[episode 20] Klein space?
Rudolf Polzer
I know a mathematical object named "Klein'sche Flasche"
that is the 4D equivalent to the Moebius' tape: as Moebius'
take does not have an inner or outer side, "Klein'sche
Flasche" does not have an inner or outer part of space.
Even if it is closed. But it does not exist in 3D space
(as the Moebius tape does not in the 2D plane - two lines
would cross each other) - in 3D space it would intersect
itself.
But if this object is meant, the expression "trapped in Klein
space" is a contradiction because you cannot be trapped
in something that does not have an inner part. So is
"Klein Space" really the same thing as Germans call "Klein'sche
Flasche"?
BTW: What is the background of the "sea of Dirac"? What did
Dirac discover that is meant here?
For non-Mathematicians: you can easily make a Moebius tape.
Make a long strip of paper and call its two sides "A" and "B".
Perhaps you could mark them. Then connect its ends so that the
A side of one end is on the same side as the B side of the
other end. You'll get a ring that is rotated by 180° at some
point. Then you can easily check that it only has one side:
try to make one side black using a pen by just drawing in one
direction. When you get back to your starting point, you
will notice that "both" sides are black!
If you want, I could try to get drawings of the Moebius
tape and the "Klein'sche Flasche".
Another interesting experiment is halving the Moebius tape
in tangential direction (it is a bit difficult not to make
a mistake, but it is possible). You then get two rings that
are wound into each other.
Unfortunately, a "Klein'sche Flasche" is much harder to explain.
--
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Disaster
news:slrn9udngc.6...@www42.durchnull.de...
That does not sound like a sea of dirac!? I'm no expert here, we'd have to
drag Michael over and get him to make a comparison but I seem to remember
his periouvs explanation and that of Ritsuko on NGE to mean something other
then what you have described!
--
Kind regards
Disaster
JAE FAQ - www.evafaq.com
Rudolf Polzer
I meant two different things. The sea of dirac was only mentioned in BTW
and does not have anything to do with the Klein space.
I just found this:
,--- http://www.mccallie.org/myates/The%20Shape%20of%20Space/Student%20Projects/group10.htm
| Klein Space is a bizarre universe that has plagued the minds of great
| intellectuals over the years. Now some light shall be thrown upon the
| troublesome topic. As in a 3-Taurus, one may exit the left and return on the
| right in the exact same position it exited on the left side. If one were to
| leave the back he would come back in the front in the same fashion. However, thetwo topologies differ concerning the top and bottom. It has what might be called a "mirror effect" as one exits one side and returns in the other. For example,
| if one exits in the left back of the top, he will return in the right back of
| the bottom. From the right middle of the bottom, he will come back in the left
| middle of the top. Therefore, a Klein bottle in effect is a 2-torus with the
| third pair of sides acting like a mobius strip. Note: It is impossible for us to invision a Klein bottle exactly, because it exists in four dimensions, and we
| are only able to perceive three dimensions.
`---
which implies that Klein Space and Klein bottle are the same thing
(and the "Kleinsche Flasche" is translated as Klein bottle even if
the word Flasche in it does not have anything to do with a bottle
but originally meant "Fläche" ie area. Somehow it became Flasche...)
so nothing can be trapped in it because there's no inside. Unless
you restrict space to the Klein bottle, because the three-dimensional
Klein space (which is the 4D bottle's surface) does exist. That's like
the two-dimensinal Moebius strip:
,--------------------------------------------.
| V >>>>>>|
|_________________________V__________________|
|>>>>>>>>>>>>>>>>> V |
| V |
`--------------------------------------------'
The left and right are connected with a 180° rotation, while the top
and bottom are connected directly.
So nothing that has a 4D volume can be trapped in Klein space which
does not have a 4D volume. But in physics, there aren't any 2D
objects in 3D space (according to the current model, there is a
minimum width and a minimum delta T) and there aren't any 3D objects
in 4D space. So Shinji's Ego must be 3D or less and be in 4D space
and is physically not noticable, which contradicts to it being
physically monitored. But then what is an AT field? If it is a hard-to-
penetrate 2D-like wall in 3D space - what's its thickness then? Or
is it more like an electric or magnetic field (and can therefore
be neutralized, i.e. pushed away, by a "stronger" field)?
But from what I found out about the sea of dirac, it is a negative
electron sea that does not have anything to do with "teleporting"
or black holes, but more with Dirac's prediction of positrons.
