This page contains 32 emails that relate to a discussion of different types of coordinate systems on the Geometry Research Forum.
Do Points Have Area?
Subject: Points with no area
Author: Jesse Yoder jesse@flowresearch.com
Organization: Epigone
Date: 10 Nov 1997 10:51:36 -0500
Candice -
Your posting comes like a breath of fresh air. Could it be that, in reality, the emperor has no clothes? Certain ideas, such as those in mathematics that attempt to create a mathematics out of fictional arealess points, have become so deeply entrenched that it is difficult to even find an appropriate language to challenge them. How could it, after all, even be possible that all those calculus professors with tenure could be teaching a false philosophy?
I am glad to hear that you are willing to say that these mathematical constructs such as arealess points and infinity make no sense, despite the centuries that "learned scholars" have spent repeating them and drilling them into students.
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/mimkholyah/jq0h0xcdnm49@forum.swarthmore.edu
Subject: Re: Points with no area
Author: John Conway <conway@math.Princeton.EDU>
Date: Tue, 18 Nov 1997 12:36:20 -0500 (EST)
d
On 10 Nov 1997, Jesse Yoder wrote:
> Candice -
>
> Your posting comes like a breath of fresh air. Could it be that, in
> reality, the emperor has no clothes? Certain ideas, such as those in
> mathematics that attempt to create a mathematics out of fictional
> arealess points, have become so deeply entrenched that it is difficult
> to even find an appropriate language to challenge them. How could it,
> after all, even be possible that all those calculus professors with
> tenure could be teaching a false philosophy?
>
> I am glad to hear that you are willing to say that these mathematical
> constructs such as arealess points and infinity make no sense, despite
> the centuries that "learned scholars" have spent repeating them and
> drilling them into students.
>
> Jesse
Even not-particularly-learned scholars have long been aware that
Euclidean geometry is merely an idealized version of the geometry
of real space. We go on teaching it because it's the simplest kind of
geometry there is, and because it's really a very good fit, and has
enormous practical value, and is very interesting.
In other words, the emperor HAS very elegant clothes that fit him
very well indeed, and are nicer to wear than skin-tight ones would be.
John Conway
Subject: Re: Reply to "Do Points Have Area?
Author: John Conway <conway@math.Princeton.EDU>
Date: Tue, 16 Dec 1997 12:29:28 -0500 (EST)
On 16 Dec 1997, Jesse Yoder wrote:
> Hi Candice -
>
> It was very nice to hear from you again.
>
> You (Candice) wrote the following:
>
> >" Since circles (as I believe) do not exist (I believe they are just
> polygons with many many sides and angles) it makes sense to have the
............................
> exist??? Jesse, I don't understand how a circle can exist...What
> exactlly is a point??? That is what geometry is really based on...an
> assumption that makes no sense..."
............................
and Jesse replied:
> Since circles exist, there is a need to find their areas. And this can
> be done by means of the formula 4*r*r, where 'r' equals the radius,
> (or, alternatively, d*d, where d equals the diameter). And the result
> will be in round inches, instead of square inches.
[Candice again]:
> ">What
> exactlly is a point??? That is what geometry is really based on...an
> assumption that makes no sense..."
[Jesse again]:
> In reference to 2, in Euclidean geometry, a point has no area. In
> circular geometry, it is the smallest unit area--hence, a point has
> area in circular geometry.
These discussions all seem very confused to me. Neither of the
participants seems to "believe" in Euclidean geometry. That's fine,
but they don't say what they MEAN by such statements as "circles
don't really exist they are just polygons with many sides" or "points
really have area".
What ARE these "circles", "polygons", and "points" being spoken of?
Are we talking about points in real physical space, or in some purely
conceptual one? All the statements are nonsense for real physical space,
which behaves very strangely indeed when dimensions get small, and is,
in particular, so unlike Euclidean 3-dimensional space that all these
terms are utterly meaningless. To learn the appropriate questions to
ask about real physical space, you first have to learn a lot of physics.
Euclidean 3-space is only an approximation that's valid when no dimensions
are two large or too small.
If we're just talking about some purely conceptual space then the
assertions are meaningless until that space is somehow defined.
Jesse speaks of "circular geometry", in which a "point" is the
smallest unit area, and in other statements he's made it clear
that he thinks of these "points" as little circles and lines
as like strings of beads: oooooooooooooooooo, in which
any two adjacent ones touch each other at a point.
But in this second use, the word "point" seems to be used in
something like its Euclidean sense! The double use is confusing.
Since "point" is well-established with its Euclidean meaning,
Jesse should use a new term, say "spot", for his new object.
Now I want to know how these spots are arranged. I presume
they can't overlap (for otherwise the area of the overlap of two
would be smaller than either). Are they arranged hexagonally,
like this:
o o o o o o
o o o o o o o
o o o o o o
o o o o o o
(but magnified so as to touch each other)? More importantly
than any particular such question is the meta-question : where
do we get all this information from? How does Jesse know that
these spots touch each other, are circular, and all have the
same area? I presume this is not by examination of physical space,
but somehow by pure thought.
> It is tempting to view a point as the limiting case of circle (a
> circle with no area). Is it contradictory to say "A circle has no
> area, yet it is solid"? kirby has taken me to task for using the
> phrase "radius of a point", yet if a point has area, it should be
> possible to meaningfully use this phrase.
It's an example of the same kind of confusion. "Radius" has
a well-defined meaning in Euclidean geometry, as the distance
from the center of a circle to any point on its periphery. It
has no meaning in Jesse's geometry until he gives it one. What's
your definition, Jesse?
> I agree with you that the key to unlocking the mysteries of geometry
> lie in a correct understanding of the concept of a point.
What does "CORRECT" understanding MEAN? Just what kind of
system are you talking about? We know what "point" means in
Euclidean geometry, but you seem to think that this word has a
life of its own, and also means something outside of Euclidean
geometry. Well, I don't know what meaning you intend, and so
have no idea what it could possibly mean for a statement about
your new kind of "point" to be correct.
It's as if you started to deny the truth of Lewis Carroll's
poem by saying that no snark is a boojum. Until you've given
meanings to the terms involved, it's silly to say that this
statement is either "correct" or "incorrect".
> Have a supergreat holiday season!
>
> Jesse
and the same from me!
John Conway
Reply to John Conway
Subject: Reply to "Do Points Have Area?"
Author: Jesse Yoder < jesse@flowresearch.com>
Date: 18 Dec 97 17:59:08 -0500 (EST)
Hi John -
You began by quoting some back and forth comments between me and
Candice, then said:
>" These discussions all seem very confused to me. Neither of the
participants seems to "believe" in Euclidean geometry. That's fine,
but they don't say what they MEAN by such statements as "circles
don't really exist they are just polygons with many sides" or "points
really have area".
Response: John, possibly you have not seen my effort to better define
my terms, but in response to some of your earlier posts I have
proposed nine axioms as a replacement for the first nine axioms of
Euclid. I could reiterate these here, but I would simply refer you to
this earlier post. I'm not sure of the exact title (I believe it was
"Re: Pi", and it was around 11/21/97).
The idea that circles don't really exist is Candice's so I won't
comment on that (since I disagree with that anyhow). But on whether
points have area, I believe that if we define a point as having area,
we can avoid Zeno-like paradoxes (as I've said before). A point is the
smallest allowable round unit of measure in a system. This definition
is more coherent than Euclid's, which is "A point is that which has
no part."
You then continue:
>"What ARE these "circles", "polygons", and "points" being spoken of?
Are we talking about points in real physical space, or in some purely
conceptual one? All the statements are nonsense for real physical
space, which behaves very strangely indeed when dimensions get small,
and is, in particular, so unlike Euclidean 3-dimensional space that
all these terms are utterly meaningless. To learn the appropriate
questions to ask about real physical space, you first have to learn a
lot of physics. Euclidean 3-space is only an approximation that's
valid when no dimensions are two large or too small."
Response: I think it's pretty clear here that we're talking about
mathematical conceptual space, not just physical space which
apparently is more Riemannian than Euclidean. Even in Euclidean
geometry when I measure a trianglular object, I bestow 180 degrees on
it even though the physical object may not be perfectly triangular.
Likewise, "straight" lines like ropes are not perfectly straight, but
we treat them as straight when we measure (even a physical ruler isn't
perfectly straight).
You then continue:
>"If we're just talking about some purely conceptual space then the
assertions are meaningless until that space is somehow defined.
Jesse speaks of "circular geometry", in which a "point" is the
smallest unit area, and in other statements he's made it clear
that he thinks of these "points" as little circles and lines
as like strings of beads: oooooooooooooooooo, in which
any two adjacent ones touch each other at a point."
Response: You seem to understand pretty well what I mean. Here is how
a plane would look, with lots of points;
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
The above points are circular, solid, and touching horizontally as
well as vertically. I can't draw a solid circle with this email
system. A point, as you say, is the smallest, allowable round unit
area in a system.
You then continue:
>"But in this second use, the word "point" seems to be used in
something like its Euclidean sense! The double use is confusing.
Since "point" is well-established with its Euclidean meaning,
Jesse should use a new term, say "spot", for his new object."
Response: I'm not keen on giving up the term 'point', since I am
simply responding to what I see as a paradox in the traditional
concept of 'point'. Also, I don't want to have to invent a new term
for 'circle', etc. But I will accept your idea for now (though perhaps
I would prefer the term 'ball' to 'spot.'), if it would help clarify
the discussion and avoid ambiguity.
You then continue as follows:
>" Now I want to know how these spots are arranged. I presume
they can't overlap (for otherwise the area of the overlap of two
would be smaller than either). Are they arranged hexagonally,
like this:
o o o o o o
o o o o o o o
o o o o o o
o o o o o o
(but magnified so as to touch each other)? More importantly
than any particular such question is the meta-question : where
do we get all this information from? How does Jesse know that
these spots touch each other, are circular, and all have the
same area? I presume this is not by examination of physical space,
but somehow by pure thought."
