Getting a tiling inflation to work in the hyperbolic plane turns out to be harder than it looked at first sight --- because, of course, there is no such thing as similarity --- so the Penrose tiling analogy is probably a red herring. Maybe there's a nice canonical form for the associated hyperbolic (symmetry) group? On 8/8/07, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
Someone came up to me the other day and said, ``I'm Don Knuth'', to which I answered, ``I see you are.'' He first showed how all integers, including the negative ones, could be uniquely represented using base -2 and also, under the Zeckendorf condition of no two adjacent one-bits, using the negative Fibs, F_-n, i.e., 1, -2, 3, -5, 8, -13, ... . I can't, off the top of my head, give the correspondence twixt triads of digits and the below-mentioned tiling of the hyperbolic plane with pi/2, pi/4, pi/5 triangles. R.
What "triads of digits" are these --- could you give an example? WFL On 8/7/07, James Propp <jpropp@cs.uml.edu> wrote:
Did anyone go to Knuth's talk at MathFest this past weekend?
It sounded interesting, but I wasn't able to go and couldn't find a write-up on the web; I'm hoping one of you can summarize.
Here's what's on the web:
PI MU EPSILON J. SUTHERLAND FRAME LECTURE
NEGAFIBONACCI NUMBERS AND THE HYPERBOLIC PLANE
Donald E. Knuth, Stanford University, Professor Emeritus of the Art of Computer Programming
Saturday, August 4, 8:00 pm - 8:50 pm
All integers can be represented uniquely as a sum of zero or more "negative" Fibonacci numbers F-1 = 1, F-2 = -1, F-3 = 2, F-4 = -3, provided that no two consecutive elements of this infinite sequence are used. The NegaFibonacci representation leads to an interesting coordinate system for a classic infinite tiling of the hyperbolic plane by triangles, where each triangle has one 90 degree angle, one 45 degree angle, and one 36 degree angle.
Jim Propp