Re: [math-fun] {4, infinity} tiling of H^2 and a generalization of the mediant
This sounds extremely interesting, but I'm not familiar enough with the concepts; do you have an actual pictorial diagram? At 12:34 PM 7/4/2012, Mike Stay wrote:
In the Poincare half-plane model of hyperbolic 2-space, if we draw the {3, infinity} regular tiling of the hyperbolic plane and put the three points of one triangle at 0, 1, and infinity, then the rest of the points land on Farey fractions. Since the sides are semicircles, each triangle has one high side and two low sides. By placing a vertical line at x, we can read off the continued fraction for x by looking at which of the two low sides of each hyperbolic triangle it passes through---that is, a sequence like LRRRLRRLLL becomes [0,1,3,1,2,3].
What points do we get with the {4, infinity} tiling? For every two points, we'll get two new ones, a "mediant pair". In general, the {n+2, infinity} tiling will have a notion of mediant that gives an n-tuple of points.
What's the corresponding notion of continued fraction? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On Wed, Jul 4, 2012 at 3:32 PM, Henry Baker <hbaker1@pipeline.com> wrote:
This sounds extremely interesting, but I'm not familiar enough with the concepts; do you have an actual pictorial diagram?
Here are a bunch of pictures of the usual way of doing things: http://www-bcf.usc.edu/~fbonahon/STML49/FareyFord.html
At 12:34 PM 7/4/2012, Mike Stay wrote:
In the Poincare half-plane model of hyperbolic 2-space, if we draw the {3, infinity} regular tiling of the hyperbolic plane and put the three points of one triangle at 0, 1, and infinity, then the rest of the points land on Farey fractions. Since the sides are semicircles, each triangle has one high side and two low sides. By placing a vertical line at x, we can read off the continued fraction for x by looking at which of the two low sides of each hyperbolic triangle it passes through---that is, a sequence like LRRRLRRLLL becomes [0,1,3,1,2,3].
What points do we get with the {4, infinity} tiling? For every two points, we'll get two new ones, a "mediant pair". In general, the {n+2, infinity} tiling will have a notion of mediant that gives an n-tuple of points.
What's the corresponding notion of continued fraction? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On 7/4/12, Mike Stay <metaweta@gmail.com> wrote:
Here are a bunch of pictures of the usual way of doing things: http://www-bcf.usc.edu/~fbonahon/STML49/FareyFord.html
Those are great illustrations, but the label positioning isn't working. Here is what I get in Chrome 18.0 on MacOS 10.6.8 (Safari gives me exactly the same thing) mrob.com/users/fbonahon/farey-broken-labels.png The "1/7", "1/6", and "1/5" should be closer together and further to the right; virtually all the rest of the labels are in the wrong places too. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
On Wed, Jul 4, 2012 at 6:01 PM, Robert Munafo <mrob27@gmail.com> wrote:
On 7/4/12, Mike Stay <metaweta@gmail.com> wrote:
Here are a bunch of pictures of the usual way of doing things: http://www-bcf.usc.edu/~fbonahon/STML49/FareyFord.html
Those are great illustrations, but the label positioning isn't working. Here is what I get in Chrome 18.0 on MacOS 10.6.8 (Safari gives me exactly the same thing)
It should: they use the same rendering engine, WebKit.
mrob.com/users/fbonahon/farey-broken-labels.png
The "1/7", "1/6", and "1/5" should be closer together and further to the right; virtually all the rest of the labels are in the wrong places too.
Yeah, the guy who put the page together undoubtedly only checked it in FireFox. Here's how it renders there: http://i.imgur.com/CO24J.png -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Robert Munafo