Response: See above -- the hexagonal idea is interesting, but what I
have in mind is simply a bunch of "spots" or "balls" that touch each
other above, below, and on the sides (also, there is an x - y
coordinate system, with one of these rows serving as an x axis and one
row serving as a y axis).
>" You then contine, quoting from an earlier post of mine, then
commenting:
> It is tempting to view a point as the limiting case of circle (a
> circle with no area). Is it contradictory to say "A circle has no
> area, yet it is solid"? kirby has taken me to task for using the
> phrase "radius of a point", yet if a point has area, it should be
> possible to meaningfully use this phrase.
It's an example of the same kind of confusion. "Radius" has
a well-defined meaning in Euclidean geometry, as the distance
from the center of a circle to any point on its periphery. It
has no meaning in Jesse's geometry until he gives it one. What's
your definition, Jesse?"
Response: I suppose this is a fair question. I would give the same
definition as in Euclidean geometry -- a raidus is the distance from
the center of a circle to any point on its periphery.
You then continue, beginning with a quote from an earlier post of
mind:
>" I agree with you that the key to unlocking the mysteries of
geometry
> lie in a correct understanding of the concept of a point.
What does "CORRECT" understanding MEAN? Just what kind of
system are you talking about? We know what "point" means in
Euclidean geometry, but you seem to think that this word has a
life of its own, and also means something outside of Euclidean
geometry. Well, I don't know what meaning you intend, and so
have no idea what it could possibly mean for a statement about
your new kind of "point" to be correct.
It's as if you started to deny the truth of Lewis Carroll's
poem by saying that no snark is a boojum. Until you've given
meanings to the terms involved, it's silly to say that this
statement is either "correct" or "incorrect"."
Response: Again, John, I would refer you to my nine axioms. But as for
a point, I will stick with this definition: "A point (or spot, or
ball) is the smallest round unit area allowable in a system." This
seems to be more informative than the Euclidean "area with no part,"
which you feel is imbued with so much meaning. I am not resorting to
uttering meaningless phrases, as in Carroll's poem.
You tnen quote me as saying, and then comment:
">> Have a supergreat holiday season!
>
> Jesse
and the same from me!"
Response: Since you are displaying such good will, I will apologize
if I seemed sardonic in some of my earlier responses. You are forcing
me to state my case as clearly as possible. And thanks for your
holiday wishes!
I will be out till 12/23, but will return then.
Best wishes,
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/3e68ctbx0k4o@forum.swarthmore.edu
Subject: Re: Reply to "Do Points Have Area?"
Author: DGoncz <dgoncz@aol.com>
Organization: AOL http://www.aol.com
Date: 1 Jan 1998 09:41:19 -0500
I can only add that "meaning" can be ignored if "properties" are specified. I
am thinking of the high-level computer languages my girlfriend uses at work. I
never mastered anOO (object oriented) language. But we worked with definitions
in college geometry. It is possible for a circle to consist only of four
points. But it wouldn't be a conventional circle.
It's important to say where the unconventional usage begins, and specify it
from then on in an unambiguous way. But dialog can develop ambiguous
definitions into clear ones. It's just not guaranteed that dialog (or polylog)
will do this. The only one who can make certain it happens is the inventor.
Nobody will take on someone else's problem and clear it up completely. It just
doesn't happen.
Golly, I'm tired.
Happy New Year!
Yours,
DGoncz@aol.com
Inventor of Topologically Correct Video
(As far as I know)
Please include "DGoncz" in your posted reply so I can find it with Deja News.
AOL's newsreader doesn't index these posts.
Subject: Re: Reply to "Do Points Have Area?
Author: Kirby Urner <pdx4d@teleport.com>
Date: 18 Dec 97 20:07:13 -0500 (EST)
> = Conway
= Urner
>meaningless. To learn the appropriate questions to ask about
>real physical space, you first have to learn a lot of physics.
>Euclidean 3-space is only an approximation that's valid when no
>dimensions are two large or too small.
Re: "Euclidean 3-space" I find it confusing when people into
the standard academic notions of dimensionality and real numbers
appropriate the adjective "Euclidean" for their exclusive use.
As I've posted above (or below, as the case may be), I don't
see how serious students of Euclid's Elements are suddenly
less serious if they don't buy that volume is "three
dimensional" for example. Nowhere in The Elements is
volume so defined.
I say the linear algebra conventions which treat "positive" and
"negative" spokes of the Cartesian six-spoked "jack" asymmetrically,
calling only the former "basis vectors" and the latter not,
because the result of an operation (direction reversal by
means of multiplication by -1) is all conceptual apparatus
which we might want to take with a grain of salt. And
Euclid should not be saddled with necessarily arguing on
behalf of such conventions.
Kirby
http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/ht95m5q5jlla@forum.swarthmore.edu
Subject: Reply to "Do Points Have Area?"
Author: Jesse Yoder < jesse@flowresearch.com>
Date: 8 Jan 98 07:25:57 -0500 (EST)
Hi John -
On December 23, 1997, John Conway wrote, in response to some comments
I made on circular geometry:
>" OK, so what does "touch" mean, for your "points"? I'm not
going back to your old postings, but I definitely recall
that in one of them, it was said that adjacent ones touched
at a point!
Again, what does "circular" mean? Euclid's definition
is that a circle consists of all points at a given distance
from another point, called the center of the circle. What's
your definition?"
Response: This is a good question, but I don't know why the idea of
two mathematical points touching is any more obscure in my geometry
than it is anywhere else, e.g. when two physical objects touch. And
I'm not sure what it means for two physical objects to touch -- do
they have a point in common, or do they share any molecules or
electrons? For two mathematical points to touch is for them to stand
in the same relation two spheres do when they touch (e.g., two
baseballs touching each other), whatever that is.
Thinking along these lines, do the points on a Euclidean number line
ever touch? I assume the answer is "No", since you can always put
another point between two points on a Euclidean number line. Could
they ever touch, and how much space is between them?
I don't have a problem with the standard definition of a circle, as
all points equidistant from a center point, except that this
definition doesn't insure that the circle is continuous. Four points
at north, south, east, and west from the center don't make a circle,
even though they are equidistant from the center of the circle. Also,
the points must somehow form a circular pattern, and I'm not sure how
to say that without smuggling the concept of "circular" in the
definition, e.g., "a closed, continuous, circular line whose points
are all equidistant from a fixed point."
You then said:
Your figure above suggests that your "points" in a given
plane are arranged in a square array. Is this true?
In Euclidean geometry, when circles are arranged like this,
there are some spaces in between. Is this also the case in
your new geometry? If so, what are these spaces "made of"? Points???
Response: The points are conceived as existing in an array, somewhat
like the Cartesian Coordinate system. The plane they are on extends
indefinitely in all directions, so I don't see why it has to been seen
as "square".
There are spaces in the this coordinate system, since you can't "tile
a plane" with circles i.e., fill up an entire plane with circles,
leaving no area uncovered. No, this empty space is not made up of
points -- it is simply empty mathematical space. Again, you seem to be
implying that there is some unexplained phenomenon in circular
geometry -- what is space 'made up of' -- that doesn't exist in
Euclidean geometry. But this question can also be asked of Euclidean
geometry.
A better way to view this is as 3-dimensional, in which case the
points or "spots" as you prefer to call them become 3-dimenensional
balls with physical space between them.
You then continue:
>"Aha! So your geometry fails to be isotropic, and has a preferred
system of coordinate-axes! But I thought the point of your system
was that it gave in some sense a better fit to our native ideas
about the world, or at least about geometry. Was this my
misapprehension?
Was it really supposed just to be a better fit to the pixels on
a computer screen?"
Response: Please explain what you mean by "isotropic" and by saying
that my geometry is not isotropic. I'm not sure what direction of
measurement has to do with this, unless you're using the term in some
special sense. Yes, my geometry does give a better fit to our native
ideas about the world and about geometry.
You then continue, beginning with a quote from an earlier post:
> "Radius" has
> a well-defined meaning in Euclidean geometry, as the distance
> from the center of a circle to any point on its periphery. It
> has no meaning in Jesse's geometry until he gives it one. What's
> your definition, Jesse?"
>
> Response: I suppose this is a fair question. I would give the same
> definition as in Euclidean geometry -- a raidus is the distance from
> the center of a circle to any point on its periphery.
> And in what sense is the word "point" being used in that last
clause? Yours, or Euclid's? Also, what's "periphery" mean in
your new system? As far as I can see, there are NONE of your
"points" that are actually ON the periphery of a given one,
and exactly FOUR "points" that are adjacent to it. But the
distance from any of these to the given one is what Euclid
and I would call its DIAMETER, rather than its RADIUS (and I
might remark, that since you seem to be adopting Euclid's
definition of a circle, that this circle seems to consist
just of four of your "points"!)"
Response: 'Point' is being used in my sense, not in Euclid's. Let me
try to give a better definition without using the term 'periphery': "A
circle is a continuous, closed line, all of whose points are
equidistant from a fixed point. This fixed point is called the center
of the circle, and the radius of the circle is the distance from the
center to any point on the continuous, closed line. This continuous,
closed line is called the circumference. The fixed point or 'spot' is
the smallest allowable unit area."
You then wrote:
"> I read your nine axioms when you posted them, and, then as now,
found your language replete with tacit assumptions from the very
Euclidean geometry you were trying to replace. I've since deleted
them, but will happily respond to them if you'll send me another
copy."
Response: OK, I will resend them in a separate post.
You continue:
>" Some time ago, you were critical of the logic of the calculus,
and now you have some similar criticisms of Euclidean geometry.
