^^Coordinate systems. John Conway. Quadrays coordinates.
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Coordinate systems
11 post di 4 autori
Clifford J. Nelson
08/02/98
On Wed, Feb 4, 1998 7:01 AM, John Conway
wrote:
.... In fact it's a mere matter of convenience whether
we use orthogonal or hexagonal coordinates - neither is intrinsically
better than the other; it's just that some coordinate-systems are better
suited to some problems than others. I've used dozens of different
coordinate-systems in my life, as have most other professional
mathematicians,
...
John Conway
-----------------------------------------
I have spent about fifty dollars a month for about twenty years on math
books and some time in libraries and I have only run across two coordinate
systems for the uncurved plane: the perpendicular XY system and polar
coordinates. I discovered what I call the Synergetics or simplex
coordinates for the plane in 1994, but I can't find any books about them.
Could you tell me the names of the coordinate systems for the uncurved
plane and maybe steer me to some books? Thank you.
Cliff Nelson
John Conway
08/02/98
On 8 Feb 1998, Clifford J. Nelson wrote:
> I have spent about fifty dollars a month for about twenty years on math
> books and some time in libraries and I have only run across two coordinate
> systems for the uncurved plane: the perpendicular XY system and polar
> coordinates. I discovered what I call the Synergetics or simplex
> coordinates for the plane in 1994, but I can't find any books about them.
>
> Could you tell me the names of the coordinate systems for the uncurved
> plane and maybe steer me to some books? Thank you.
>
> Cliff Nelson
It's a bit hard to answer this, precisely because switching to a new
coordinate system is a pretty trivial business; people just say something
like "use the following as coordinates" rather than "use So-and-so
coordinates".
First, there are various systems of trilinear coordinates for the
plane (becoming (n+1)-linear in n-dimensions), which were used
particularly in projective geometry and so are often called "projective
coordinates". These were popularized by Mobius in the middle of the
last century in his little book "The barycentric calculus", and
one particular variety of them is called "barycentric" coordinates.
They are still very much used in the geometry of a triangle, and
since they shade off into a variety of other systems, I'll describe
them first.
Mobius' idea was to use (x,y,z) for the center of gravity you get
by putting masses x,y,z at the vertices A,B,C of a fixed triangle.
Obviously you get the same CG if you use masses kx,ky,kz, so that
the coordinates (kx,ky,kz) (k not 0) represent the same point as (x,y,z).
This is the "projective" property, so that barycentric coordinates
are a particular case of projective coordinates. We call the
coordinates normalized if x+y+z = 1. There are some triples
(x,y,z) that you can't normalize, because x+y+z = 0 : then you can think
of (x,y,z) as representing a vector, and the set of all (kx,ky,kz)
as either a "direction" or a "point at infinity".
Suppose you take Euclidean 3-dimensional coordinates. Then the
condition x+y+z = 1 determines a plane, and so in this case the normalized
barycentrics can be thought of as using 3 Euclidean coordinates in this
plane. As k varies, the points (kx,ky,kz) represent all the points of
a line through the origin, so making them represent the same point
is really centrally projecting the rest of 3-space onto this plane
from the origin. But we could use other projections - for instance
orthogonal projection, under which (x+k,y+k,z+k) would represent
the same point as (x,y,z). Now you could normalize instead by
taking x+y+z = 0 if you like. Sometimes the name "simplicial"
coordinates has been used for this, so that using n+1 coordinates
for an n-space with the condition that their sum is zero would
be "normalized simplicial coordinates", from which you get the
unnormalized ones by letting (x+k,y+k,...) represent the same
point as (x,y,...).
This is the same as projecting in the direction of the vector
(1,1,1,...) - in generalized simplicial coordinates you'd project
in the direction of some other vector. You can combine the two
types of projection by using n+2 coordinates for an n-space, with
the understanding that (Kx+k,Ky+k,...) should represent the
same point as (x,y,...). The name "pentahedral coordinates" is
used for this in the case n = 3. Of course pentahedral coordinates
would be the natural choice to use for a problem that involved 5
particular planes.
