[math-fun] {4, infinity} tiling of H^2 and a generalization of the mediant
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 12:34 PM, Mike Stay <metaweta@gmail.com> 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?
The nearest integer continued fraction! The four sides correspond to the transformations x |-> x - 2 x |-> 1/x x |-> -1/x x |-> x + 2 -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Mike Stay