New fractal based on the Golden mean/ Fibonacci numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg65290] New fractal based on the Golden mean/ Fibonacci numbers*From*: Roger Bagula <rlbagulatftn at yahoo.com>*Date*: Fri, 24 Mar 2006 00:59:47 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

I was thumbing through my book on Continuum Mechanics looking for a specific fact when I saw Mohr circles. I said:" wow, those are half plane modular form tessellation just like in the Elliptic curves book." They are forming half plane "triangles" in the Moebius/ bilinear transform sense. A little thought and a minimum of consultation in the Mathematica reference book and I had a Mohr circle based Fibonacci tessellation that represents an actual physical application of the golden mean. I've done a Goggle search and it appears nobody has thought of this before or put this set of 2 plus 2 together and got 4! I tried it for the next higher Bonaccis and it doesn't give this pleasing well formed set of curves. As most people in complex analysis know there is a classical half plane to disk transform that makes tessellations like this into tiling of the unit disk. These remind one of Ford circles, but they are definitely their kind that seems to be both unique and new. Here is the very simple notebook for generating this marvel: Clear[c, r, n] (* centers of Mohr stresses*) c[n_, 3] := (Fibonacci[n] + Fibonacci[n + 1])/2 c[n_, 2] := (Fibonacci[n] + Fibonacci[n + 2])/2 c[n_, 1] := (Fibonacci[n + 1] + Fibonacci[n + 2])/2 (* radius of Shear Stress*) r[n_, 0] := Abs[(Fibonacci[n + 1] - Fibonacci[n + 2])/2] r[n_, 1] := Abs[Fibonacci[n] - (Fibonacci[n + 1] + Fibonacci[n + 2])/2] r[n_, 2] := Abs[(Fibonacci[n] - Fibonacci[n + 2])/2] r[n_, 3] := Abs[Fibonacci[n + 1] - (Fibonacci[n] + Fibonacci[n + 2])/2] r[n_, 4] := Abs[(Fibonacci[n] - Fibonacci[n + 1])/2] r[n_, 5] := Abs[Fibonacci[n + 2] - (Fibonacci[n] + Fibonacci[n + 1])/2] a = Flatten[Table[{Circle[{c[n, i + 1], 0}, r[n, 2*i + j], {0, Pi}]}, {i, 0, 2}, {j, 0, 1}, {n, 1, 25}]]; Show[Graphics[a], AspectRatio -> Automatic, PlotRange -> All] Roger L. Bagula { email: rlbagula at sbcglobal.net or rlbagulatftn at yahoo.com } 11759 Waterhill Road, Lakeside, Ca. 92040 telephone: 619-561-0814