--
#!/usr/bin/perl
no warnings;$,=('~'x80)."\r";END{print"\n"}print
~_^ 1004 ^_^ 1818, m }^_^{ } *_* KGBENTE=>5423683
,'nu...@durchnull.de'
Disaster
> > That does not sound like a sea of dirac!? I'm no expert here, we'd have
to
> > drag Michael over and get him to make a comparison but I seem to
remember
> > his periouvs explanation and that of Ritsuko on NGE to mean something
other
> > then what you have described!
>
> I meant two different things. The sea of dirac was only mentioned in BTW
> and does not have anything to do with the Klein space.
Ah I see. My mistake then. It was late when I read it. :)
[SNIP]
> But from what I found out about the sea of dirac, it is a negative
> electron sea that does not have anything to do with "teleporting"
> or black holes, but more with Dirac's prediction of positrons.
This is definately Michael's area! One day I'm gonna look into all of this,
perhaps when I expand my mind a little more! I once remember being able to
comprehend E=MC2, but that was a while ago and my mind has shrunk since!
David Scarlett
>
> I know a mathematical object named "Klein'sche Flasche"
> that is the 4D equivalent to the Moebius' tape: as Moebius'
> take does not have an inner or outer side, "Klein'sche
> Flasche" does not have an inner or outer part of space.
> Even if it is closed. But it does not exist in 3D space
> (as the Moebius tape does not in the 2D plane - two lines
> would cross each other) - in 3D space it would intersect
> itself.
Wait a minute..... Do you consider a mobius strip to be 2D or 3D?
So do you mean Klein space is like a portion of 3D space with the
ends joined? (And hence wrapped up in 4D, like how a mobius strip
is effectively 2D, but can only be envisioned in 3D?)
Reminds me of a good one-liner... "Why did the chicken cross the mobius
strip?"
--
David Scarlett
dscarlett [AT] optushome [DOT] com [DOT] au
http://members.optushome.com.au/dscarlett/
"Why am I talking to myself? More importantly, why am I expecting a
reply?"
-Daria
Watchman
You mean I have to get a more powerful microscope to see it now?! :p
--
'Anyone who isn't confused doesn't really know what's going on'
Disaster
this,
> > perhaps when I expand my mind a little more! I once remember being able
to
> > comprehend E=MC2, but that was a while ago and my mind has shrunk since!
>
You'll have to stop aiming it at your pants before you'll see any of MY
brain matter Watchman! :P
Disaster
> strip?"
*Blinks*
David Scarlett
>
> *Blinks*
> Why did the chicken cross the mobius strip?
You can't cross a Mobius strip... it only has one side. Hence thew question is a joke in itself. ;-)
--
David Scarlett
dscarlett [AT] optushome [DOT] com [DOT] au
http://members.optushome.com.au/dscarlett/
"Damn it, Kif, where's the little umbrella? That's what makes it a
scotch on the rocks!"
-Capt. Zapp Brannigann, Futurama
Michael Wignall
>
From a post of mine from March last year.
> Well, on the noteboard beside Ritsuko, there's written "the existence
> of the string and the space that moves around" "the macrocosmos" "the
> movement dimensions of the strings", and Jap Ency said that Ritsuko
> try to explain the Angel using the "super strings theory" that make
> possible to join the 4 atomic forces. The "super strings theory" says
> hat cosmos have 10 dimension (never heard before...) "super"is short
> for "super simmetric".
Ok... This may take a while :)
Firstly there really is no relationship between string theory and
the real world. It appears to be a theory of quantum gravity,
I just don't think it applies to "our" universe :)
The "goal" of Super String Theory is to "unify" the four fundamental
forces: Gravity, the Electromagnetic force, the Strong force and the
Weak force.
We are used to thinking of fundamental particles (like electrons) as
point-like 0-dimensional objects. A generalisation of this is fundamental
strings which are1-dimensional objects. They have no thickness but
do have a length, typically 10^(-33) cm. This is very small compared
to the length scales that we can reasonably measure, so these strings
are so small that they practically look like point particles.
Now these strings have certain vibrational modes which can be
characterised by various quantum numbers such as mass, spin, etc.
The basic idea is that each mode carries a set of quantum numbers
that correspond to a distinct type of fundamental particle. This is the
ultimate unification: all the fundamental particles we know can be
described by one object, a string.