The above are the most common systems of "linear" coordinates,
that adjective meaning that lines, planes, etc., are determined by
linear equations. People studying such subjects as potential theory,
fluid dynamics and the like use all sorts of non-linear coordinate
systems determined by the particular shapes that concern them. So
for instance you'd use spherical polar coordinates (r,theta,phi) for
a problem involving spheres, cylindrical polars (r,theta,z) for one
involving cylinders, ellipsoidal coordinates (often called "confocal"
coordinates" for one involving ellipsoids, and so on. Confocal
coordinates are so called because their level surfaces are a confocal
system of quadrics (or conics in 2 dimensions, where they are also
called "elliptic coordinates").
John Conway
Jesse Yoder
08/02/98
John Conway wrote -
>"People studying such subjects as potential theory,
fluid dynamics and the like use all sorts of non-linear coordinate
systems determined by the particular shapes that concern them. So
for instance you'd use spherical polar coordinates (r,theta,phi) for
a problem involving spheres, cylindrical polars (r,theta,z) for one
involving cylinders, ellipsoidal coordinates (often called "confocal"
coordinates" for one involving ellipsoids, and so on."
RESPONSE: I find your comments on fluid dynamics very interesting, since
I claim that Circular Geometry has implications for the measurement of
fluid flow in pipes. So what you are saying, then, is that engineers
measuring circular pipes use coordinate systems that are appropriate for
the particular circular shapes they are measuring. I find this very
encouraging to hear.
I also find it interesting to hear you say this, since you have
indicated that you don't find the idea of selecting a Point size to suit
a particular measurement to be a worthwhile concept -- yet you claim
that other people already do something even more extreme -- select a
different coordinate system to suit a particular measurement. So
according to you, it's just fine to shift coordinate systems to make a
particular measurement, which is implied by what I have been saying all
along.
Jesse Yoder
John Conway
08/02/98
On Sun, 8 Feb 1998, Jesse Yoder wrote:
> RESPONSE: I find your comments on fluid dynamics very interesting, since
> I claim that Circular Geometry has implications for the measurement of
> fluid flow in pipes. So what you are saying, then, is that engineers
> measuring circular pipes use coordinate systems that are appropriate for
> the particular circular shapes they are measuring. I find this very
> encouraging to hear.
Thanks for your thanks. I think it's not really in "measuring
circular pipes" so much as in considering (say) fluid flow around
such pipes, that a special coordinate-system would be used.
> I also find it interesting to hear you say this, since you have
> indicated that you don't find the idea of selecting a Point size to suit
> a particular measurement to be a worthwhile concept -- yet you claim
> that other people already do something even more extreme -- select a
> different coordinate system to suit a particular measurement. So
> according to you, it's just fine to shift coordinate systems to make a
> particular measurement, which is implied by what I have been saying all
> along.
>
> Jesse Yoder
I think that what you call "selecting a Point size to suit a
particular measurement" is obviously very sensible, but that it's
silly to use "Point" and "point" with different meanings. Have you
any strong reason for doing this rather than using the same language
as everyone else - for instance saying something like "working to
within a tolerance of 1/100 of an inch"? I thought you had, namely
that you felt Euclid's "points" didn't exist, but perhaps some
absolute "Points" did.
I'm surprised that you regard changing a coordinate-system as
"extreme": to me it seems a rather trivial and practical matter.
I have a suggestion to make. If it's really the case that
your distinguishing between "points" and "Points" is practical
rather than theoretical, why not just drop it and use a more
traditional language in the interests of better communication
even if (like many other people) you'd really prefer it if language
hadn't developed in the way it has? I say this because talking
about the use of words is much less valuable than talking about
the things they represent, and your present terminology has made
it very difficult (at least for me) to understand what you're
really trying to say.
John Conway
Jesse Yoder
08/02/98
John Conway wrote -
> Thanks for your thanks. I think it's not really in "measuring
> circular pipes" so much as in considering (say) fluid flow around
> such pipes, that a special coordinate-system would be used.
>
RESPONSE - You're welcome. I guess I was speaking carelessly -- I didn't
mean measurement of circular pipes so much as measuring fluid flow
within such pipes (I don't know where you would measure flow around
pipes, unless you're talking about orifice plate measurement or open
channel flow measurement, where you use weirs or flumes).
