Reply to Candice
 
Subject: RE: REPLY TO "RE: REPLY TO POINTS HAVE AREA?"
Author: Jesse Yoder < jesse@flowresearch.com>
Date: Mon, 2 Feb 1998 08:39:03 -0500
 
Hi Candice -
 
First of all, let me say that I like your convention of capitalizing
your ENTIRE ANSWER, which is both consistent with the spirit of Circular
Geometry, and seems to give your comments added importance. 
 
On SUNDAY, FEBRUARY 1, 1998, at 6:35 PM, YOU WROTE, BEGINNING WITH A
QUOTE FROM ME ABOUT THE DEFINITION OF CIRCLES:
 
> [YOU (i.e. Jesse) WROTE ON JANUARY 31, 1998]
> 
> >A circle by its very nature (in other words, by definition), is a
> >continuous circular line. This is what's wrong with the traditional
> >definition of a circle as "a set of points equidistant from a fixed
> >point." If these points aren't "continuous", there is no circle, but
> >merely a set of points arranged in a circular fashion. I believe that
> >the Euclidean tendency to identify a line with "infinitely many 
> >points" tends to obscure the requirement that the points lying on a 
> >circle must be continuous in order for a circle to exist.
> 
> RESPONSE (from Candice): WHERE DID YOU EVER READ THAT A CIRCULAR HAD
> TO BE A
> CONTINUOUS CIRCULAR LINE???  DEPENDING ON WHAT FORM OF GEOMETRY YOU'RE
> USING, A CIRCLE COULD CONSIST OF FOUR POINTS. IF YOU WERE TO HAVE
> TAXICAB GEOMETRY WHERE POINTS COULD ONLY EXIST ON THE "CORNERS", THEN,
> IF YOU USE EUCLIDEAN'S DEFINITION OF A CIRCLE (ALL POINTS EQUIDISTENT
> FROM A FIXED POINT), YOU GET A CIRCLE THAT CONSISTS OF FOUR POINTS.
> 
RESPONSE: I didn't read it anywhere that a circle has to be a continuous
circular line. Instead, I take this to be implicit in the very concept
of a circle. And you example, from taxicab geometry, simply shows the
total bankruptcy of the Euclidean definition of a circle as a set of
points equidistant from a fixed point. If four points equidistant from a
circle can actually BE a circle, then I suppose the four corners of a
square and actually BE a square, and the three tips of a triangle can
actually FORM a triangle. 
 
Let me also say that after someone mentioned the idea of taxicab
geometry a few months ago, I went out and bought a book on taxicab
geometry. While I don't have it here to refer to, I remember enough of
this to understand what you mean be saying that you have points that are
the intersections of streets (and note that these points are actually
squares or rectangles, NOT circles). And what I would say to you is that
there are no circles in taxicab geometry, and there are no Circles
either, because there is no circular area (unless the streets happen to
be circles, in which case they will be Circles, since they have width). 
 
To reiterate, circles and Circles are continuous closed loops or Loops
and not merely a set of points equidistant from a fixed point. The
Euclidean definition only arises because a line is analyzed as being
made up of infinitely many points with no dimension. But this analysis
of the line has to be rejected because it leads to a paradox. In its
place, I propose the Line, which is created by putting a Point in
motion.
 
Thanks for your comments, and I look forward to your response!
 
Jesse

http://forum.swarthmore.edu/epigone/geometry-research/dwodeldsal

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