Now this hasn't been proven, its just a theory, it does seem right
when you look at it, it really can't be shown to be right.
Ok, now Supersymmetry, is a theory which unifies bosons and
fermions. Every known particle would be paired with a "superpartner"
of the opposite type(boson vs. fermions). The major goal of the
newest high energy accelerators is to discover these superpartners
and to find evidence for supersymmetry. It is an essential ingredient in
realistic string theory models, hence the "super" in "superstring".
What this all means.... hmmm...
I guess the "the movement dimensions of the strings" means the
vibrational modes... Um... :) I haven't done "heaps" of study on
string theory, just general stuff...
It seems that in its final form string theory will be able to provide
answers to answer questions like: Where do the four forces that
we see come from? Why do we see the various types of particles
that we do? Why do particles have the masses and charges that
we see? Why do we live in 4 spacetime dimensions? What is the
nature of spacetime and gravity? Stuff like that :) Which would help
to explain what is happening within the universe, ie, a universe in
which the string theory holds true.
> I hope that could be a "SF rational explanation"...
HAHA well... :)
I hope I have explained string theory well enough, basically it
allows us to know why things are related, why things are. How this
is related to Eva, well if Ritsuko is using string theory then that means
basically they she can explain a lot of stuff about the "other universe"
that we can't do now about our own, so therefore she could get a
lot more info about it and make more informed choices about what
it is. Ritsuko and Co. didn't seem to question how Unit-01 got to
the "Shadow" they were more interested in the fact that it was moving
when it shouldn't have been able to :) I think that the info the a proven
string theory would give probably would explain how Unit-01 got from
A to B, but I don't know that much physics so... :)
-------------
Ok so how is all that related to Moebius strips Klein bottles? (I was
actually going to buy a glass Klien bottle the other day, they look
so cool, however it was much to expensive ^^)
Well the Klein bottles or rather the Kaluza-Klein bottle is the basis
of modern string theory. Which links into hyperspace and all this
really cool stuff.
So as to being "trapped" in Klein space. Well just before Ritsuko
mentions is Maya says "No good. His ego border is in a fixed loop."
A Moebius loop perhaps? Without a beginning or an end?
I'm not sure. It has been a while since I have done any serious physics,
I've moved onto other things since then. Of course we always have to
allow for the possibility that the creators just threw in some
terms that sounded cool, were vaguely correct technically but
otherwise had no meaning at all.
--
Michael Wignall
"Look at you, Hacker. A pathetic creature of meat and bone,
panting and sweating as you run through my corridors. How
can you challenge a perfect, immortal machine?" - SHODAN
Disaster
> > Why did the chicken cross the mobius strip?
>
> You can't cross a Mobius strip... it only has one side. Hence thew
question is a joke in itself. ;-)
You can cross on one side! Besides, why would a chicken even think of trying
to cross the strip if you can't cross it huh? That's just silly!
Disaster
*Raises hand*
Uh, what was that? :P
Nice Michael, I think there's more in here then last time! :)
David Scarlett
>
> From a post of mine from March last year.
>
> > The "super strings theory" says hat cosmos have 10 dimension (never
> > heard before...)
I thought the cosmos was 11 dimensional....
There was this bumper sticker the Uni bhysics club made which said,
"Physicists do it in 11 dimensions." ;-)
>
> The "goal" of Super String Theory is to "unify" the four fundamental
> forces: Gravity, the Electromagnetic force, the Strong force and the
> Weak force.
Baka, there are 5 fundamental forces: Gravity, electromagnetic, strong,
weak and of course, AT. ;-)
Rudolf Polzer
> "Rudolf Polzer" <adsg...@durchnull.de> wrote in message news:slrn9udngc.6...@www42.durchnull.de...
> > "It won't work. The signals are trapped in Klein space."
> >
> > I know a mathematical object named "Klein'sche Flasche"
> > that is the 4D equivalent to the Moebius' tape: as Moebius'
> > take does not have an inner or outer side, "Klein'sche
> > Flasche" does not have an inner or outer part of space.
> > Even if it is closed. But it does not exist in 3D space
> > (as the Moebius tape does not in the 2D plane - two lines
> > would cross each other) - in 3D space it would intersect
> > itself.
>
> Wait a minute..... Do you consider a mobius strip to be 2D or 3D?
It is 3D but has a 2D surface. Klein's bottle is 4D but has a 3D
surface.
