Circular Inches

 
 
Subject: RE: Coordinate systems
Author: John Conway <conway@math.Princeton.EDU>
Date: Tue, 10 Feb 1998 15:28:43 -0500 (EST)
 
 
 
On Tue, 10 Feb 1998, Jesse Yoder wrote:
    • 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 ...

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.

> 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.

Well, that at least is VERY clear - NO two distinct points are "next to each other" in Euclidean geometry.

> 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.

But it seems utterly ludicrous for an opponent of Euclidean geometry to base his rival to it on - guess what? - Euclidean geometry!

> 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.

 

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.

> > 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.'

Well, I was thinking of Euclidean geometry, with "round inch" the above described non-primitive concept.

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.

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.

> You continue:

> (Conway): 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: (Yoder): Agreed. Points 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.

So Points can and do overlap.

> > (Conway) 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.

 
 So it's what Euclidean folk would call an annulus of width 1 (taking
that the to the diameter of a Point).
 
> Circles can overlap, but I cannot allow Points to overlap 
 
 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.
 
(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.'
 
 So a Line is just a Euclidean strip of width 1.
 
> 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
 
 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.
 
 John Conway

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