Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.



Unexpected alignment of grid points on Poincar� circle
Posted:
May 31, 2014 8:54 AM


Background: for our grades34 Math Circle, I needed a picture of what I thought was a standard image in geometry: grid of squares where 5 squares meet at every corner (so these are 72°squares). However, in several minutes of Googling I could not find a ³photorealistic² image (neither in Poincaré, nor in Klein models; only ³topologicallycorrect² like images), so I needed to make one myself.
(Above, a square is a regular 4gon.)
(First of all, writing down the program turned out to be much trickier than what I naively expected.) Second, inspecting the output for bugs, I found that on the Poincaré flavor picture, many point look aligned along (Euclidean!) lines.
I modified the program to output guide lines, and mark points which are on a guide line with high precision (100 decimal places). The result is on http://ilyaz.org/math/squaregrid/squaregrid5guidelines.pdf
If you zoom in, you can see small blue circles about points which are on the red guidelines (with high precision). The guidelines go through ALL the grid points near the origin.
Can somebody see the reason for this alignment? (I asked a few people around, and it does not look like this is something widely known which I just missed!)

On the Lobachevsky plane, the guidelines are unclosed curves of constant curvature (less than the curvature of horocycles) (sometimes called hypercycles). The fact that the marked points are on SOME hypercycle is obvious. What is unclear is the precise relationship between the curvature of this hypercycle, and its distance to the origin (the relationship which makes the picture of this hypercycle a Euclidean lineas opposed to Euclidean arc of a circle).

If you think you know the answer, try to deduce whether the same would happen with 60°squares, and 360°/7squares. (The pictures are in the same directory, but some people may prefer not to see them until they can deduce the answer.) (For 7 squares meeting at every corner, the coordinates are in a number field of slightly higher degree!)
Thanks, Ilya



