Re: a grid at (1/3, 1/3, ...)

On Mar 6, 8:39 am, Rainer Rosenthal <r.rosent...@xxxxxx> wrote:
Bob Jenkins wrote:
of the other. Apparently you'd need (in the limit) as many 17x17
blocks as all the other blocks combined.

Once again I'm without a clue. I do not understand, of what help
these rectangles shall be. What can be learned from an mxn rectangle
which will help with the bigger m'xn' rectangle?

But in the meantime I slowed down my investigations and made
some corrections by hand. That is: I used the basic pairing as
described inhttp://nagel-klaus.homepage.t-online.de/Gitter.pdf
and looked where there are bridges longer than 0.85.

In each case I was able to find a small local environment
where only 4 points p1, p2, p3, p4 from grid G had to
change partners (from grid H). This reduced the maximum
bridge length to a value smaller than 0.85.

The great thing is that the computation becomes easy because
of the few points needed and the low number of permutations
to be checked for max distance.

The minimal upper bound for the bridge length will probably
provable as something near 0.85.

I will repeat the relevant definitions in short:

G = grid of integer pairs (i,j)
H = rotated grid (angle 45°)

For each bijection b: G --> H let's define

mixtance_b(G,H) = sup{distance(g,b(g))|g in G)

I am searching for

mixtance(G,H) = inf{ mixtance_b(G,H) | b:G->H bijection}

We already know

mixtance(G,H) <= cos(Pi/8) = 0.924

because of Klaus Nagel's pairing function p:G->H which satisfies

mixtance_p(G,H) = cos(Pi/8).

The new explorations suggest mixtance(G,H) < 0.9.

Best regards,
Rainer .
. .
_________________________.___.___._______________________
Rainer Rosenthal . . r.rosent...@xxxxxx
.

Matchings in mxn block wouldn't help in defining matchings in an m'xn'
block. But any covering of the plane would use just four types of
rectangles: mxm, mxn, nxm, and nxn. For instance, m=7 n=17, starting
at the origin and walking along the x axis,

First you'd first go out 17, giving an error of -0.029.
Next you'd go out 7, giving an error of -0.029+0.071=0.041.
Next you'd go out 17, giving an error of 0.041-0.029=0.012.
Next you'd go out 17, giving an error of 0.012-0.029=-0.017.
And so forth. Always you'd keep the error under 0.05. The "error" is
how far the gridpoints in the 45-degree grid are from the 0-degree
grid along the x axis.

Same story for the y axis.

So, along the x axis, you'd put a 17x17, then a 17x7, then a 17x17,
then a 17x17, and so forth. Along the y you'd put a 17x17 (which is
on the x axis too), then a 7x17, then a 17x17, then a 17x17, and so
forth.

That defines a grid with lines all 7 or 17 apart that divides up the
whole plane into rectangles of dimension either 17x17, 7x7, 17x7, or
7x17.

.

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