But those who live in glass houses should at least be careful
when they throw stones! In particular, you really shouldn't
give a word two meanings in the same sentence (as I believe I
caught you doing with "point"). If you do so, then you are
clearly the one to blame if other people misunderstand you as
a result. If you intend to reject some of Euclid's ideas
and definitions while accepting others, then you must be just
as careful to say what you accept as well as what you reject.
Also, you cannot allow yourself to make tacit assumptions
from classical geometry in the way that you repeatedly have;
for it's improper to do so if your reader may not; but if you
allow your reader to make such assumptions from classical geometry
in the way that you do, then he might well make so many of them that
in effect he assumes ALL of Euclidean geometry. [To tell you
the truth, I think that you are effectively doing this, while
appearing to deny it.]"
Response: I understand more or less what you are saying here. But the
problem is that the terms we use like 'point' and 'line' are deeply
embedded in our conceptual framework, complete with Euclidean
interpretations and definitions. To see how completely Euclidean
geometry has penetrated our consciousness, notice how completely
architecture, furniture, and just about every other physical object
constructed by man conforms to straight-line geometry. This is not
"natural" -- it is the result of the extent to which Euclidean
assumptions have been adopted as if they were "common sense".
Now if I come along and say "But there are paradoxes hidden in these
assumptions -- here are the assumptions I want to propose instead" --
then I can't even say this without using the terms I am trying to
propose better definitions for--or in some cases, the definitions may
remain the same, but the terms may be interpreted differently.
Nonetheless, I agree that the result can be confusing, so I will try
to be more aware of when I am using words in a Euclidean vs.
concentric geometry sense -- and specify the difference when this is
necessary.
Happy New Year!
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/thyspenddwox/48r9qp1x40im@forum.swarthmore.edu
Atomic Points
Subject: Re: Reply to "Do Points Have Area?"
Author: Kirby Urner <pdx4d@teleport.com>
Date: Thu, 08 Jan 1998 10:56:15 -0800
> = Jesse Yoder
>> = John Conway
= Kirby Urner
>Thinking along these lines, do the points on a Euclidean number line
>ever touch? I assume the answer is "No", since you can always put
>another point between two points on a Euclidean number line. Could
>they ever touch, and how much space is between them?
>
Again, I'm thinking it possibly a misattribution to take relatively
late-in-the-game real number lines and the associated Cartesian-style
R^n n-tuple games and backloading these onto Euclid, in effect piggy-
backing them on his good name. Did anyone get his permission for
this?
I find it highly relevant to remember another Greek thinker at
this juncture: Democritus. His atomic picture of reality is a
source of countervailing imagery vis-a-vis all these 1/infinity
"no gaps continuity" dogmas.
If updating our picture of Greek thought as a whole, perhaps we
should recast it with a Euclidean "wrapper" encapsulating
Democritus-style atomic "insides". We could still do stick-
tracings in the sandy beach, ala Euclid's Elements, but the
sand grains would continually remind us why we never bought
into "real number lines" as later concocted. We have (and might
as well again) mention Zeno in this connection -- his explorations
of our concepts around motion presaged many later debates about
the ultimate "smoothness" (not!) of all phenomena.
Bishop Berkeley is another name to remember in this connection,
as his brief was we could just as well do the calculus under the
heading of discrete mathematics and not hinge anything central
on any anti-Democritus teachings. I find many in computer science
agreeing with this view, as their job has been to implement nuts
and bolts calculus in an engineering sense, and it always turns
out that dx is a definite increment, relatively miniscule vis-a-vis
a larger scale phenomenon, but never "infinitely small" -- a concept
with no operational definition on a computer (computer = "metaphor
for energetic reality" as well).
Lots of schools of thought have fought "continuity" (lets call
them the "pro Democritus" schools). What I'm arguing is that
it is intellectually dishonest to simply presume that Euclid
would have been in the "anti Democritus" camp for all eternity,
given that an atomic or discontinuous model *can* be reconciled
with his original explorations, as operationally practiced
(e.g. on sandy beaches).
>A better way to view this is as 3-dimensional, in which case the
>points or "spots" as you prefer to call them become 3-dimenensional
>balls with physical space between them.
>
>You then continue:
>>"Aha! So your geometry fails to be isotropic, and has a preferred
>>system of coordinate-axes!
Jesse, I don't much like this square-packed circles model as a home
base "peg board" for doing geometry, and moving to spheres -- I'd
encourage you to think about laminating pool balls, stacking layers
upon layers in hcp or, more isotropically, in fcc format. You've
likely already done this I realize -- so far I like your game BTW,
though I don't claim to understand it completely.
In synergetics, the fcc packing defines the vertices for a skelatal
lattice of edges dubbed the "isotropic vector matrix", with tetrahedral
and octahedral voids (or cells) in a population ratio of 2:1, with
octahedra having volume 4 vis-a-vis the unit volume tets (so far the
only aspect original with synergetics here is the assignment of unity
to the tet's volume). Synergetics also investigates the shapes of the
voids when we use baseballs (vs. just their centers), mapping the
interspheric voids to the vertices of the space-filling rhombic
dodecahedron of volume 6. You have more deactivated sphere packings
latent in these voids: the long diagonals of the rh dodeca define
one, and the short diagonals define two alternates, for a total of
four (including the currently active one).
The term "isotropic" in relation to fcc sphere packing can be found
elsewhere in the literature than in Synergetics certainly e.g.
Bonnie DeVarco copied the following to Syn-L last August:
"...We can obtain an isotropic point-lattice in space in
starting from four series of equidistant planes cutting each
other under the same angle -- that is, parallel to the four
faces of a tetrahedron. But oddly enough the partitions thus
obtained (Plate 28) do not correspond to a division of space
into close-packed tetrahedra (this in euclidian space cannot
be realized) but to a division into tetrahedra and octahedra
(twice as many tetrahedra as octahedra), or in cuboctahedra
and octahedra in equal numbers. The point-lattice is identical
to the one obtained in filling space by the most dense possible
system of equal tangent spheres, and in taking either all their
centers, or their centers and points of contact.
"As in the plane we can place outside a circle six tangent
circles identical to the first (figure 58) and repeat the
process indefinitely, (the centers of the circles are then part
of a triangular or hexagonal isotropic point-lattice), so in
space we can ahve twelve spheres tangent to an identical inner
sphere (figure 59). It is in this perfectly isotropic close-
packing of thirteen spheres indefinitely repeated, that the
centers (or else the centers and points of contact) produce
the cuboctahedral point-lattice, because in relation to the
center of each sphere the centers of the twelve surrounding
tangent spheres (also the twelve points of contact) coincide
with the vertices of a cuboctahedron (figure 59).
[from Matila Ghyka's The Geometry of Art and Life; first published
by Sheed and Ward, New York, in 1946 -- she was quizzing us as to
who might have written this (clearly not Fuller), and Richard
Hawkins took the prize].
Note that Cartesian axes are also "handed" or "chiral" in that you
have to pick which of the 8 sectors you plan to make your (+,+,+)
region. Usually people go for what's called "left handed" Cartesian
coordinates, but this is of course an arbitrary convention. Four-
dimensional quadrays are more balanced in this respect, as all
four quadrants are expressed by {+,+,+,0} where { } means "all
permutations of" -- none of the quadrants are "more positive" or
"more negative" than the others. In fact, one of Coxeter's
criticisms of synergetics is it wasn't chiral enough (right Bonnie?)
-- but we should member the inside-outing transformation through
the origin, into the space of the dual tetrahedron, has the effect
of making a right handed glove into a left handed one, and vice
versa.
Kirby
Curriculum writer
4D Solutions
Points & points
Subject: Re: Reply to "Do Points Have Area?"
Author: John Conway <conway@math.Princeton.EDU>
Date: Thu, 18 Dec 1997 18:16:28 -0500 (EST)
On 18 Dec 1997, Jesse Yoder wrote:
> Hi John -
>
> Response: I think it's pretty clear here that we're talking about
> mathematical conceptual space, not just physical space which
> apparently is more Riemannian than Euclidean. Even in Euclidean
> geometry when I measure a trianglular object, I bestow 180 degrees on
> it even though the physical object may not be perfectly triangular.
> Likewise, "straight" lines like ropes are not perfectly straight, but
> we treat them as straight when we measure (even a physical ruler isn't
> perfectly straight).
>
> You then continue:
>
> >"If we're just talking about some purely conceptual space then the
> assertions are meaningless until that space is somehow defined.
> Jesse speaks of "circular geometry", in which a "point" is the
> smallest unit area, and in other statements he's made it clear
> that he thinks of these "points" as little circles and lines
> as like strings of beads: oooooooooooooooooo, in which
> any two adjacent ones touch each other at a point."
>
> Response: You seem to understand pretty well what I mean. Here is how
> a plane would look, with lots of points;
>
> oooooooooooooooooooooooooooooooooooooooooooooooooo
> oooooooooooooooooooooooooooooooooooooooooooooooooo
> oooooooooooooooooooooooooooooooooooooooooooooooooo
> oooooooooooooooooooooooooooooooooooooooooooooooooo
> oooooooooooooooooooooooooooooooooooooooooooooooooo
>
> The above points are circular, solid, and touching horizontally as
> well as vertically. I can't draw a solid circle with this email
> system. A point, as you say, is the smallest, allowable round unit
> area in a system.
OK, so what does "touch" mean, for your "points"? I'm not
going back to your old postings, but I definitely recall
that in one of them, it was said that adjacent ones touched
at a point!
Again, what does "circular" mean? Euclid's definition
is that a circle consists of all points at a given distance
from another point, called the center of the circle. What's
your definition?
Your figure above suggests that your "points" in a given
plane are arranged in a square array. Is this true?
In Euclidean geometry, when circles are arranged like this,
there are some spaces in between. Is this also the case in
your new geometry? If so, what are these spaces "made of"? Points???
