but don’t appear in the gasket. however, one can obtain nice raster images from SVG files Every couple minutes the screen would go blank and refresh itself with a completely different fractal.

An Apollonian gasket fractal is generated by starting with a Or a tree in a forest may grow until its canopy touches another tree, and then stop.

Every circle in the gasket is generated by repeated inversions of the first six circles through these curved mirrors. As mentioned before, they could study random Apollonian gaskets. Rather sensibly, the institute assigned me an office on the east side, with a view of nothing much but my computer screen. Fractals made of circles do funny things to mathematiciansIn the spring of 2007 I had the good fortune to spend a semester at the Mathematical Sciences Research Institute in Berkeley, an institution of higher learning that takes “higher” to a whole new extreme.

This creates 12 triangular pores; insert a new circle into each one of them, just touching each side. may create four different Apollonian gaskets, one for each

For example, which numbers actually appear as bends in a given gasket?