`Reply to Candice`
` `
`Subject:      Reply to Do Points Have Area?`
`Author:       Jesse Yoder jesse@flowresearch.com`
`Date:         20 Jan 98 17:39:32 -0500 (EST)`
` `
`Hi Candice -`
` `
`Good to hear from you again! You recently asked a couple of questions`
`about my geometry, as follows:`
` `
`I have two questions about your circular geometry.  On December 18,`
`you posted`
` `
`[John Conway]`
`>"If we're just talking about some purely conceptual space then the`
`assertions are meaningless until that space is somehow defined.  `
`Jesse speaks of "circular geometry", in which a "point" is the`
`smallest unit area, and in other statements he's made it clear`
`that he thinks of these "points" as little circles and lines`
`as like strings of beads:  oooooooooooooooooo, in which `
`any two adjacent ones touch each other at a point."`
` `
`[You]`
`"Response: You seem to understand pretty well what I mean. Here is how`
`a plane would look, with lots of points;`
` `
`oooooooooooooooooooooooooooooooooooooooooooooooooo`
`oooooooooooooooooooooooooooooooooooooooooooooooooo`
`oooooooooooooooooooooooooooooooooooooooooooooooooo`
`oooooooooooooooooooooooooooooooooooooooooooooooooo`
`oooooooooooooooooooooooooooooooooooooooooooooooooo`
` `
`The above points are circular, solid, and touching horizontally as`
`well as vertically. I can't draw a solid circle with this email`
`system. A point, as you say, is the smallest, allowable round unit`
`area in a system."`
` `
`        What do you call the area between the points?  Isn't there always a`
`smaller sized point?`
` `
`Response: First off, let me take the second question. John Conway has`
`suggested I adopt a convention for indicating when I am using 'point'`
`in my sense, so I am capitalizing Point and Line. The answer is No,`
`there isn't always a smaller sized Point, since when a measurement is`
`made, you have to specify a frame of reference that says how small the`
`points are allowed to go. Thsi is often implicitly understood. For`
`example, if I'm measuring miles from work to home, I measure in tenths`
`of a mile. When I measure the amount of gas put in my car, I measure`
`in tenths of a gallon. The distance from here to the sun is measured`
`in miles. The positions of computer chips on a board might be measured`
`to the ten thousandth of an inch. Deciding what your frame of`
`reference is determines the size of your Points. Of course, there is`
`always ROOM FOR another point, but all that means is that you are`
`shifting to a different frame of reference, in which case again there`
`will be no smaller sized Points within this new frame of reference.`
` `
`As for the space between points, the answer is that this is`
`mathematical space that can be referenced in relation to Points on the`
`coordinate system.`
` `
`I hope this helps. I just read an account of the Euclidean idea that`
`points have no area, yet somehow make up a line in a book called The`
`Non-Euclidean Revolution by Richard Trudeau. This convinces me once`
`again that it is simply paradoxical to say, on the one hand, that`
`points have no dimension, and, on the other hand, that a line, which`
`has length, is made up of infinitely many of these dimensionless`
`points. Mutiplying 0 by infinity still equals 0. As far as I can see,`
`this remains an unresolved problem for Euclid's Axiom One (definition`
`of point), and I believe that ascribing area to points is the only way`
`around it.`
` `
`Yours,`
` `
`Jesse`

http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/gmf94q05o3xo@forum.swarthmore.edu