`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