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Do Points
Have Area? 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 suggestion 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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