`Circular Inch`
` `
`Subject:      RE: Coordinate systems`
`Author:       Jesse Yoder < jesse@flowresearch.com>`
`Date:         Tue, 10 Feb 1998 16:44:37 -0500`
` `
`On Feb. 10, 1998, John Conway wrote:`
` `
`> > RESPONSE[Yoder]: Possibly I did say at some point that Points touch`
`> at points.`
`> > But I also said that the relation between touching Points is modeled`
`> on`
`> > the relation between two physical objects ...`
`> `
`>     Conway: Well, yes, but it doesn't help us to say that, because it`
`> doesn't`
`> tell us precisely which usages of your words you'll consider correct`
`> ones.  Allowing yourself to import the usual language for physical`
`> ideas`
`> into your geometry robs it of any precise meaning, partly because the`
`> usual language is imprecise anyway, and partly because its more`
`> precise parts tend to embody Euclidean ideas.`
`> `
`RESPONSE: Maybe not, but I don't think that Euclidean ideas are all that`
`well defined either. `
` `
`You continue:`
` `
`> But it seems utterly ludicrous for an opponent of Euclidean`
`> geometry to base his rival to it on - guess what? - Euclidean`
`> geometry!`
`> `
`RESPONSE: I repeat that, just as Riemann built his whole geometry on a`
`different Axiom 5, I am at the very least starting with a different`
`axiom 1 (since I say that Points have area) -- and I have provided 11`
`other axioms as well which are not based on looking at Euclid's axioms`
`and rewriting them to suit myself.`
` `
`You continue:`
` `
`> Fine - so (in Euclidean terms) your "round inch" is the area of`
`> a circle of diameter 1 inch, and the area of a circle of diameter d`
`> is d^2 round inches.  `
`> `
`Response: Agreed.`
` `
`You continue:`
` `
`>    Well, I was thinking of Euclidean geometry, with "round inch"`
`> the above described non-primitive concept.`
`> `
`RESPONSE: This is another possible geometry, (as I say in an earlier`
`post today -- option #3), but what I am arguing for also says that`
`Points have area.`
` `
`        Conway: Of course it's trivial just to rescale things, and`
`trivial to`
`> remark that when you do so, you don't need pi to compare areas of`
`> circles.  In fact Euclid didn't use pi - he just has a theorem that`
`> "[the areas of] circles are to each other as the squares on their`
`> diameters".`
`> `
`>    Are you rejecting anything of Euclid, and if so, precisely what?`
`> How can you hope to persuade anyone to understand you if on the`
`> one hand you criticize him very strongly, and on the other hand,`
`> feel free to accept whichever Euclidean concepts you like?   If`
`> indeed you feel free to accept ALL of Euclid, then indeed everything`
`> you've said becomes a triviality, because, as I've pointed out, we`
`> can give Euclidean models for all your concepts that make all your`
`> assertions true.  If you DON'T want to accept all of Euclid, you`
`> have an obligation to point out what you reject (for instance his`
`> notion of points having no area), and then to be honest and not`
`> make any use of whatever ideas you reject.`
`> `
`Yoder: I think I've made it clear that I reject the idea that points`
`have no area, that lines have no width, and that planes have no depth.`
`The no-pi part is the anti-Cartesian part, which is why I say that`
`Circular Geometry has an anti-Cartesian and an anti-Euclidean component`
` `
`        Conway: So Points can and do overlap.`
` `
`Yoder: No, I cannot allow Points to overlap. Circles can overlap,`
`however.`
` `
`        Conway: So it's what Euclidean folk would call an annulus of`
`width 1 (taking`
`> that the to the diameter of a Point).`
`> `
`> > [Yoder] Circles can overlap, but I cannot allow Points to overlap `
`> `
`>  Conway:  This contradicts what you said earlier, that every disc of`
`> diameter 1`
`> is a Point, because discs of diameter 1 whose centers differ by less`
`> than 1 DO overlap.`
`> `
`>    I am afraid you must drop one or other of the two assertions that`
`> every Euclidean disc of diameter 1 is a Point and that two Points`
`> cannot`
`> overlap. If you don't do this, your ideas are inconsistent and I won't`
`> bother to listen to them any more.`
`> `
`Yoder: It sounds like I still haven't made myself completely clear. A`
`Point does not have diameter 1. The unit Circle has diameter 1. The unit`
`Circle is generated by rotating a Point around a fixed Point of the same`
`size (e.g., 1/16th of an inch, or 1/100th. of an inch). A "disc" is a`
`Point only if it is the smallest unit of measurement within a system (or`
`reflects the level of precision chosen for a particular measurement). I`
`believe that you did not see this distinction between Points and`
`Circles, and this is why you thought what I said was contradictory. When`
`I say "every disc is a Point", I didn't mean to include Circles -- A`
`dics is a Point only if it's the smallest unit of measurement -- and it`
`also can be used to generate a Circle. So this is how I would revise the`
`claim "Every disc is a Point" to avoid the contradiction you feel I have`
`fallen into.`
` `
`        Conway: There may be some confusion here.  I was locally asking`
`you to give `
`> the meanings of your concepts in Euclidean terms, which appear to be`
`> `
`>            Yoder         Euclid`
`> `
`>            Point  = disc of diameter 1`
`> `
`>            Circle = annulus of width 1`
`> `
`> (straight) Line   = strip of width 1`
`> `
`> except that I STILL don't know exactly WHICH Euclidean discs, annuli`
`> and`
`> strips of width 1 you're counting, since you say that Points cannot`
`> overlap.`
`> `
`>    Let me make it clear WHY I was asking for the meanings in Euclidean`
`> terms - it's because you haven't been able to give any coherent `
`> descriptions that don't presuppose the Euclidean ideas, and since you `
`> give yourself the freedom of using any Euclidean ideas you like.`
`> `
`Yoder: I think the confusion is in saying unit Circle is annulus 1 (or`
`one inch in diameter) and then also saying that the Point is diameter 1.`
`If the unit Circle is 1 inch in diameter (and note also that there's an`
`inside and outside diameter), then the Point is going to be less than an`
`inch, because it is by rotating the Point that you get the unit Circle.`
`So the Point would be something like 1/16th. of an inch, 1/100th. of an`
`inch, or 1/200th. of an inch -- whatever degree of precision you want.`
`The width of the Line making up the unit Circle = the diameter of the`
`Point.`
` `
`Ditto with the Line. Again, the Line will be the same width as the Point`
`(e.g., 1/100th. of an inch, or whatever). The Point remains the smallest`
`unit of measure, and it is used to generate the unit Circle and a Line.`
`These are as wide as the diameter of the Point.`
` `
`Jesse Yoder`

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