Circular Inch
Subject: RE: Coordinate systems
Author: Jesse Yoder <>
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,
 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
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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