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Subject:      RE: Coordinate systems
Author:       John Conway <conway@math.Princeton.EDU>
Date:         Tue, 10 Feb 1998 14:11:15 -0500 (EST)



On Tue, 10 Feb 1998, Jesse Yoder wrote:
> RESPONSE: As I have pointed out in previous discussion, there is an
> anti-Eucldean and an anti-Cartesian element to what I call Circular
> Geometry:
> 
> The anti-Euclidean element consists of saying that Points have area
> (unlike points, which have no area), and that Lines have width and
> length (unlike lines, which have only length. 

   Yes, this is the sort of thing I thought you were trying to say.
But with "Points" being defined in Euclidean terms as certain discs it's
a tautology - obviously discs have area.  Also, what purpose is there in 
defining your anti-Euclidean ideas in Euclidean terms?  You don't seem
to realise how silly it sounds to say that "Points touch at points"
when half of your purpose is to abolish the use of points.  An anti-Euclidean
geometry that can only be built on a foundation of Euclidean geometry
doesn't sound to me to be a very successful opponent!   Aren't you capable
of developing it in its own terms?

> The anti-Cartesian element consists of analyzing circular area in terms
> of round inches rather than square inches.

    I don't know why you want to do this, and why it isn't any more
than a triviality.  In Euclidean terms we can define "a circular inch"
(I deliberately use a term other than "round inch" because I'm not
quite sure what you want that to mean) to be the area of a circle
of radius 1.  Then it follows from Euclid's theorems that the area
of a circle of radius R is  R^2 round inches.  

    I think that this means that Euclid provides a foundation that
can do what you want about round inches (and, of course, can also
do much more, that you don't want).  So again we come to the 
question you haven't really faced: it seems to be trivial to define
your kind of geometry on the foundation of Euclidean geometry - can
you do it WITHOUT assuming this foundation?

    I may remark that even in Euclidean terms I still haven't got
much of a clue about the meaning of your proposed terms.  This may
be just because I've forgotten answers you may have given to some
of my questions, so I'll repeat them. 

   At one time we agreed that Points could be taken to be discs of
diameter one.  Is every such disc a Point, or only those centered 
at points (x,y) with integer coordinates; or maybe some other set?
In particular, can Points overlap without being equal?

  Also, can you remind me what Circles and straight Lines are; and
if they are not made up of Points, what it means for a Point to
be "on" a Line or Circle?

     John Conway

http://forum.swarthmore.edu/epigone/geometry-research/sarswimpbli/Pine.3.07.9802101415.A3052-c100000@broccoli.princeton.edu

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