`Circular Coordinate System`
` `
`Subject:      Re: Reply to Do Points Have Area?`
`Author:       Jesse Yoder < jesse@flowresearch.com>`
`Date:         26 Jan 98 18:20:29 -0500 (EST)`
` `
`Hi John -`
` `
`I have read through your expression of frustration regarding my`
`discussion of circular geometry. After offline discussions with you, I`
`have come to realize that I am addressing two separate issues in the`
`new geometry I am developing, and I haven't been distinguishing them`
`adequately. These issues are as follows:`
` `
`1. Finding a new geometry that provides a rational value for the area`
`of a circle and does not rely on pi. This has to do with CIRCULAR`
`GEOMETRY, and it involves developing an alternative to the Cartesian`
`Coordinate System.`
` `
`2. Finding an analysis of the number line that avoids the paradoxes`
`generated by the assumption that a line is made up of infinitely many`
`dimensionless points.  This has to do with developing the concept of`
`Points (i.e, points with area), and it involves developing an`
`alternative to Euclidean geometry.`
` `
`I now believe that it is not possible to easily develop a Circular`
`Geometry (#1) that provides an alternative to the Cartesian Coordinate`
`system in terms of Points -- instead, I believe it should be done in`
`terms of a series of circles that provide an alternative to the X-Y`
`Cartesian Coordinate system. Once this is done, one could give either`
`a Euclidean analysis of the lines contained in this geometry, saying`
`it is made up of Euclidean points, or could then go on to analyze`
`these lines as made of of Points, with the number of Points changing`
`as the unit of measurement changes. I prefer the second of these two`
`options.`
` `
`In terms of developing Circular Geometry, an alterntative to the`
`Cartesian Coordinate system (#1), I would suggest the followng:`
` `
`Replacing the X axis in the Cartesian Coordinate system with a series`
`of unit circles laid out end to end in an east and west direction,`
`each with an area of one round inch, and a radius of 1/2 inch. These`
`unit circles INTERSECT (share a common point) at the interger points--`
`1, 2, 3, etc., and likewise on the negative side (-1, -2, -3, etc.),`
`as well as at the point of origin. These are circles with an area of`
`one round inch (not solid Points). `
` `
`Likewise, replacing the Y axis in the Cartesian Coordinate system with`
`a series of unit circles laid out end to end in a north and south`
`direction, each with an area of one round inch, and a radius of 1/2`
`inch. These unit circles INTERSECT (share a common point) at the`
`integer points 1,2, and 3 (in the positive direction) and -1, -2, -3,`
`etc. (in the negative direction), as well as at the point of origin.`
`These are circles with an area of one round inch (not solid Points).`
` `
`Once this structure is set up, it is possible to use these unit`
`circles to give the area of any circle in this circular coordinate`
`system, using the formula 4*r*r. Here r = the radius of the circle,`
`defined in the usual way (the distance from the center to the edge of`
`the circle). So a circle with a radius of 1/2 inch has an area of one`
`inch. A circle with a radius of 2 inches (and diameter = 4) has an`
`area of 4 round inches. The formula d*d, where d = diameter, also`
`works to find round inches.`
` `
`Once this structure is set up, it is possible to take the FURTHER step`
`of saying the lines making up the radius are made up of finitely many`
`Points with area, rather than being made of of infinitely many`
`dimensionless points. To say this is to give a non-Euclidean`
`interpretation of this Circular Geometry. While I want to say this, I`
`believe that the Circular Geometry described above (as many unit`
`circles laid end to end replacing the x and y axes) can stand on its`
`own with a Euclidean or a non-Euclidean interpretation.`
` `
`Once the two sets of intersecting series of circles are drawn, they`
`can be used as a frame of reference for describing other circles`
`within the geometric plane, much as the x and y axes are currently`
`used in Cartesian Coordinate geometry.`
` `
`I hope this helps describe more clearly what I have in mind. I will`
`desribe the second alternative (geometry with Points) in a separate`
`post.`
` `
`Jesse`

http://forum.swarthmore.edu/epigone/geometry-research/khulstaymerm/yl3btpequju7@forum.swarthmore.edu