Date: Tue, 7 Jun 2005 10:01:49 -0700 (PDT) From: "R. William Gosper" <rwg@osots.com> cheesy old mail While looking for something else, Rich kindly exhumed one of my old blurtings: Received: from SWEATHOUSE.macsyma.com ([192.233.166.105]) by NEWTON.Macsyma.COM via INTERNET with SMTP id 267086; 28 Nov 1993 01:37:52-0500 Date: Sat, 27 Nov 1993 22:28-0800 From: Bill Gosper <rwg@NEWTON.macsyma.com> Subject: Zero calorie Swiss cheese To: math-fun@cs.arizona.edu Message-Id: <19931128062850.2.RWG@SWEATHOUSE.macsyma.com> Let C be the circle |z| = (1 + sqrt(3))/2 and let D be the disk |z| < (1 + sqrt(3))/2, and 2 i pi 1 ------ z - - 3 2 f1(z) := e z, f2(z) := (2 - sqrt(3)) ----- z + 1 be the generators of a group G of homographic transformations. Then the distinct images of C under G cover C (except for a Cantor set) with non-overlapping circles. No, the distinct images of D under G cover D (except for a Cantor set) with non-overlapping disks. Mandelbrot calls this Cantor set "Apollonian gasket", and wrote several years ago that its fractal dimension remained unknown. "2D suds" is more descriptive, although the bubbles are unrealistically round. Define three different gaskets: K := the open disk D minus the (proper) images of closure(D); O := closure(D) minus the (proper) images of D; and U := the union of all the circles. It might seem that K would be empty, and O = U, but both of these intuitions are wild underestimates. In fact, O = closure(U) = closure(K) = boundary(U) = boundary(K), yet U is of measure zero in both O and K. K + U = O, where "+" = disjoint union. Still, U, O, and K have the same Hausdorff dimension and graphical appearance. (The really correct graphical appearance would be completely invisible!) [Ford circle pix I found on the Web were too arty for clarity. The Book of Numbers covers them, but I can't find mine.] Plotting the circles to high order [Thanks to the kind auspices of Stephen M. Jones, the circular gasket is http://gosper.org/suds.gif, the strip is http://gosper.org/stripcheese.gif, and the Ford circle subset is http://gosper.org/ford10.gif .] entails several annoyances, e.g., a tendency toward "unfloatable" numbers: 1 --------------------------------------------- 72010600134783751 - 41575339372323900 sqrt(3) gives 1/.0, even with double precision. (Molly Brown's antithesis? Unthinkable.) Another problem is whittling the collisions down to "the distinct images of C". I.e., it appears that the number of distinct group elements divided by the number of distinct circles they generate goes to infinity with increasing word length. Crudely, the smaller the circle, the more words which generate it. E.g., abbreviate f1 as W, f2 as G, and f2^-1 as g. Then the only relation is W^3 = 1. But obviously WWC = WC = C. Less obviously, G^n WGC = WGC, G^n WWGC = WWGC, W^n GGC = GGC, GWggC = WGWWGGGC. Is there a name (e.g., "projected group") for this sort of structure? Problem: is there a regular expression (or even a subgroup) which generates each circle only once? (If distinct group elements do generate the same circle c, they do not do so pointwise, but rather are related by a continuous "permutation" of the points of c. Is this permutation of finite order?) Exercise: Exhibit an explicit point of K. (It's infinitely bigger than U, so how can you miss?-) JHC's method for the triangular gasket should work, if you can find his old mail. Another approach might start with finding that limit point on the Soddy spiral page of www.tweedledum.com/rwg/ . The rest of the message was i pi 5 i pi - ---- - ------ 6 6 (sqrt(3) + 1) e z + e f(z) := - ----------------------------------- (sqrt(3) + 1) z + 1 maps the gasket onto the strip between the real axis and imag(z)=1, with the circles now including the Ford circles: |z - n/d - i/d^2| = 1/d^2, for every reduced rational n/d. These are non-overlapping circles which cluster against the real axis from above. This strip gasket is simply generated by 1 f1(z) := z + 1, f2(z) := ----- z - i acting on |z - i/2| = 1/2. --rwg