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Circular Inch Subject: RE: Coordinate systems Author: Jesse Yoder < jesse@flowresearch.com> Date: Tue, 10 Feb 1998 15:32:15 -0500 On Feb. 10, John Conway wrote: > Yes, this is the sort of thing I thought you were trying to say [that > points have area].But with "Points" being defined in Euclidean terms > as certain discs it's > a tautology - obviously discs have area. Also, what purpose is there > in > defining your anti-Euclidean ideas in Euclidean terms? You don't seem > to realise how silly it sounds to say that "Points touch at points" > when half of your purpose is to abolish the use of points. > RESPONSE: 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 -- (e.g., two baseballs), and I don't believe they share any points (though according to the physics book I was reading this morning, their molecules may influence each other). So I don't believe that I'm committed to saying that Points touch at points. Also, I don't think that the Euclidean account of the relation between two points that are "next" to each other is all that clear. You continue: > An anti-Euclidean > geometry that can only be built on a foundation of Euclidean geometry > doesn't sound to me to be a very successful opponent! Aren't you > capable > of developing it in its own terms? > RESPONSE: I'm glad you said this again, because it has recently occurred to me that other non-Euclidean geometries are built up just by denying ONE Eucldean axiom, viz. the Fifth or Parallel postulate. Yet these geometries (e.g., Riemannian) accept much of the rest of Euclidean geometry. Circular Geometry is based in part on denying the first axiom (A point is that what has no part). So I disagree with you that to be interesting or worthwhile, a geometry has to start completely from scratch and proceed on totally independent terms. You continue: > > (Yoder:)The anti-Cartesian element consists of analyzing circular > area in terms > > of round inches rather than square inches. > > (Conway:)I don't know why you want to do this, and why it isn't > any more > than a triviality. In Euclidean terms we can define "a circular inch" > (I deliberately use a term other than "round inch" because I'm not > quite sure what you want that to mean) to be the area of a circle > of radius 1. Then it follows from Euclid's theorems that the area > of a circle of radius R is R^2 round inches. > RESPONSE: I prefer to define a round inch with a radius of 1/2 inch and use the formula 4 * r^2, or simply d * d. > > > I think that this means that Euclid provides a foundation that > can do what you want about round inches (and, of course, can also > do much more, that you don't want). So again we come to the > question you haven't really faced: it seems to be trivial to define > your kind of geometry on the foundation of Euclidean geometry - can > you do it WITHOUT assuming this foundation? > RESPONSE: I believe you are referring to circular geometry, i.e., a geometry that uses the round inch as a primitive, but accepts the Euclidean definitions of 'point', 'line', and 'circle.' In your example, if we use the round inch and define the area of circles using 4*(r^2), we can then eliminate pi from consideration (at least as long as all we're doing is describing the areas of circles). I'm not sure why this is so trivial. Again, I don't feel compelled to reject everything Euclidean to develop a geometry -- Riemann certainly didn't. You continue: > I may remark that even in Euclidean terms I still haven't got > much of a clue about the meaning of your proposed terms. This may > be just because I've forgotten answers you may have given to some > of my questions, so I'll repeat them. > > At one time we agreed that Points could be taken to be discs of > diameter one. Is every such disc a Point, or only those centered > at points (x,y) with integer coordinates; or maybe some other set? > In particular, can Points overlap without being equal? > RESPONSE: Agreed. Pints are discs of diameter one, that are considered "unbreakable" for a particular measurement. That means they are the smallest unit area allowed for a particular measurement. Every such disc is a Point -- though some delineate integer coordinates, viz. those located at intersecting Circles at integer Points. > Also, can you remind me what Circles and straight Lines are; and > if they are not made up of Points, what it means for a Point to > be "on" a Line or Circle? > Circles are not discs or Points. Circles have area (they are not solid), and they are created by revolving a Point around a fixed Point of the same size. What's more, the width of the Lines making up the Circles is the diameter of the Points. So if a Point is 1/16th. of an inch, then a Circle is generated by rotating a Point around a Point of 1/16th. of an inch. Circles can overlap, but I cannot allow Points to overlap (but then Euclidean points don't overlap either). If I allow Point to overlap, then I can't really describe the overlapping area without shifting to a different frame of reference. So instead of overlapping Points, I'd say let's just shift to a more precise or smaller frame of reference up front, and not allow Points to overlap. Straight Lines are formed by moving a Point in a uniform direction. The width of the Line is equal to the diameter of the Point. It pains me greatly to have to admit straight lines in this geometry, but I don't know how to avoid it since I need the definition of 'diameter' and 'radius.' I don't know how to directly answer the question what it means for a Point to lie on a Line, except to say that it's like a cup sitting on a table. I am trying to avoid the paradoxes of continuity that arise by saying that a line is MADE UP OF points. So I choose to define a Line as the path of a moving Point, and say that Points lie on the Line, rather than in the line. The number of Points lying on a Line will vary with the measurement, and with the degree of precision selected. So points are like clothes hanging out to dry on a clothesline -- the Line is always there, but the density of the Points varies with the particular wash (measurement). Jesse Yoder
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