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Topic: Unexpected alignment of grid points on Poincar� circle
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Ilya Zakharevich

Posts: 152
Registered: 12/13/04
Unexpected alignment of grid points on Poincar� circle
Posted: May 31, 2014 8:54 AM
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Background:
  for our grades-3­4 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 ³photo-realistic²
  image (neither in Poincaré, nor in Klein models; only
  ³topologically-correct² like images), so I needed to make one myself.

    (Above, a square is a regular 4-gon.)

(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/square-grid/square-grid-5-guidelines.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 line---as 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°/7-squares.  (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



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