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From: Archimedes Plutonium on 17 Mar 2010 06:47
Now in the case of Euclidean maximum tiler in 2D, some may think that
for the 100-
Model where 99.99 is the largest finite number and where 100 is
infinity, that the
square box of sides 99.99 can be 100% tiled by rectangles of length
99.99 and width
0.01 (where we make allowances that such is a unit area rectangle).

So the mistake here is that, yes, those rectangles of unit area 100%
tile the Euclidean
100-Model and likewise the 10^500 Model, but, the problem begins,
because in the
Conjecture, the maximum tiler has to tile not just Euclidean geometry
but at least one
of the other two geometries of Elliptic and Hyperbolic and be a winner
in one of those.
So when we transfer this rectangle of 99.99 by 0.01 to that of the
circle of diameter
99.99 we get a 0% tiling in Elliptic geometry and a 0% tiling in
Hyperbolic geometry.
So this tells us that the unit rectangle tiler has to be shorter and
wider than 99.99 by
0.01.

So to be the winning tiler, the best tiler or the maximum tiler, it
has to win in two of the
three geometries. And the restrictions that the tilers are unit area
tilers, and that the
containers in the three geometries is the 10^500 Model (or we can use
the 100-Model
since it is more convenient then writing out all those 9s).

So the above also is an arguement in the favor of the equilateral
triangle, because if you
make the rectangles too skinny and long, they cause too many gaps and
holes in the
Elliptic and Hyperbolic geometry containers and thus lose out.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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