> So do you mean Klein space is like a portion of 3D space with the
> ends joined? (And hence wrapped up in 4D, like how a mobius strip
> is effectively 2D, but can only be envisioned in 3D?)
Exactly. If you live in Klein space, it behaves exactly like
a Moebius strip - it just has one dimension more.
But the only way of entering Klein space (a portion of 4D that
behaves like Klein space) you need to reduce your physical
dimension from 4 to 3. It's like the fact that a flea cannot
be trapped in a Moebius strip: it'd have to reduce its apparent
dimension from 3 to 2, becoming an area.
> Reminds me of a good one-liner... "Why did the chicken cross the mobius
> strip?"
I don't know it...
--
LATIN LESSON:
I hear, Audio,
I see, ---> Video,
I learn! Disco! [Robin Koch in de.talk.jokes, translated]
Watchman
All of which begs the question; what is the '4th dimension' in relation to the
Klein bottle?
David Scarlett
>
> All of which begs the question; what is the '4th dimension' in
> relation to the Klein bottle?
What is a Klein bottle?
--
David Scarlett
dscarlett [AT] optushome [DOT] com [DOT] au
http://members.optushome.com.au/dscarlett/
"If you have a thing for parentheses, then LISP is the programming
language for you!"
-My Computing Fundamentals lecturer
Rudolf Polzer
> "Watchman" <str...@SPAMPHOBICwhyalla.net.au> wrote in message news:3BE8AB43...@SPAMPHOBICwhyalla.net.au...
> >
> > All of which begs the question; what is the '4th dimension' in
> > relation to the Klein bottle?
>
> What is a Klein bottle?
I try it...
A self-intersecting 3D model of it can be seen on
http://www.kleinbottle.com (no, you don't need to buy one).
Especially the one on the top. You see that the "inner" and
"outer" side of the glass are connected so there is no "inner"
and "outer" side and/or space.
--
#!/usr/bin/perl -w # Das "Menschliche Genom Projekt" in Perl. [x] #
# original by joc...@plumeyer.org # shortened by nu...@durchnull.de #
for$0(1..23){print"\n\n\nDas $0te Chromosom:\n";sleep 4;for(0..int((
109065435.697619/(exp(($0-1)/4)+1))-.5)){print qw(A T G C)[rand 4]}}
Disaster
> > > relation to the Klein bottle?
> >
> > What is a Klein bottle?
>
> I try it...
>
> A self-intersecting 3D model of it can be seen on
> http://www.kleinbottle.com (no, you don't need to buy one).
> Especially the one on the top. You see that the "inner" and
> "outer" side of the glass are connected so there is no "inner"
> and "outer" side and/or space.
That is so cool! I'm thinking about getting the glass one! Uh, I mean the
drinking glass one! *GRINS*
Watchman
David Scarlett wrote:
>
> "Watchman" <str...@SPAMPHOBICwhyalla.net.au> wrote in message news:3BE8AB43...@SPAMPHOBICwhyalla.net.au...
> >
> > All of which begs the question; what is the '4th dimension' in
> > relation to the Klein bottle?
>
> What is a Klein bottle?
>
> --
A 3 dimensional object with only one side.
Nevermann
Can someone explain this to me like I'm 8 years old?
James Nevermann II
David Scarlett
> David Scarlett wrote:
> >
> > What is a Klein bottle?
> >
>
>
Like a normal cup? ;-)
--
David Scarlett
dscarlett [AT] optushome [DOT] com [DOT] au
http://members.optushome.com.au/dscarlett/
"Why am I talking to myself? More importantly, why am I expecting a
reply?"
-Daria
David Scarlett
>
> Can someone explain this to me like I'm 8 years old?
No. But here's the consolation prize:
"Should reality be something that only a handful of the world's most
advanced physicists understand? One would expect at least a majority
of people to understand it. Should reality be expressible only in
only in symbols that require university-level mathematics to
manipulate? Should it be something that changes from year to year as
new scientific theories are formulated? Should it be something about
which different schools of physics can quarrel for years with no firm
resolution on either side? If this is so then how is it fair to
imprison a person in a mental hospital for life with no trial and no
jury and no parole for "failing to understand reality"? By this
criterion shouldn't all but a handful of the world's most advanced
physicists be locked up for life? Who is crazy here and who is sane?"
-Robert M. Pirsig
I.e. The whole Klein space thing is full of shit so don't worry about
it. ;)
Disaster
> >
> > A 3 dimensional object with only one side.
