Subject: Re: Reply to Do Points Have Area

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
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