`Reply to Jesse`
` `
`Subject:      Reply to "Reply to Do Points Have Area?"`
`Author:       Candice Hebden <dreamy_aurora@hotmail.com>`
`Date:         21 Jan 98 08:54:05 -0500 (EST)`
` `
`Hey Jesse, `
` `
`On January 20, 1998 `
` `
`[You said]`
`>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. This 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.`
` `
`So,  if this is all true, then there would be much "empty" space in`
`certain frames of reference and less "empty" space in others.  Not`
`everyone measures the amount of gas in their car by tenths of a`
`gallon.  In fact, most of the world doesn't even know what a gallon`
`is!  Every frame of reference you make will have to be stated before`
`any work is done on the problem.  Still then, many people might not`
`understand your frame of reference!  `
` `
`Sometimes the simpler theory is more "correct" because it makes`
`sense.  I certainly am not a believer of Euclid's arealess point, but`
`it does have it's merits.  People once thought that the Earth was the`
`center of the Universe.  Aristotle made all kinds of rules to support`
`his theory in respect to the "strange" orbits of Jupiter's satellites`
`and moons.  But Copernicus's idea of the Heliocentric galaxy`
`(although not widely accepted at first) was simpler and makes more`
`sense.`
` `
`I am not doubting the accuracy for your circular geometry.  It seems it`
`will make sense once certain things are worked out.`
` `
`[You then continue, answering my first question]`
`>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 still don't understand.  All space must contain points right? `
`Aren't points supposed to define the space of something?  If this is`
`true then there is space unaccounted for... making an infinite amount`
`of non-space!  If it isn't true then state it.  And then explain to`
`me, please, how space could go from empty to not-empty with a change of`
`frame of reference.`
` `
`John Conway earlier posed the sujestion that you lay your points on a`
`hexagonal frame.  As the frame of reference decreases, there is always`
`a model for the arrangement of the points so that there is less empty`
`space.  Would you want to use something like that?  Or would that`
`further confuse the issue?`
` `
`[You continue again]`
`>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.`
` `
`I am in complete agreement with the failure of Euclidean's arealess`
`points.  I also am in agreement with you that a new type of geometry`
`should be devised.  However, I do not believe it has been completed`
`yet!  Good luck.  I really hope you can "fix" what the new geometry of`
`yours needs!`
` `
`Yours, `
` `
`Candice Hebden`
`@`
`dreamy_aurora@hotmail.com`
` `

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