>
> Like a normal cup? ;-)
It is a 4D object in a 3D space. It is not like a normal cup.
Disaster
>
> No. But here's the consolation prize:
>
> "Should reality be something that only a handful of the world's most
> advanced physicists understand? One would expect at least a majority
> of people to understand it. Should reality be expressible only in
> only in symbols that require university-level mathematics to
> manipulate? Should it be something that changes from year to year as
> new scientific theories are formulated? Should it be something about
> which different schools of physics can quarrel for years with no firm
> resolution on either side? If this is so then how is it fair to
> imprison a person in a mental hospital for life with no trial and no
> jury and no parole for "failing to understand reality"? By this
> criterion shouldn't all but a handful of the world's most advanced
> physicists be locked up for life? Who is crazy here and who is sane?"
> -Robert M. Pirsig
That is stupid! Those committed to a mental hospital are those who can not
perceive reality as the rest of society does on an average basis. This makes
them incompatible with the majority. Thus leaving them to their own devices
would be dangerous to both themselves and others.
Further, on occasion, those in the hospital have a more advanced grasp of
reality.
> I.e. The whole Klein space thing is full of shit so don't worry about
> it. ;)
It is NOT full of shit! In the pictures presented, it has colored water in
it! ;P
David Scarlett
>
> It is NOT full of shit! In the pictures presented, it has colored
> water in it! ;P
LMAO! But those are Klein _bottles_, not Klein _space_. I got the
impression that the two are different.... Especially considering how
Klein space cannot exist within 3D space. ;-)
Rudolf Polzer
> "Disaster" <disa...@disfanfic.net> wrote in message news:tuk8opq...@corp.supernews.com...
> >
> > It is NOT full of shit! In the pictures presented, it has colored
> > water in it! ;P
>
> LMAO! But those are Klein _bottles_, not Klein _space_. I got the
> impression that the two are different.... Especially considering how
> Klein space cannot exist within 3D space. ;-)
Klein space is easy to explain from the Klein bottle: Klein space is
the 3D surface of the 4D Klein bottle. The bottle does not exist
in 3D... you see it is self-intersecting in 3D.
In 4D, however, it exists. Its surface is 3D. And any 3D surface is a
'space'.
Let me try to demonstrate this: imagine the earth were a sphere. On
it there is a planar geometry, of course - earth is a plane and you,
being rather small compared to earth, do not notice a difference
between earth surface and a real plane. Many centuries people thought
earth was a disc...
But there are some things on earth that cannot happen on a disc. For
example, if you go ca. 40000 kilometers in any direction, you return
at your starting point. This could not happen in Euclidean plane
geometry.
Another thing: if you go 10000 kilometers in one direction, then
turn by 90°, then go x kilometers, then turn by 90° and go
10000 kilometers, you return at your starting point. To see this,
imagine you start at the south pole. Then after 10000 kilometers you are
at the equator. The equator is orthogonal to any meridian, so after
turning by 90°, you go in equator's direction. You stay on it while
walking x kilometers. After turning by 90° again, you walk on a meridian
and will return at the starting point - the south pole - after 10000
kilometers. Impossible on Euclid's plane.
Like the difference between the geometry on a sphere's surface and
Euclidean planar geometry, there's a difference between the well-known
Euclidean 3D geometry and the one on a Klein bottle's 3D surface.
Both are 3D, but they differ anyway.
--
bash$ telnet post.strato.de 110 # POP3 at its best
[...]
LIST
+OK 0 messages, listing follows
Watchman
Disaster wrote:
>
> > > > What is a Klein bottle?
> > >
> > > A 3 dimensional object with only one side.
> >
> > Like a normal cup? ;-)
>
> It is a 4D object in a 3D space. It is not like a normal cup.
> --
...and the fourth dimension is?
Disaster
> > > >
> > > > A 3 dimensional object with only one side.
> > >
> > > Like a normal cup? ;-)
> >
> > It is a 4D object in a 3D space. It is not like a normal cup.
>
Due to the limitations of our brains and the lack of any sensory organ
designed to perceive a 4th dimension. It's a little hard to grasp the
concept. We live in a 3D space, we have no need to perceive or understand
anymore dimensions so our brains happily evolved, using a mere 3 dimensions
to get around in. A 4th dimension is just like a 1st, 2nd and 3rd dimension,
only we are ill equipped to see it or understand it.