Then you wrote:
> I think that what you call "selecting a Point size to suit a
> particular measurement" is obviously very sensible, but that it's
> silly to use "Point" and "point" with different meanings. Have you
> any strong reason for doing this rather than using the same language
> as everyone else - for instance saying something like "working to
> within a tolerance of 1/100 of an inch"? I thought you had, namely
> that you felt Euclid's "points" didn't exist, but perhaps some
> absolute "Points" did.
>
RESPONSE: I'm glad you finally agree that selecting a Point size to suit
a particular measurement is obviously very sensible. As to the idea that
it's silly to use "Point" and "point" with different meanings, I have
two comments:
a) This whole idea of using the terms in Circular Geometry with
different notation arose as a result of your repeated objections to what
you said was my conflating the terms used in a Euclidean sense vs. a
Circular Geometry sense when discussing Circular Geometry, as I note in
a footnote to the article I recently posted. It sounded like a good idea
at the time.
b) I believe that the meaning of a term is defined by its rules of use.
Obviously the rules for the use of the term 'Point' are different from
those for 'point', since Points have area, while points do not. So they
do have somewhat of a different meaning or connotation. Also, as I
mention in a footnote, it is possible to create a circular geometry that
is somewhat parallel to the one I am proposing by taking the terms
'point', 'line', 'circle', etc. in their Euclidean senses. In fact, I
more or less credit you in the footnote for suggesting this idea
(perhaps this is an incorrect attribution, but I believe in giving
credit where credit is due). So capitalizing these terms does call
attention to the differences in these two possible circular geometries.
If my language of Points can be replaced by something like "measuring to
within the tolerances of 1/100th. of an inch" (I claim that there is
some implicit limit in any measurement, even though the measurer may not
think about his assumptions in each case), then that's fine with me. In
that case, the language of Points could be reserved for times when you
are explaining the complete foundations of measurement, which is, after
all, something of a philosophical question.
You then continue:
> I'm surprised that you regard changing a coordinate-system as
> "extreme": to me it seems a rather trivial and practical matter.
>
> I have a suggestion to make. If it's really the case that
> your distinguishing between "points" and "Points" is practical
> rather than theoretical, why not just drop it and use a more
> traditional language in the interests of better communication
> even if (like many other people) you'd really prefer it if language
> hadn't developed in the way it has? I say this because talking
> about the use of words is much less valuable than talking about
> the things they represent, and your present terminology has made
> it very difficult (at least for me) to understand what you're
> really trying to say.
>
RESPONSE: If dropping the use of "Points" will help you understand
better what I am saying, then I am certainly willing to do this. But at
the same time, I find your comments that you "find it very difficult to
understand what [I'm] really trying to say" somewhat frustrating at this
point.
I realize that what I am saying may be difficult to understand when it
is viewed in isolation, rather than as a complete system. Up to this
point, then, I have been fully in sympathy with your comments that you
"do not understand" whay I am trying to say," even to the point of being
willing to alter my terminology so that you would be better able to
comprehend my remarks. In fact, I viewed your repeated claims not to
understand what I am saying as a reason to further elaborate and attempt
to explain myself. You also said that I don't define my terms and also
don't understand what I mean or even adequately think through what I am
saying before I say it.
So to meet your objections, I have done the following four things:
1. Provided a discussion of the basis for my proposal (namely, what I
claim are flaws in the foundations of Euclidean and Cartesian geometry).
2. Provided a list of the axioms of Circular Geometry that defines the
fundamental concepts of 'Point', 'Line', and 'Circle' that I am using
and gives rules for their use. What's more, this list of axioms does not
rely overly much on Euclidean axioms or definitions, as you said my last
effort to axiomatize my system did.
3. Provided three practical consequences of the use of Circular Geometry
to the measurement of fluid flow in closed pipes.
4. Provided a graphic that depicts the Coordinate System I am talking
about, together with the Point that can be used to generate this system.
Please note that I have done all the above in the space of seven pages,
that I have done so in plain English and have not relied on the
introduction or the use of words that people of common sense can't
understand (with the possible exception of the discussion of fluid flow
measurement, which might presuppose some technical understanding), and
that I have not surreptitiously taken your ideas and smuggled them in,
claiming them as my own but rather have given credit where credit is
due, including not only to you, but to other members of the geometry
forum from whose comments I have benefited.