.................................
> for 'circle', etc. But I will accept your idea for now (though perhaps
> I would prefer the term 'ball' to 'spot.'), if it would help clarify
> the discussion and avoid ambiguity.
It certainly would. A very big problem with all your descriptions
is that they presuppose an underlying Euclidean geometry. For instance,
you say you want to use the term "circle", a concept that, to the rest
of the world, is defined in terms of Euclid's notion of "point" rather
that your new one. They are perfectly comprehensible if you do allow
yourself to use Euclid's notions, but in that case it's confusing,
and also to my mind improper, to use some of the Euclidean terminology
with different meanings to his.
> Response: See above -- the hexagonal idea is interesting, but what I
> have in mind is simply a bunch of "spots" or "balls" that touch each
> other above, below, and on the sides (also, there is an x - y
> coordinate system, with one of these rows serving as an x axis and one
> row serving as a y axis).
Aha! So your geometry fails to be isotropic, and has a preferred
system of coordinate-axes! But I thought the point of your system
was that it gave in some sense a better fit to our native ideas
about the world, or at least about geometry. Was this my misapprehension?
Was it really supposed just to be a better fit to the pixels on
a computer screen?
> >" You then contine, quoting from an earlier post of mine, then
> commenting:
>
> > It is tempting to view a point as the limiting case of circle (a
> > circle with no area). Is it contradictory to say "A circle has no
> > area, yet it is solid"? kirby has taken me to task for using the
> > phrase "radius of a point", yet if a point has area, it should be
> > possible to meaningfully use this phrase.
>
> It's an example of the same kind of confusion. "Radius" has
> a well-defined meaning in Euclidean geometry, as the distance
> from the center of a circle to any point on its periphery. It
> has no meaning in Jesse's geometry until he gives it one. What's
> your definition, Jesse?"
>
> Response: I suppose this is a fair question. I would give the same
> definition as in Euclidean geometry -- a raidus is the distance from
> the center of a circle to any point on its periphery.
And in what sense is the word "point" being used in that last
clause? Yours, or Euclid's? Also, what's "periphery" mean in
your new system? As far as I can see, there are NONE of your
"points" that are actually ON the periphery of a given one,
and exactly FOUR "points" that are adjacent to it. But the
distance from any of these to the given one is what Euclid
and I would call its DIAMETER, rather than its RADIUS (and I
might remark, that since you seem to be adopting Euclid's
definition of a circle, that this circle seems to consist
just of four of your "points"!)
> You then continue, beginning with a quote from an earlier post of
> mind:
>
> What does "CORRECT" understanding MEAN? Just what kind of
> system are you talking about? We know what "point" means in
> Euclidean geometry, but you seem to think that this word has a
> life of its own, and also means something outside of Euclidean
> geometry. Well, I don't know what meaning you intend, and so
> have no idea what it could possibly mean for a statement about
> your new kind of "point" to be correct.
>
> It's as if you started to deny the truth of Lewis Carroll's
> poem by saying that no snark is a boojum. Until you've given
> meanings to the terms involved, it's silly to say that this
> statement is either "correct" or "incorrect"."
>
> Response: Again, John, I would refer you to my nine axioms. But as for
> a point, I will stick with this definition: "A point (or spot, or
> ball) is the smallest round unit area allowable in a system." This
> seems to be more informative than the Euclidean "area with no part,"
> which you feel is imbued with so much meaning. I am not resorting to
> uttering meaningless phrases, as in Carroll's poem.
I read your nine axioms when you posted them, and, then as now,
found your language replete with tacit assumptions from the very
Euclidean geometry you were trying to replace. I've since deleted
them, but will happily respond to them if you'll send me another
copy.
Some time ago, you were critical of the logic of the calculus,
and now you have some similar criticisms of Euclidean geometry.
But those who live in glass houses should at least be careful
when they throw stones! In particular, you really shouldn't
give a word two meanings in the same sentence (as I believe I
caught you doing with "point"). If you do so, then you are
clearly the one to blame if other people misunderstand you as
a result. If you intend to reject some of Euclid's ideas
and definitions while accepting others, then you must be just
as careful to say what you accept as well as what you reject.
Also, you cannot allow yourself to make tacit assumptions
from classical geometry in the way that you repeatedly have;
for it's improper to do so if your reader may not; but if you
allow your reader to make such assumptions from classical geometry
in the way that you do, then he might well make so many of them that
in effect he assumes ALL of Euclidean geometry. [To tell you
the truth, I think that you are effectively doing this, while
appearing to deny it.]
I hope you read this before you get out of touch for the
season, because you may need quite some time to think out your
position!
Regards, John Conway
Comment to John Conway
Subject: Re: Reply to "Re: Reply to "Do Points Have Area?"
Author: Candice Hebden <dreamy_aurora@hotmail.com>
Date: 16 Jan 98 15:21:02 -0500 (EST)
Hi John,
sorry I haven't written in a while. On December 16, you said
(referring to me and Jesse Yoder)
"These discussions all seem very confused to me. Neither of the
participants seems to "believe" in Euclidean geometry. That's fine,
but they don't say what they MEAN by such statements as "circles
don't really exist they are just polygons with many sides" or "points
really have area".
What ARE these "circles", "polygons", and "points" being spoken of?
Are we talking about points in real physical space, or in some purely
conceptual one? All the statements are nonsense for real physical
space, which behaves very strangely indeed when dimensions get small,
and is, in particular, so unlike Euclidean 3-dimensional space that
all these terms are utterly meaningless. To learn the appropriate
questions to ask about real physical space, you first have to learn a
lot of physics. Euclidean 3-space is only an approximation that's
valid when no dimensions are two large or too small."
You're right to an extent. I don't believe that the Euclidean world
exists in the real world. But I do believe that it exists in a like
"Parallel" world. I've been talking about the real world; trying to
relate it to the Euclidean one. When I say that circles do not exist,
I mean that in both the Euclidean world and the real world. Holding
the same definitions as Euclidean named, things in the real world are
closly related to polygons, but there is nothing in the real world
that resembles a circle (closer than it does a polygon).
I hope this clarifies things,
Candice Hebden
http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/bl8vkgw7k0ej@forum.swarthmore.edu
Questions for Jesse
Subject: Reply to "Reply to Do Points Have Area?"
Author: Candice Hebden <dreamy_aurora@hotmail.com>
Date: 16 Jan 98 15:27:59 -0500 (EST)
Hi Jesse,
I have two questions about your circular geometry. On December 18,
you posted
[John Conway]
>"If we're just talking about some purely conceptual space then the
assertions are meaningless until that space is somehow defined.
Jesse speaks of "circular geometry", in which a "point" is the
smallest unit area, and in other statements he's made it clear
that he thinks of these "points" as little circles and lines
as like strings of beads: oooooooooooooooooo, in which
any two adjacent ones touch each other at a point."
[You]
"Response: You seem to understand pretty well what I mean. Here is how
a plane would look, with lots of points;
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
The above points are circular, solid, and touching horizontally as
well as vertically. I can't draw a solid circle with this email
system. A point, as you say, is the smallest, allowable round unit
area in a system."
What do you call the area between the points? Isn't there always a
smaller sized point?
Candice
http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/i53lox7o60kn@forum.swarthmore.edu
Reply to Candice
Subject: Reply to Do Points Have Area?
Author: Jesse Yoder < jesse@flowresearch.com>
Date: 22 Jan 98 13:57:31 -0500 (EST)
Hi Candice -
Referring to the idea I suggested that the number of Points in a line
might vary with the frame of reference, you said:
>"So, if this is all true, then there would be much "empty" space in
certain frames of reference and less "empty" space in others. Not
everyone measures the amount of gas in their car by tenths of a
gallon. In fact, most of the world doesn't even know what a gallon
is! Every frame of reference you make will have to be stated before
any work is done on the problem. Still then, many people might not
understand your frame of reference!"
RESPONSE: Perhaps you're right about the empty space. But I don't
really understand why this is so different from the Cartesian
coordinate system, where you have an x and y axis serving as a frame
of reference for the points on the plane. The area in the plane is
empty, until you put in some additional lines or points. If you draw a
curve on a Cartesian Coordinate system, what is the rest of the area
if not "empty space"?
Let me address the issue about shifting frames of reference. I realize
that more people use the metric system of liters than the American
system of gallons, so I don't expect everyone to measure gas in tenths
of gallons. The point of saying you have to specify a refernce system
is that it provides a way to avoid the paradox of arealess points. If
you specify a frame of reference, by which I mean saying what your
unit of measurement is, then you can picture the x and y axes as
composed of a corresponding number of Points. For example, if you are
measuring in tenths, then a Point can be 1/10 of an inch (I realize
I'm switching from gallons to inches here). And if you are measuring
to the 1/100 position, then the Points are 1/100th of an inch. This
avoids the paradoxes that arise from arealess points. And if this
seems awkward or unduly complex, keep in mind that every time you make
a measurement, some unit of reference is explicitly or implicitly
implied. If you say "It's 93 million miles to the sun," you're using
miles as your unit. I'm just saying "Let's make this assumption
explicit, and we avoid the paradoxes that arise form arealess points."
Candice, you then continue as follows:
>" Sometimes the simplier theory is more "correct" because it makes
sense. I certainly am not a believer of Euclid's arealess point, but
it does have it's merits. People once thought that the Earth was the
center of the Universe. Aristole made all kinds of rules to support
his theory in respect to the "strange" orbits of Jupiter's satilites
and moons. But Copernicus's idea of the Heliocentric gallaxy
(although not widley accepted at first) was simplier and makes more
sense."