Disaster
> > water in it! ;P
>
> LMAO! But those are Klein _bottles_, not Klein _space_. I got the
> impression that the two are different.... Especially considering how
> Klein space cannot exist within 3D space. ;-)
*GRINS*
Well, that's why the bottle intersects itself. The idea is that in 4D space
this bottle would have it's surface broken in order to achieve the one sided
surface.
Disaster
I think my explanation was easier!
Watchman
To paraphrase, it doesn't exist. :)
Disaster
> > > > > >
> > > > > > A 3 dimensional object with only one side.
> > > > >
> > > > > Like a normal cup? ;-)
> > > >
> > > > It is a 4D object in a 3D space. It is not like a normal cup.
> > >
> > > ...and the fourth dimension is?
> >
> > Due to the limitations of our brains and the lack of any sensory organ
> > designed to perceive a 4th dimension. It's a little hard to grasp the
> > concept. We live in a 3D space, we have no need to perceive or
understand
> > anymore dimensions so our brains happily evolved, using a mere 3
dimensions
> > to get around in. A 4th dimension is just like a 1st, 2nd and 3rd
dimension,
> > only we are ill equipped to see it or understand it.
>
*Shrug*
The bottle exists! I've never seen the real thing, due to comments above. I
have not seen anyone say it does or does not exist, or that it's proven or
just a theory. I've only ever heard of people talking about it. I would
venture to guess that it is a theory in the working like a lot of the
physics stuff.
Rudolf Polzer
It's more like this:
- in 3D space, it can be proven that a Klein bottle does not exist.
- in 4D space, it can be proven that a Klein bottle does exist.
- in physics, time is often referred to as the forth dimension.
- so we can never see a Klein bottle, but we could perhaps see
moving 3D slices of it.
The same thing is with the sphere:
- in a 2D plane it can be proven that it does not exist.
- in 3D space, it does exist.
- if you would only see 2D and time was the 3rd dimension, you
could perhaps see a point which becomes a circle, a bigger circle,
achieves a maximum diameter and then shrinks again to a point which
disappears.
--
The easiest way to kill a Linux system:
www42:~ # rm /dev/null
Also possible:
www42:~ # chmod 000 /dev/null
David Scarlett
> - in 3D space, it can be proven that a Klein bottle does not exist.
Then perhaps you should report www.kleinbottle.com to the office of
fair trading. ;-)
--
David Scarlett
dscarlett [AT] optushome [DOT] com [DOT] au
http://members.optushome.com.au/dscarlett/
"Damn it, Kif, where's the little umbrella? That's what makes it a
Disaster
>
> Then perhaps you should report www.kleinbottle.com to the office of
> fair trading. ;-)
LOL
No way! Those things are cool!
Rudolf Polzer
> "Rudolf Polzer" <adsg...@durchnull.de> wrote in message news:slrn9usfe0.e...@www42.durchnull.de...
> > - in 3D space, it can be proven that a Klein bottle does not exist.
>
> Then perhaps you should report www.kleinbottle.com to the office of
> fair trading. ;-)
They do write on their site that a 3D Klein bottle does not exist. What
they sell is a self-intersecting Klein bottle - which does exist. But
without the intersection it does not.
--
POST = Personen ohne sinnvolle Tätigkeit
[Dieter Bruegmann in dtj]
David Scarlett
>
> They do write on their site that a 3D Klein bottle does not exist.
> What they sell is a self-intersecting Klein bottle - which does
> exist. But without the intersection it does not.
I see.... so does a 2D equivalent exist?
ie. Just as a self-intersecting Klein bottle is a 3D object which has
no edges and no enclosed space, does a 2D shape exist that has no end
points and no enclosed area?
Rudolf Polzer
> "Rudolf Polzer" <adsg...@durchnull.de> wrote in message news:slrn9usi9p.r...@www42.durchnull.de...
> >
> > They do write on their site that a 3D Klein bottle does not exist.
> > What they sell is a self-intersecting Klein bottle - which does
> > exist. But without the intersection it does not.
>
> I see.... so does a 2D equivalent exist?
>
> ie. Just as a self-intersecting Klein bottle is a 3D object which has
> no edges and no enclosed space, does a 2D shape exist that has no end
> points and no enclosed area?
If you allow self-intersection, yes. But then you get edges both
in 2D and 3D...
Even a Moebius strip has to self-intersect if it should be drawn in
2D.