If after all this, you still say to me: "I don't understand what you are
talking about," then I would say to you: "What is it that you don't
understand?" And where specifically do I make a statement that you
disagree with or find hard to grasp?
Jesse Yoder
Kirby Urner
08/02/98
Re: Coordinate Systems
Lets not forget "latitude and longitude" as an important
coordinate system. I know some would argue these are a
subclass of "polar coordinate" and they'd have a case,
but lots of the specifics are unique to the actual
practice of navigation, including the real time use of
global positioning devices (not just the paper and
pencil nomenclature, but the gizmos applied, has a
bearing on what we mean by "coordinate system" I would
argue).
Cliff has some simplex coordinates he associates with
Synergetics. In keeping with my philosophy that what
Fuller meant by '4D' was quite simply Platonic-Euclidean
space of the ordinary kind (volumetric conceptuality),
but minus any 0,1,2,3- D 'dimensional ladder', I've
been offering quadray coordinates as yet another
coordinate system for service as a pedagogical tool
for math teachers, aimed at keeping our minds flexible
and open to "new gizmos" generally (the future will
doubtless have many new games for us to learn).
In quadrays, we spoke out in 4 directions from the center
of a tetrahedron to its vertices, and label these basis
rays (definitional move):
a (1,0,0,0)
b (0,1,0,0)
c (0,0,1,0)
d (0,0,0,1)
You can then clone and translate these basis vectors, scaled
by floating point numbers, and add them tip-to-tail, as per
usual, such that any address (fp, fp, fp, fp) will signify a
point in volume surrounding the origin. However, because any
given point only needs to make use of at most 3 basis vectors,
and because shrink/grow scaling can take care of spanning any
quadrant without making use of the 'vector reversal' operator
(i.e. negation or - ), we will always have a 'lowest terms'
expression of a coordinate address of the form {fp, fp, fp, 0}
where all fp are positive floating point numbers at least one
of which will be zero -- the curly braces indicate that we're
not tacking down which.
I've derived an alternative distance formula for dealing with
quadrays, thereby giving myself a metric, and written object
oriented computer code for translating to/from xyz. This gives
me what I need to bring polyhedra to the screen using a database
of 4D coordinates, with the xyz conversion happening 'on the fly'
as I write out to my ray tracer engine, which of course expects
input using the time-tested Cartesian protocol. I also have
an alternative volume expression, which syncs with Synergetics.
Plug in the four coordinates of any tetrahedron, and get back
its volume in terms of the unit-volume tet defined by the
centers of 4 adjacent IVM spheres. Turns out any tetrahedron
with IVM vertices has a whole-number volume by this method
of reckoning, no matter how skew.
What I can use quadrays for is to challenge the idea that
the "linear independence" necessarily gets us to "3-D" as
the only logical result. I claim that my system in many
ways streamlines, by making negative scalars unnecessary,
and by getting by with 4 spokes, omnisymmetrically or
spherically arranged, instead of the Cartesian 6 (which
includes 3 positive and 3 negative). Plus I can derive
the Cartesian apparatus by adding all pairs of my basis
rays, to get vectors poking the the 6 mid-edges of my home
base tetrahedron. These vector sums have the form {1,0,1,0}
and I can tell kids to "paint them black and relabel with
positive and negative numbers" to play the standard textbook
xyz games, which of course they still need to learn and will
for the foreseeable future.
Kirby
Cite:
http://www.teleport.com/~pdx4d/quadrays.html
Jesse Yoder
10/02/98
On Feb. 8, John Conway wrote:
> I think that what you call "selecting a Point size to suit a
> particular measurement" is obviously very sensible, but that it's
> silly to use "Point" and "point" with different meanings. Have you
> any strong reason for doing this rather than using the same language
> as everyone else - for instance saying something like "working to
> within a tolerance of 1/100 of an inch"? I thought you had, namely
> that you felt Euclid's "points" didn't exist, but perhaps some
> absolute "Points" did.