RESPONSE: Agreed about simpler theories sometimes being true, but
simple theories can also contain hidden paradoxes that aren't
immediately obvious. It sounds easy to say "OK, I'm standing here on
a dimensionless point. Now if I move to the other side of the room,
I'll be located at another dimensionless point." It's not until you
bring out Zeno's paradox, which seems ot show you can never reach the
other side of the room, that you realize that it may not be such a
great idea to say that a 3-dimensional object can be located at an
arealess or dimensionless point.
In general, I'm not a big fan of the "Simpler, and therefore more
likely to be true" theory. If the world is complex, it may take a
complex theory to adequately explain it!
You then say:
>"I am not doubting the accuracy for you circular geometry. It seems
it will make sense once certain things are worked out."
RESPONSE: Thanks for the vote of confidence. I realize there are
problems with the theory I'm presenting too, but I think that
sometimes certain people lose track of the problems in their own
positions as they get carried away criticizing the positions of others
(I realize this applies equally well to me).
Best wishes,
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/79jyd7fzl79y@forum.swarthmore.edu
Reply to Candice
Subject: Reply to Do Points Have Area?
Author: Jesse Yoder jesse@flowresearch.com
Date: 20 Jan 98 17:39:32 -0500 (EST)
Hi Candice -
Good to hear from you again! You recently asked a couple of questions
about my geometry, as follows:
I have two questions about your circular geometry. On December 18,
you posted
[John Conway]
>"If we're just talking about some purely conceptual space then the
assertions are meaningless until that space is somehow defined.
Jesse speaks of "circular geometry", in which a "point" is the
smallest unit area, and in other statements he's made it clear
that he thinks of these "points" as little circles and lines
as like strings of beads: oooooooooooooooooo, in which
any two adjacent ones touch each other at a point."
[You]
"Response: You seem to understand pretty well what I mean. Here is how
a plane would look, with lots of points;
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooooooooooooooooooooooooooo
The above points are circular, solid, and touching horizontally as
well as vertically. I can't draw a solid circle with this email
system. A point, as you say, is the smallest, allowable round unit
area in a system."
What do you call the area between the points? Isn't there always a
smaller sized point?
Response: First off, let me take the second question. John Conway has
suggested I adopt a convention for indicating when I am using 'point'
in my sense, so I am capitalizing Point and Line. The answer is No,
there isn't always a smaller sized Point, since when a measurement is
made, you have to specify a frame of reference that says how small the
points are allowed to go. Thsi is often implicitly understood. For
example, if I'm measuring miles from work to home, I measure in tenths
of a mile. When I measure the amount of gas put in my car, I measure
in tenths of a gallon. The distance from here to the sun is measured
in miles. The positions of computer chips on a board might be measured
to the ten thousandth of an inch. Deciding what your frame of
reference is determines the size of your Points. Of course, there is
always ROOM FOR another point, but all that means is that you are
shifting to a different frame of reference, in which case again there
will be no smaller sized Points within this new frame of reference.
As for the space between points, the answer is that this is
mathematical space that can be referenced in relation to Points on the
coordinate system.
I hope this helps. I just read an account of the Euclidean idea that
points have no area, yet somehow make up a line in a book called The
Non-Euclidean Revolution by Richard Trudeau. This convinces me once
again that it is simply paradoxical to say, on the one hand, that
points have no dimension, and, on the other hand, that a line, which
has length, is made up of infinitely many of these dimensionless
points. Mutiplying 0 by infinity still equals 0. As far as I can see,
this remains an unresolved problem for Euclid's Axiom One (definition
of point), and I believe that ascribing area to points is the only way
around it.
Yours,
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/gmf94q05o3xo@forum.swarthmore.edu
Volumetric Points
Subject: Re: Reply to Do Points Have Area?
Author: Kirby Urner <pdx4d@teleport.com>
Date: Tue, 20 Jan 1998 20:53:33 -0800
>points. Mutiplying 0 by infinity still equals 0. As far as I can see,
>this remains an unresolved problem for Euclid's Axiom One (definition
>of point), and I believe that ascribing area to points is the only way
>around it.
>
At the risk of being redundant, I'd prefer to ascribe volume to points,
since your pancake points, if as flat as the ghostly "2D plane" won't
stack to create volume, any more than ghostly "0D points" you criticize
would make a line.
So I'd go with centers floating in an isotropic matrix or lattice, ala
the centers of fcc spheres -- thinking of an ideal gas (Avogadro) with
this being a snap shot of atoms in their "averaged home position" (of
course in reality everything is moving like crazy).
The stuff you say about a reference frame being bracketed by frequency
limits sounds fine to me.
I'm also willing to have a zerovolume in my philosophy -- but it's
conceptually a tetrahedron, because is in the event of four planes
approaching one another and passing on through (an "inside outing"
operation wherein a tetrahedron is instantaneously zerovolume).
This is a departure from the axiomatic Euclidean concept of point
I suppose, but still supports Euclidean-style geometry ala the many
constructions in The Elements.
Kirby
Reply to Jesse
Subject: Reply to "Reply to Do Points Have Area?"
Author: Candice Hebden <dreamy_aurora@hotmail.com>
Date: 21 Jan 98 08:54:05 -0500 (EST)
Hey Jesse,
On January 20, 1998
[You said]
>Response: First off, let me take the second question. John Conway
>has suggested I adopt a convention for indicating when I am using
>'point' in my sense, so I am capitalizing Point and Line. The answer
>is No, there isn't always a smaller sized Point, since when a
>measurement is made, you have to specify a frame of reference that
>says how small the points are allowed to go. This is often
implicitly >understood. For example, if I'm measuring miles from work
to home, I >measure in tenths of a mile. When I measure the amount of
gas put >in my car, I measure in tenths of a gallon. The distance from
here to >the sun is measured in miles. The positions of computer chips
on a >board might be measured to the ten thousandth of an inch.
Deciding >what your frame of reference is determines the size of your
Points. Of >course, there is always ROOM FOR another point, but all
that means >is that you are shifting to a different frame of
reference, in which >case again there will be no smaller sized Points
within this new frame >of reference.
So, if this is all true, then there would be much "empty" space in
certain frames of reference and less "empty" space in others. Not
everyone measures the amount of gas in their car by tenths of a
gallon. In fact, most of the world doesn't even know what a gallon
is! Every frame of reference you make will have to be stated before
any work is done on the problem. Still then, many people might not
understand your frame of reference!
Sometimes the simpler theory is more "correct" because it makes
sense. I certainly am not a believer of Euclid's arealess point, but
it does have it's merits. People once thought that the Earth was the
center of the Universe. Aristotle made all kinds of rules to support
his theory in respect to the "strange" orbits of Jupiter's satellites
and moons. But Copernicus's idea of the Heliocentric galaxy
(although not widely accepted at first) was simpler and makes more
sense.
I am not doubting the accuracy for your circular geometry. It seems it
will make sense once certain things are worked out.
[You then continue, answering my first question]
>As for the space between points, the answer is that this is
>mathematical space that can be referenced in relation to Points on
>the coordinate system.
I still don't understand. All space must contain points right?
Aren't points supposed to define the space of something? If this is
true then there is space unaccounted for... making an infinite amount
of non-space! If it isn't true then state it. And then explain to
me, please, how space could go from empty to not-empty with a change of
frame of reference.
John Conway earlier posed the sujestion that you lay your points on a
hexagonal frame. As the frame of reference decreases, there is always
a model for the arrangement of the points so that there is less empty
space. Would you want to use something like that? Or would that
further confuse the issue?
[You continue again]
>I hope this helps. I just read an account of the Euclidean idea that
>points have no area, yet somehow make up a line in a book called >The
Non-Euclidean Revolution by Richard Trudeau. This convinces >me once
again that it is simply paradoxical to say, on the one hand, >that
points have no dimension, and, on the other hand, that a line, >which
has length, is made up of infinitely many of these >dimensionless
points. Mutiplying 0 by infinity still equals 0. As far as I >can see,
this remains an unresolved problem for Euclid's Axiom One >(definition
of point), and I believe that ascribing area to points is the >only
way around it.
I am in complete agreement with the failure of Euclidean's arealess
points. I also am in agreement with you that a new type of geometry
should be devised. However, I do not believe it has been completed
yet! Good luck. I really hope you can "fix" what the new geometry of
yours needs!
Yours,
Candice Hebden
@
dreamy_aurora@hotmail.com
http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/vrzyo0c0cjq7@forum.swarthmore.edu
Points & points
Subject: Re: Reply to Do Points Have Area?
Author: John Conway <conway@math.Princeton.EDU>
Date: Wed, 21 Jan 1998 16:31:01 -0500 (EST)
On 20 Jan 1998, Jesse Yoder wrote:
> [John Conway]
> >"If we're just talking about some purely conceptual space then the
> assertions are meaningless until that space is somehow defined.
> Jesse speaks of "circular geometry", in which a "point" is the
> smallest unit area, and in other statements he's made it clear
> that he thinks of these "points" as little circles and lines
> as like strings of beads: oooooooooooooooooo, in which
> any two adjacent ones touch each other at a point."
>
> [Jesse]
> "Response: You seem to understand pretty well what I mean. Here is how
> a plane would look, with lots of points;
It still surprises me that you didn't even notice the double
use of the word "point" in the sentence I obliquely quoted from you!
How can two points touch at a point?
Of course, you've now agreed to distinguish between "Points"
and "points", but it really seems to me that in a fundamental
sense this vitiates your system, because it bases it on the
traditional notions. Surely you should be able to describe
the structure and arrangement of your Points without using
Euclid's points? If not, it can hardly be true that "a Point
is the smallest allowable unit of area".
I have difficulty in following your comments about switching to
new frames of reference. Do you think this is legal, or were you
really saying it was impossible? It seems to me that it's obviously
impossible in your system. If a Point is really the smallest
allowable unit of area, then no kind of changing frames of
reference can possibly produce a smaller Point.
John Conway
Dimensional Lines
Subject: Re: Reply to "Re: Reply to Do Points Have Area?"
