`Points & points`
` `
`Subject:      Re: Reply to Do Points Have Area?`
`Author:       John Conway <conway@math.Princeton.EDU>`
`Date:         Thu, 22 Jan 1998 16:44:27 -0500 (EST)`
` `
` `
`  Jesse, I have tried, and tried very hard,  to understand what you're `
`saying, but have reached the point at which I'm about to give up.  Before `
`I do so, I'm making one last try (I will make more "last tries" if I`
`get something out of this one!).  `
` `
`  Let me say that I am entirely happy with your basic idea of getting rid `
`of Euclid's fiction of "points with no magnitude".  It's just that I have `
`not managed to get any kind of understanding of what you think you are `
`putting in its place.  Several times I have asked you direct questions, `
`but the answers have been no help in telling me what you're thinking about.`
` `
`   The closest I got was when you told me that the Points in your plane `
`looked like:`
` o o o o o o o o o o o o o  `
` o o o o o o o o o o o o o `
` o o o o o o o o o o o o o `
` o o o o o o o o o o o o o `
`(but magified so that they touch).  But then you went on to define a Circle`
`to be a continuous string of these, from which I deduced that in fact `
`there couldn't be any non-trivial Circles.  Then it turned out that it`
`wasn't the set of all Points in your plane that looked like the above `
`figure, but only those in the coordinate-system (or something). Forgive `
`me if I'm getting this wrong, but but I really am confused.  `
` `
`   So I got the idea that there were more Points besides those in the `
`coordinate system.  It seemed to me that (taking a suitable unit), the `
`points in the coordinate system were discs of unit diameter centered at `
`(Euclid's) points with integer coordinates, while perhaps there were`
`also other Points (which were also discs of unit diameter) centered at `
`other points.  This would then allow there to be Circles in the sense in `
`which you defined that term, i.e., continuous closed loops of Points all `
`at the same distance from a given Point, namely the discs of unit `
`diameter centered at the vertices of one of Euclid's regular polygons of `
`edgelength 1.  So I asked you explicitly whether there was any `
`difference between this model and your geometry, and you said something `
`like "well, let's try that".  Well, I don't want to just try something.`
`I'm perfectly capable of studying all kinds of geometry and working out `
`their properties; but what I want to know is precisely what you are `
`thinking about, and you don't seem to be capable of telling me.  I really `
`don't know just what it is. `
` `
`   I ask again.  Are all the Points of your plane arranged in an array `
`like the above, or are there others?  Can two Points overlap withouyt `
`being equal?  Is there any difference between your kind of plane geometry `
`and the set of all unit discs in Euclid's geometry, and if so, just what `
`is this difference?  Or have you not yet understood your own ideas in `
`enough detail to be able to give answers to these questions?`
` `
`    John Conway`

http://forum.swarthmore.edu/epigone/geometry-research/khulstaymerm/Pine.SUN.3.91.980122154603.23265C-100000@math.princeton.edu