--
/dev/null
Für die einen ist es das Nulldevice, für die anderen das größte
Archiv der Welt. [Sebastian Enke in dtj]
David Scarlett
> David Scarlett <lo...@my.signature> wrote:
> >
> > ie. Just as a self-intersecting Klein bottle is a 3D object which
> > has no edges and no enclosed space, does a 2D shape exist that has
> > no end points and no enclosed area?
>
> If you allow self-intersection, yes. But then you get edges both
> in 2D and 3D...
>
....so what would it look like? I'm really having trouble trying to
picture what it could look like....
Rudolf Polzer
> "Rudolf Polzer" <adsg...@durchnull.de> wrote in message news:slrn9usqi0.t...@www42.durchnull.de...
> > David Scarlett <lo...@my.signature> wrote:
> > >
> > > ie. Just as a self-intersecting Klein bottle is a 3D object which
> > > has no edges and no enclosed space, does a 2D shape exist that has
> > > no end points and no enclosed area?
> >
> > If you allow self-intersection, yes. But then you get edges both
> > in 2D and 3D...
>
> ....so what would it look like? I'm really having trouble trying to
> picture what it could look like....
There is one problem with that: such objects have an area of 0
only by definition. You can define the area as a postive number,
too.
____________________
/ \
| ______________ |
| / \ |
| \__ __________/ |
| \/ |
\_____/\_____________/
You can define both objects as having an area of 0, if you want. The
first one is a Moebius strip projected into 2D, and its area cannot be
found using integration without using the |absolute| value when
integrating - but sometimes an area is defined as a negative value
just because of this. But you cannot set the + and - signs without
getting a contradiction.
___ ___
/ \ / \
/ \ / \
| \/ |
| /\ |
\ / \ /
\___/ \___/
If you use the "directed area" here, you will always get the value
0.
But if you define "no enclosed area" as "all points are accessible from
the outside using a Jordan curve", there is no such 2D object.
But without usage of a directed area and directed angles, there
is no 2D equivalent of the Klein bottle. The 3D one has edges, so
it also does not fulfill your requirements.
--
#!/usr/bin/perl -w -- sequences
$0=++$|;for(;;){print-length$0;$0=~s/(.)\1*/$1.length$&/ge;}print"\n";
David Scarlett
>
> But if you define "no enclosed area" as "all points are accessible
> from the outside using a Jordan curve", there is no such 2D object.
Aww, that's a shame. ;-)
>
> But without usage of a directed area and directed angles, there
> is no 2D equivalent of the Klein bottle. The 3D one has edges, so
> it also does not fulfill your requirements.
Where are the edges of the 3D one?
Rudolf Polzer
> "Rudolf Polzer" <adsg...@durchnull.de> wrote in message news:slrn9usu2e.6...@www42.durchnull.de...
> >
> > But if you define "no enclosed area" as "all points are accessible
> > from the outside using a Jordan curve", there is no such 2D object.
>
> Aww, that's a shame. ;-)
>
> >
> > But without usage of a directed area and directed angles, there
> > is no 2D equivalent of the Klein bottle. The 3D one has edges, so
> > it also does not fulfill your requirements.
>
> Where are the edges of the 3D one?
At the self-intersection. The edge is the intersection circle...
--
#!/usr/bin/perl -W -- WARNING: This will print 22,307 bytes! <strictsafe!>
use strict;for(my$y=-1;$y<1;$y+=.1){for(my$x=-1.9;$x<.4;$x+=.03){print'+';
my$X=my$Y=0;for(0..99){($X,$Y)=($X*$X-$Y*$Y+$x,2*$X*$Y+$y);print"\b "if$X*
$X+$Y*$Y>9;}}print"\n"};print''.reverse"\nHPAJ \a!rezloP .R yb torblednaM"
David Scarlett
>
> At the self-intersection. The edge is the intersection circle...
Ahh, yes. You could call that an edge.... I was thinking more along
the lines of a sharp edge, where a plane abruptly ends.
Watchman
Ah, you mean it's all quantum.
Right.
No problem.
O_o;;
Disaster
> > The bottle exists! I've never seen the real thing, due to comments
above. I
> > have not seen anyone say it does or does not exist, or that it's proven
or
> > just a theory. I've only ever heard of people talking about it. I would
> > venture to guess that it is a theory in the working like a lot of the
> > physics stuff.
>
>
> Right.
>
> No problem.
>
> O_o;;
..... Ah, maybe!?