Author: Kirby Urner <pdx4d@teleport.com>
Date: Wed, 21 Jan 1998 16:58:06 -0800
>No matter how much sense it makes, there is nothing that states that
>if you lay a bunch of lines together you will have height! But, to
>make a cube, you need 12 line segments (which could have zero width)
>because of the angles of the line segments, the construction will be
>three dimentional! Even though lines with width makes sense, it is
>not necessary in order to (visually) construct in the 3D
>
I understand you.
Coming from a computer background, say a ray tracing world, I'm
used to stuff having definite dimension, as otherwise light has
nothing to bounce off, so it might as well not be there.
So for me, in this world, a point is a relatively tiny entity,
too small to have its details make any difference, but there's
no quantum leap to some lower dimensional state (i.e. 0).
Lines and planes are likewise slender/thin, but I experience
no intellectual pressure to alter their dimensionality relative
to any old ordinary, light-reflective substance in my ray
traced world. A line and a cube look different, I can always
tell which is which -- but I don't go by "dimension number"
as this is the same for both (see below).
So for me, points, lines and planes are all shapes with properties
(e.g. planes are "razor thin"), but don't sit on different rungs
of the "dimension ladder". What is usually called Euclidean space
(or volume) is for me a space of "lumps" and the "point", "line"
and "plane" characterizations still make sense, but minus the
0D,1D,2D claptrap.
If I want to bring D ("dimension") into it, then I note that volume
is containment, the logical space of things with inside/outside
concave/convex attributes -- the space of hulls, shells, rooms,
cells...
Then I do what you do, I go with thin lines (edges) and figure
out what simplest model of inside/outside I can conceive --
realizing that my lines themselves have insides/outsides (but
that doesn't mean I have to consider them my paradigm
"containments").
The answer I come up with is the tetrahedral wireframe: four
windows, four corners, six edges. No shape is simpler. "Spheres"
as such turn out to be high frequency porous membranes -- just
as my planes turn out to be networks as well (mostly space).
I'm in Euler's world of V, F and E -- but my F is more a W (window).
Vs are where Es cross, but they don't even have to go through each
other exactly -- no two things occupy the same space at the same
time.
So I say volume is 4D. I get my 4 from the 4 windows and 4
corners of the tetrahedron. 0D, 1D, 2D and 3D are all undefined
in this philosophical language.
The aesthetics here trace to Democritus. Discontinuity, discrete,
empty space versus substance, emptiness between things, islands,
events with novent surroundings, holes, voids... I'm not looking
for anything to fill the holes, now that I've got them.
I claim I can do Euclidean geometry in this logical space. So I
say Euclidean space is 4D, realizing this sounds all wrong, very
dissonant, to ears trained in the 1900s.
>I can't wait till a new geometry that makes more sense, maybe Jesse
>Yoder's circular geometry, but I don't want to disclaim things in
>Euclidean geometry yet if they still make some sense!
>
I don't want to disclaim stuff in Euclidean geometry either.
My curriculum makes use of The Elements, the kind of logic that
goes on in these proofs, but tosses some of the definitional
beginnings. Euclidean constructions "float" in 4D space without
needing "support from below" in the form of "a bedrock of
axioms" -- especially where this funny concept of "dimension"
is concerned.
Also, I'm not trying to push the old logic off stage with this
newfangled talk (as if I could, even if I wanted to). I know the
standard lingo and would expect kids learning my meaning of 4D
to also learn the standard "dim talk".
I say "three dimensional" just like everyone else when talking
about volume (when in Rome) even if that's not what I'm thinking
(I translate my thinking for backward compatibility with my peers).
Kirby
Arealess Points
Subject: Reply tp "Reply to "Re: Do Points Have Area""
Author: Clifford J. Nelson <nelsoncj@gte.net>
Date: 22 Jan 98 03:23:34 -0500 (EST)
A point is a location. How can a location have an area?
An area has more than one location!!!
Cliff Nelson
Volumetric Points
Subject: Reply to "Re: Reply to Do Points Have Area?"
Author: Candice Hebden <dreamy_aurora@hotmail.com>
Date: 21 Jan 98 09:06:17 -0500 (EST)
Kirby,
On January 20, 1998
[You wrote]
At the risk of being redundant, I'd prefer to ascribe volume to
points,
since your pancake points, if as flat as the ghostly "2D plane" won't
stack to create volume, any more than ghostly "0D points" you
criticize would make a line.
Response: By assigning points area, hence giving lines length, the 3D
can be arrised... with out even giving lines thickness (although I
don't personally believe that lines can really have zero thinckness)
Let me explain:
No matter how much sense it makes, there is nothing that states that
if you lay a bunch of lines together you will have height! But, to
make a cube, you need 12 line segments (which could have zero width)
because of the angles of the line segments, the construction will be
three dimentional! Even though lines with width makes sense, it is
not necessary in order to (visually) construct in the 3D
I can't wait till a new geometry that makes more sense, maybe Jesse
Yoder's circular geometry, but I don't want to disclaim things in
Euclidean geometry yet if they still make some sense!
Candice Hebden
@
dreamy_aurora@hotmail.com
http://forum.swarthmore.edu/epigone/geometry-research/khulstaymerm/48rtibaa4j2z@forum.swarthmore.edu
Volumetric Points
Subject: Reply to Do Points Have Area?
Author: Jesse Yoder < jesse@flowresearch.com>
Date: 22 Jan 98 14:08:14 -0500 (EST)
Hi Kirby -
You said, beginning with a quote about dimensionless points::
">points. Mutiplying 0 by infinity still equals 0. As far as I can
see,
>this remains an unresolved problem for Euclid's Axiom One (definition
>of point), and I believe that ascribing area to points is the only
way
>around it.
>
At the risk of being redundant, I'd prefer to ascribe volume to
points,
since your pancake points, if as flat as the ghostly "2D plane" won't
stack to create volume, any more than ghostly "0D points" you
criticize would make a line.
RESPONSE: Boy, am I glad you said that! I think that volume is a good
way to go, if you are operating in 3-dimensional space. This makes the
Points into small Spheres or Balls. For the most part, I have confined
my discussion to the 2-dimensional plane of circles, rather than the
3-dimensional area that involves volume. I can avoid your issues about
"pancake Points" by ascribing height to planes. But in general, once
we switch to 3 dimensions, points become Spheres and hence have
volume.
I'm sorry I can't really follow the rest of your comments relating to
an isotropic matrix or lattice, although it sounds like you are
suggesting some type of link between geometry and physical theory. If
I sometimes don't respond to your comments, it's only because I
haven't mastered the language of your Fullerian geometry.
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/khulstaymerm/uzfpsp2ko14l@forum.swarthmore.edu
Measuring Points
Subject: Reply to ""Reply to "Re: Do Points Have Area?""
Author: Candice Hebden <dreamy_aurora@hotmail.com>
Date: 22 Jan 98 21:51:33 -0500 (EST)
Hey Jesse---
In respect to your Points, when you measure the distance between two
Points, do you measure from???
a. the center of the two points,
b. from the sides facing each other, or
c. from the opposite sides?
If your answer is a, how can you have a center to the smallest
circular mesurement??? Wouldn't that center have to be at a Point???
So then the center of a Point is a Point which is a Point to infinity!
If your answer is b, how can you have area that doesn't exist???
Because if you want to find the length between Point A and Point B,
and there exists a Point C (which is colinear with A and B) which is
between Point A and Point B. Because AC+CB=AB when the points are
colinear, we must account for the length in Point C... which is not
measured with AC or with CB, so then, in your circular geometry, AC +
CB does not equal AB. AC + CB < AB
If your answer is c, how can you count up area twice... in the above
scenario with this answer, AC + CB > AB
Candice Hebden
@
dreamy_aurora@hotmail.com
Zeno’s Paradox
Subject: "Reply to "Re: Do Points Have Area?"
Author: Jesse Yoder < jesse@flowresearch.com>
Date: 22 Jan 98 14:43:27 -0500 (EST)
Hi Cliff -
Let me attempt to comment on your comment, which was:
>"A point is a location. How can a location have an area?
An area has more than one location!!!"
RESPONSE: You have put your finger on the problem that generates
Zero's paradox. If you say "Here I am at point A. Now I will walk
across the room to point B". Then you reflect "But to do this, I have
to go halfway from point A to point B, then halfway again, etc. How is
this possible?" The problem comes in when you imagine that a
3-dimensional object can be located at a dimensionless area. Once you
admit this, since you can always interpose a point between any two
other points, you open the door to the possibility of an infinite
series. The way around this is to say not that you are located at a
point, but that you are at a Point, i.e., a point that has dimension
(area).
At the same time, you have to specify what is to count as moving to a
new location. This is parallel to specifying a unit of measurement.
Once you see specify what is to count as a unit of motion for a
3-dimensional object (such as your human body), you realize that
moving ahead 1/1000th of an inch is not a motion -- you are still
located at the same (3-dimensional) place. This defeats the
possibility of introducing an infinite series of motions, which is the
idea that Zeno's paradox is based on.
To avoid paradox, we must say that points are Points! (i.e., what
appear to be dimensionless points are really points with area i.e.
Points)
Jesse
Reply to John Conway
Subject: Re: Reply to Do Points Have Area?
Author: Jesse Yoder < jesse@flowresearch.com>
Date: 22 Jan 98 14:32:16 -0500 (EST)
Hi John -
On January 21, 1998, you wrote:
>" [John Conway]
> >"If we're just talking about some purely conceptual space then the
> assertions are meaningless until that space is somehow defined.
> Jesse speaks of "circular geometry", in which a "point" is the
> smallest unit area, and in other statements he's made it clear
> that he thinks of these "points" as little circles and lines
> as like strings of beads: oooooooooooooooooo, in which
> any two adjacent ones touch each other at a point."