>
> I'm surprised that you regard changing a coordinate-system as
> "extreme": to me it seems a rather trivial and practical matter.
>
> I have a suggestion to make. If it's really the case that
> your distinguishing between "points" and "Points" is practical
> rather than theoretical, why not just drop it and use a more
> traditional language in the interests of better communication
> even if (like many other people) you'd really prefer it if language
> hadn't developed in the way it has? I say this because talking
> about the use of words is much less valuable than talking about
> the things they represent, and your present terminology has made
> it very difficult (at least for me) to understand what you're
> really trying to say.
>
RESPONSE: As I have pointed out in previous discussion, there is an
anti-Eucldean and an anti-Cartesian element to what I call Circular
Geometry:
The anti-Euclidean element consists of saying that Points have area
(unlike points, which have no area), and that Lines have width and
length (unlike lines, which have only length.
The anti-Cartesian element consists of analyzing circular area in terms
of round inches rather than square inches.
These makes four possible geometries:
1.Euclidean and Cartesian (our current geometry)
2. NonEuclidean and Cartesian (Using Points and Lines, but sticking with
square inches instead of round inches)
3. Euclidean and Non-Cartesian (Sticking with Euclidean points and
lines, but using round inches instead of square inches)
4. Non-Euclidean and Non-Cartesian (Circular Geometry, i.e., using
Points and Lines AND starting with round inches instead of square
inches).
I have been arguing for 4, partly because I think it's the most
interesting of the four possibilities. But I could drop the terminology
of Points and Lines altogether and even accept Euclidean concepts of
point and line, and still have an interesting alternative geometry (#3).
I am willing to consider your suggestions, though, since they usually
turn out to be good ones. As for your suggestion of dropping the
terminology of Points for points, Lines for lines, and Circles for
circles, I have no problem with this (as I've said) except that since I
claim that Points have area and Lines have width, it still seems like a
useful technique (one inspired by your suggestion, as I've noted).
Does "Points have area" just mean "When any measurement is made, it's
made within a certain tolerance", the answer is basically yes, EXCEPT
THAT I want to block the possibility of saying there are infinitely many
points on a line (I claim there are finitely many, and how many there
are varies with the measurement being made).
Jesse Yoder
P.S. I think that Circular Geometry is a lot more compelling if you can
view the graphic of the Coordinate Systems, which unfortunately didn't
come across on the Geometry-Research post, so I will repeat my offer to
fax this to anyone who sends me their fax number. Eventually, I expect
to be able to post this on a website. I also have a Word for Windows
document I can email that does have the graphic--again, this didn't make
it through the forum posting.
- mostra testo citato -
John Conway
10/02/98
On Tue, 10 Feb 1998, Jesse Yoder wrote:
> RESPONSE: As I have pointed out in previous discussion, there is an
> anti-Eucldean and an anti-Cartesian element to what I call Circular
> Geometry:
>
> The anti-Euclidean element consists of saying that Points have area
> (unlike points, which have no area), and that Lines have width and
> length (unlike lines, which have only length.
Yes, this is the sort of thing I thought you were trying to say.
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. 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?
> The anti-Cartesian element consists of analyzing circular area in terms
> of round inches rather than square inches.
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.
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?
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?
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?
John Conway
Jesse Yoder
10/02/98
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 tothe physics
book I was reading this morning, their molecules may influence each
other). So I don't believe thtat 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 particualr
wash (measurement).
Jesse Yoder
John Conway
10/02/98
Traduci messaggio in italiano
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 thtat 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:
>
> > 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.
So Points can and do overlap.
> > 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 particualr
> 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
Jesse Yoder
10/02/98
Traduci messaggio in italiano
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
Jesse Yoder
Automation Research Corp.
3 Allied Drive
Dedham, MA 02026
781-461-9100 x128
Fax: 781-461-9101
[email protected]
Our website: http://www.arcweb.com
> ----------
> From: John Conway[SMTP:[email protected]]
> Sent: Tuesday, February 10, 1998 3:28 PM
> To: Jesse Yoder
> Cc: John Conway; Clifford J. Nelson;
> [email protected]
> Subject: RE: Coordinate systems
>
>
>
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