Re: Cantor Confusion

In article <34i6m255sfv5upv6nenqm6menle2gp8l8b@xxxxxxx> Lester Zick <dontbother@xxxxxxxxxxx> writes:
On Tue, 21 Nov 2006 03:04:42 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:
In article <8m8pl2pj1icbeven2hq7rp4hq1rufqh1u2@xxxxxxx> Lester Zick <dontbother@xxxxxxxxxxx> writes:
...
Why are square circles unimaginable?

They are not. With the Manhattan measure of the plane, each circle is
a square.

Then what is a square?

Pray, tell me. I would say that a straight line in the Euclidean plane
is a line of the form ax + by = c. With the standard formulas for angles
it is easy enough to get rectangles. And I would say that a rectangle is
a square when the sides have equal length (this is the point where the
measure creeps in). So we have a rectangle enclosed by the lines:
x + y = 1
x - y = 1
- x + y = 1
- x - y = 1
Now define the Manhattan measure:
d((x1,y1), (x2,y2)) = ||x1 - x2| + |y1 - y2||
and we see easily enough that the figure enclosed by the lines above is
a square with sides with length 2.

A circle is a figure where each point has the same distance to a common
centre, and it is also easy to show that the points on the boundary of
that square have the same distance to the origin: 0.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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