>
> [Jesse]
> "Response: You seem to understand pretty well what I mean. Here is
how
> a plane would look, with lots of points;
>" It still surprises me that you didn't even notice the double
use of the word "point" in the sentence I obliquely quoted from you!
How can two points touch at a point?"
RESPONSE: How can two Points touch at a Point? I don't know; perhaps
they touch at a point. But I don't understand why my position is so
much less understandable than the Euclidean one. On Euclid's account a
line if made up of infinitely many dimensionless points. So the points
are compactly packed, yet there is aways room for one more between any
two points! Does this mean there is empty space between two Euclidean
points? I'm not even sure they touch -- what I'm claiming is that when
two Points are next to each other, they have the same relation as when
two physical objects are next to each other. But I still don't even
know if two objects that are touching have a point in common, or if
they are just "up against" each other the way a baseball would be in a
glove.
You then continue:
>" Of course, you've now agreed to distinguish between "Points"
and "points", but it really seems to me that in a fundamental
sense this vitiates your system, because it bases it on the
traditional notions. Surely you should be able to describe
the structure and arrangement of your Points without using
Euclid's points? If not, it can hardly be true that "a Point
is the smallest allowable unit of area"."
RESPONSE: I don't think that distinguishing between Points and points
vitiates my system. If you say that I've reintroduced Euclidean points
by talking about intersecting Points, then I would refer you to the
above paragraph, where I say it is not yet clear how two Points next
to each other relate to each other--it may not require introducing the
idea of point.
You then continue:
>"I have difficulty in following your comments about switching to
new frames of reference. Do you think this is legal, or were you
really saying it was impossible? It seems to me that it's obviously
impossible in your system. If a Point is really the smallest
allowable unit of area, then no kind of changing frames of
reference can possibly produce a smaller Point."
RESPONSE: If my "frames of reference switching" comments are hard to
follow, I apologize. Perhaps I haven't adequately explained the idea.
But the idea is, I claim, not hard to understand, though perhaps the
term "frame of reference" is too abstract. What I am saying is that
when someone uses a coordinate system, they should specify their unit
of measurement (which itself is embedded in a frame of reference). I
believe there is a unit of measurement implicit every time a
measurement is made. For example, if I'm measuring distance to the
sun, it's miles. If it's gasoline, it's tenths of a gallon. Once the
unit of measurement is known, this determines the size of the points
in the line. If it's tenths of an inch, the Points are 1/10 of an inch
in length (or diameter).If it's 1/100th of an inch, the Points are
1/100th of an inch in length (or diameter). This is how changing
frames of reference (whcih is really just changing units of
measurement) can produce a smaller Point. I prefer this to saying that
the points are infinitely small.
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/khulstaymerm/5k30n9k6k5ss@forum.swarthmore.edu
Points & points
Subject: Re: Reply to Do Points Have Area?
Author: John Conway <conway@math.Princeton.EDU>
Date: Thu, 22 Jan 1998 16:44:27 -0500 (EST)
Jesse, I have tried, and tried very hard, to understand what you're
saying, but have reached the point at which I'm about to give up. Before
I do so, I'm making one last try (I will make more "last tries" if I
get something out of this one!).
Let me say that I am entirely happy with your basic idea of getting rid
of Euclid's fiction of "points with no magnitude". It's just that I have
not managed to get any kind of understanding of what you think you are
putting in its place. Several times I have asked you direct questions,
but the answers have been no help in telling me what you're thinking about.
The closest I got was when you told me that the Points in your plane
looked like:
o o o o o o o o o o o o o
o o o o o o o o o o o o o
o o o o o o o o o o o o o
o o o o o o o o o o o o o
(but magified so that they touch). But then you went on to define a Circle
to be a continuous string of these, from which I deduced that in fact
there couldn't be any non-trivial Circles. Then it turned out that it
wasn't the set of all Points in your plane that looked like the above
figure, but only those in the coordinate-system (or something). Forgive
me if I'm getting this wrong, but but I really am confused.
So I got the idea that there were more Points besides those in the
coordinate system. It seemed to me that (taking a suitable unit), the
points in the coordinate system were discs of unit diameter centered at
(Euclid's) points with integer coordinates, while perhaps there were
also other Points (which were also discs of unit diameter) centered at
other points. This would then allow there to be Circles in the sense in
which you defined that term, i.e., continuous closed loops of Points all
at the same distance from a given Point, namely the discs of unit
diameter centered at the vertices of one of Euclid's regular polygons of
edgelength 1. So I asked you explicitly whether there was any
difference between this model and your geometry, and you said something
like "well, let's try that". Well, I don't want to just try something.
I'm perfectly capable of studying all kinds of geometry and working out
their properties; but what I want to know is precisely what you are
thinking about, and you don't seem to be capable of telling me. I really
don't know just what it is.
I ask again. Are all the Points of your plane arranged in an array
like the above, or are there others? Can two Points overlap withouyt
being equal? Is there any difference between your kind of plane geometry
and the set of all unit discs in Euclid's geometry, and if so, just what
is this difference? Or have you not yet understood your own ideas in
enough detail to be able to give answers to these questions?
John Conway
Circular Coordinate System
Subject: Re: Reply to Do Points Have Area?
Author: Jesse Yoder < jesse@flowresearch.com>
Date: 26 Jan 98 18:20:29 -0500 (EST)
Hi John -
I have read through your expression of frustration regarding my
discussion of circular geometry. After offline discussions with you, I
have come to realize that I am addressing two separate issues in the
new geometry I am developing, and I haven't been distinguishing them
adequately. These issues are as follows:
1. Finding a new geometry that provides a rational value for the area
of a circle and does not rely on pi. This has to do with CIRCULAR
GEOMETRY, and it involves developing an alternative to the Cartesian
Coordinate System.
2. Finding an analysis of the number line that avoids the paradoxes
generated by the assumption that a line is made up of infinitely many
dimensionless points. This has to do with developing the concept of
Points (i.e, points with area), and it involves developing an
alternative to Euclidean geometry.
I now believe that it is not possible to easily develop a Circular
Geometry (#1) that provides an alternative to the Cartesian Coordinate
system in terms of Points -- instead, I believe it should be done in
terms of a series of circles that provide an alternative to the X-Y
Cartesian Coordinate system. Once this is done, one could give either
a Euclidean analysis of the lines contained in this geometry, saying
it is made up of Euclidean points, or could then go on to analyze
these lines as made of of Points, with the number of Points changing
as the unit of measurement changes. I prefer the second of these two
options.
In terms of developing Circular Geometry, an alterntative to the
Cartesian Coordinate system (#1), I would suggest the followng:
Replacing the X axis in the Cartesian Coordinate system with a series
of unit circles laid out end to end in an east and west direction,
each with an area of one round inch, and a radius of 1/2 inch. These
unit circles INTERSECT (share a common point) at the interger points--
1, 2, 3, etc., and likewise on the negative side (-1, -2, -3, etc.),
as well as at the point of origin. These are circles with an area of
one round inch (not solid Points).
Likewise, replacing the Y axis in the Cartesian Coordinate system with
a series of unit circles laid out end to end in a north and south
direction, each with an area of one round inch, and a radius of 1/2
inch. These unit circles INTERSECT (share a common point) at the
integer points 1,2, and 3 (in the positive direction) and -1, -2, -3,
etc. (in the negative direction), as well as at the point of origin.
These are circles with an area of one round inch (not solid Points).
Once this structure is set up, it is possible to use these unit
circles to give the area of any circle in this circular coordinate
system, using the formula 4*r*r. Here r = the radius of the circle,
defined in the usual way (the distance from the center to the edge of
the circle). So a circle with a radius of 1/2 inch has an area of one
inch. A circle with a radius of 2 inches (and diameter = 4) has an
area of 4 round inches. The formula d*d, where d = diameter, also
works to find round inches.
Once this structure is set up, it is possible to take the FURTHER step
of saying the lines making up the radius are made up of finitely many
Points with area, rather than being made of of infinitely many
dimensionless points. To say this is to give a non-Euclidean
interpretation of this Circular Geometry. While I want to say this, I
believe that the Circular Geometry described above (as many unit
circles laid end to end replacing the x and y axes) can stand on its
own with a Euclidean or a non-Euclidean interpretation.
Once the two sets of intersecting series of circles are drawn, they
can be used as a frame of reference for describing other circles
within the geometric plane, much as the x and y axes are currently
used in Cartesian Coordinate geometry.
I hope this helps describe more clearly what I have in mind. I will
desribe the second alternative (geometry with Points) in a separate
post.
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/khulstaymerm/yl3btpequju7@forum.swarthmore.edu
Definition of Circle
Subject: RE: Reply to "Do Points Have Area?"
Author: Jesse Yoder jesse@flowresearch.com
Date: Sat, 31 Jan 1998 12:59:34 -0500
Hi DGoncz,
I hope you have recovered from you long bout of sleeplessness.
On Jan. 1, 1998, you wrote:
>"It is possible for a circle to consist only of four
> points. But it wouldn't be a conventional circle."
>
RESPONSE: I don't see how it's possible for a circle to consist of only
four points. This would not be a circle at all (even an unconventional
one), but simply four points that lie on a circle.
A circle by its very nature (in other words, by definition), is a
continuous circular line. This is what's wrong with the traditional
definition of a circle as "a set of points equidistant from a fixed
point." If these points aren't "continuous", there is no circle, but
merely a set of points arranged in a circular fashion. I believe that
the Euclidean tendency to identify a line with "infinitely many points"
tends to obscure the requirement that the points lying on a circle must
be continuous in order for a circle to exist.
I don't understand what you mean by saying "meaning can be ignored if
properties are specified." Since you mentioned this in a computer
context, are you talking about terms that have to intrinsic meaning, yet
have a function if the rules of their use are specified?
Happy New Year to you as well!
Jesse
Definition of Circle
Subject: REPLY TO "RE: REPLY TO POINTS HAVE AREA?"
Author: CANDICE HEBDEN <DREAMY_AURORA@hotmail.com>
Date: 1 Feb 98 18:35:13 -0500 (EST)
JESSE---
[YOU WROTE ON JANUARY 31, 1998]
>A circle by its very nature (in other words, by definition), is a
>continuous circular line. This is what's wrong with the traditional
>definition of a circle as "a set of points equidistant from a fixed
>point." If these points aren't "continuous", there is no circle, but
>merely a set of points arranged in a circular fashion. I believe that
>the Euclidean tendency to identify a line with "infinitely many
>points" tends to obscure the requirement that the points lying on a
>circle must be continuous in order for a circle to exist.
RESPONSE: WHERE DID YOU EVER READ THAT A CIRCULAR HAD TO BE A
CONTINUOUS CIRCULAR LINE??? DEPENDING ON WHAT FORM OF GEOMETRY YOU'RE
USING, A CIRCLE COULD CONSIST OF FOUR POINTS. IF YOU WERE TO HAVE
TAXICAB GEOMETRY WHERE POINTS COULD ONLY EXIST ON THE "CORNERS", THEN,
IF YOU USE EUCLIDEAN'S DEFINITION OF A CIRCLE (ALL POINTS EQUIDISTENT
FROM A FIXED POINT), YOU GET A CIRCLE THAT CONSISTS OF FOUR POINTS.
HOPE TO TALK TO YOU LATER JESSE,
CANDICE HEBDEN
@
DREAMY_AURORA@HOTMAIL.COM
http://forum.swarthmore.edu/epigone/geometry-research/thyspenddwox/sp7yz32axrxy@forum.swarthmore.edu
Circular Definition
Subject: RE: Reply to "Do Points Have Area?"
Author: Bill Haloupek <haloupekb@UWSTOUT.EDU>
Date: Sat, 31 Jan 1998 13:03:51 -0600 (CST)
Jesse Yoder wrote:
>A circle by its very nature (in other words, by definition), is a
>continuous circular line.
Seems like a "circular" definition to me. :o)
Bill H.
http://forum.swarthmore.edu/epigone/geometry-research/thyspenddwox/01IT10A471AA001J7Z@UWSTOUT.EDU
Circular Definitions
Subject: RE: Reply to "Do Points Have Area?"
Author: Jesse Yoder < jesse@flowresearch.com>
Date: Mon, 2 Feb 1998 09:01:27 -0500
Bill - On the face of it, I agree. I think it is very difficult to come
up with a noncircular definition of a circle. Notice, however, that the
definition you quote is not viciously circular, since it does not
contain the word 'circle' - instead, it contains the word 'circular.' On
the other hand, as I have argued elsewhere (in a post to Candice), the
Euclidean definition of the circle as a set of points equidistant from a
fixed point is BANKRUPT, since it allows for the possibility that four
points form a circle.
In any system, some concepts or terms must be accepted as undefined. Not
every concept can be defined in terms of previously defined concepts.
Garry (Mr K K G Yau) has urged me in the past to accept 'point' as
undefined. I am unwilling to do this, because the difference between
points and Points (which are points with area) is fundamental to the
Circular Geometry I have developed.
On the other hand, it may be necessary to accept certain operations as
fundamental and undefined. In particular, it may be necessary to accept
as undefined the concept of circular motion (as in a Circle is generated
by moving a Point in a circular motion). Likewise, if we are to suffer
with the concept of a straight line (which I have it appears been forced
to accept against my wishes, since I need the concept of radius and
diameter), perhaps the concept of "straight" should also be undefined,
as in a straight Line is generated by moving a Point in a single uniform
direction--the path so generated is a straight Line. (Note that a curve
is generated by moving a Point in a nonuniform direction).
Possibly the idea of circular motion could be specified without using
the concept of circle or circularity, e.g., by moving a Point in such a
way as to maintain a constant distance from a fixed point until the
Point intersects with itself--the path so generated is a Circle. If you
accept this, then this is a noncircular definition of the circle.
Jesse
Reply to Candice
Subject: RE: REPLY TO "RE: REPLY TO POINTS HAVE AREA?"
Author: Jesse Yoder < jesse@flowresearch.com>
Date: Mon, 2 Feb 1998 08:39:03 -0500
Hi Candice -
First of all, let me say that I like your convention of capitalizing
your ENTIRE ANSWER, which is both consistent with the spirit of Circular
Geometry, and seems to give your comments added importance.
On SUNDAY, FEBRUARY 1, 1998, at 6:35 PM, YOU WROTE, BEGINNING WITH A
QUOTE FROM ME ABOUT THE DEFINITION OF CIRCLES:
> [YOU (i.e. Jesse) WROTE ON JANUARY 31, 1998]
>
> >A circle by its very nature (in other words, by definition), is a
> >continuous circular line. This is what's wrong with the traditional
> >definition of a circle as "a set of points equidistant from a fixed
> >point." If these points aren't "continuous", there is no circle, but
> >merely a set of points arranged in a circular fashion. I believe that
> >the Euclidean tendency to identify a line with "infinitely many
> >points" tends to obscure the requirement that the points lying on a
> >circle must be continuous in order for a circle to exist.
>
> RESPONSE (from Candice): WHERE DID YOU EVER READ THAT A CIRCULAR HAD
> TO BE A
> CONTINUOUS CIRCULAR LINE??? DEPENDING ON WHAT FORM OF GEOMETRY YOU'RE
> USING, A CIRCLE COULD CONSIST OF FOUR POINTS. IF YOU WERE TO HAVE
> TAXICAB GEOMETRY WHERE POINTS COULD ONLY EXIST ON THE "CORNERS", THEN,
> IF YOU USE EUCLIDEAN'S DEFINITION OF A CIRCLE (ALL POINTS EQUIDISTENT
> FROM A FIXED POINT), YOU GET A CIRCLE THAT CONSISTS OF FOUR POINTS.
>
RESPONSE: I didn't read it anywhere that a circle has to be a continuous
circular line. Instead, I take this to be implicit in the very concept
of a circle. And you example, from taxicab geometry, simply shows the
total bankruptcy of the Euclidean definition of a circle as a set of
points equidistant from a fixed point. If four points equidistant from a
circle can actually BE a circle, then I suppose the four corners of a
square and actually BE a square, and the three tips of a triangle can
actually FORM a triangle.
Let me also say that after someone mentioned the idea of taxicab
geometry a few months ago, I went out and bought a book on taxicab
geometry. While I don't have it here to refer to, I remember enough of
this to understand what you mean be saying that you have points that are
the intersections of streets (and note that these points are actually
squares or rectangles, NOT circles). And what I would say to you is that
there are no circles in taxicab geometry, and there are no Circles
either, because there is no circular area (unless the streets happen to
be circles, in which case they will be Circles, since they have width).
To reiterate, circles and Circles are continuous closed loops or Loops
and not merely a set of points equidistant from a fixed point. The
Euclidean definition only arises because a line is analyzed as being
made up of infinitely many points with no dimension. But this analysis
of the line has to be rejected because it leads to a paradox. In its
place, I propose the Line, which is created by putting a Point in
motion.
Thanks for your comments, and I look forward to your response!
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/dwodeldsal
Do Points Have Area?
Subject: RE: Reply to "Do Points Have Area"
Author: John Conway <conway@math.Princeton.EDU>
Date: Mon, 2 Feb 1998 14:28:51 -0500 (EST)
I remark that this question no longer has any significance.
Jesse has distinguished between the old "points", that don't have
area, and his new "Points", which are defined to be circles of
some fixed area. Of course with definition "Points" do have area,
but - may I say it - what's the point? If the question had been
(as it now translates) "Do circles have area?", the discussion
would have been simpler and shorter(I hope!).
John Conway
http://forum.swarthmore.edu/epigone/geometry-research/wingquandwe
Measuring Points
Subject: RE: Reply to ""Reply to "Re: Do Points Have Area?""
Author: Jesse Yoder jesse@flowresearch.com
Date: Mon, 2 Feb 1998 16:50:30 -0500
Hey Candice - Sorry I overlooked this email. Let me respond to your
questions.
You wrote:
> In respect to your Points, when you measure the distance between two
> Points, do you measure from???
>
> a. the center of the two points,
> b. from the sides facing each other, or
> c. from the opposite sides?
>
RESPONSE: In general, I would say "Use corresponding positions. So if
you measure from the center of Point A, use the center of point B. Your
choices b and c violate this principle. This is a real-world problem
that most people completely ignore.
> If your answer is a, how can you have a center to the smallest
> circular mesurement??? Wouldn't that center have to be at a Point???
> So then the center of a Point is a Point which is a Point to infinity!
>
RESPONSE: I know (or believe) you are trying to find a contradiction in
my theory here. What I have said is that a Point is the smallest unit of
measurement accepted for a given purpose or application. So you are
treating the Point as being "unbreakable" for your measurement. So in a
sense the distance between any two Points A and B is from anyplace on A
to anyplace on B. But logic would dictate using corresponding locations
on A and B, and measuring from there.
Your discussions of b and c are interesting, but I reject both of these
as answers.
> If your answer is b, how can you have area that doesn't exist???
>
> Because if you want to find the length between Point A and Point B,
> and there exists a Point C (which is colinear with A and B) which is
> between Point A and Point B. Because AC+CB=AB when the points are
> colinear, we must account for the length in Point C... which is not
> measured with AC or with CB, so then, in your circular geometry, AC +
> CB does not equal AB. AC + CB < AB
>
> If your answer is c, how can you count up area twice... in the
> above
> scenario with this answer, AC + CB > AB
>
>
Jesse
http://forum.swarthmore.edu/epigone/geometry-research/smumshenskel