Original Usenet thread from alt.ascii-art, started 3 Jul 1998.
KLEIN BOTTLE.
alt.ascii-art
· 12 messages · 3 Jul 1998 - 11 Jul 1998
I know this is a 'dumb' question...but what's a klein bottle? I've seen one before...but what are they used for? K. -- a saner side of neverland (sig and spamguard:)-- Bill Taylor <mat...@math.canterbury.ac.nz> wrote in article <6nhlvv$hhu$1...@cantuc.canterbury.ac.nz>... > Improvements or other diddles welcome... > > .--------------------. > / \ > / ,.:::::::.. \ > : ( ) : > | .\_________/. | > | . . | > Klein bottle | . . | > ============ | . . | > | . . | > - wfct | . . | > | . . | > | . . | > | . `. . | > | . `/^---------. > | `. ( : \ > | ` . . . .\________. \ > | | \ : > `. .' | | > \ / | | > \ / ' ; > \ ( / / > \ \ / / > \ \_______' / > \ / > `-------------------------' > > -------------------------------------------------------------------------- > * * *---* N.Z. graph theory is... SELF-COMPLEMENTARY ! > |\ | / <---------------------------~~~~~~~~~~~~~~~~~~ > | \ | / (Who else would compliment us...?) > | \| / > * * *---* Bill Taylor W.T...@math.canterbury.ac.nz > -------------------------------------------------------------------------- > > >
Khall wrote: > > I know this is a 'dumb' question...but what's a klein bottle? I've seen one > before...but what are they used for? > > K. > > -- a saner side of neverland (sig and spamguard:)-- > > Bill Taylor <mat...@math.canterbury.ac.nz> wrote in article > <6nhlvv$hhu$1...@cantuc.canterbury.ac.nz>... > > Improvements or other diddles welcome... > > > > .--------------------. > > / \ > > / ,.:::::::.. \ > > : ( ) : > > | .\_________/. | > > | . . | > > Klein bottle | . . | > > ============ | . . | > > | . . | > > - wfct | . . | > > | . . | > > | . . | > > | . `. . | > > | . `/^---------. > > | `. ( : \ > > | ` . . . .\________. \ > > | | \ : > > `. .' | | > > \ / | | > > \ / ' ; > > \ ( / / > > \ \ / / > > \ \_______' / > > \ / > > `-------------------------' > > > > > -------------------------------------------------------------------------- > > * * *---* N.Z. graph theory is... SELF-COMPLEMENTARY ! > > |\ | / <---------------------------~~~~~~~~~~~~~~~~~~ > > | \ | / (Who else would compliment us...?) > > | \| / > > * * *---* Bill Taylor W.T...@math.canterbury.ac.nz > > > -------------------------------------------------------------------------- > > > > > > A Klein Bottle is extremely useful for NOT storing things! Look closely, and you should see that there is only one surface. Compare to a Moebius strip. Lynn Killingbeck
You have never seen a Klein bottle, as it cannot be constructed in a 3D
space, requiring at least 4 spacial dimensions.
They are not "for" anything. Mathematicians just got to studying all the
different ways things can get connected together, and found out this was
one. It has some non-intuitive properties, so it became famous.
Mike
In article <6nj1ti$qdf$2...@news-2.news.gte.net>,
Khall <jpr...@a.saner.side.of.never.land.gte.net> wrote:
)I know this is a 'dumb' question...but what's a klein bottle? I've seen one
)before...but what are they used for?
)
)K.
--
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char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
I don't speak for DSC. <- They make me say that.
Improvements or other diddles welcome...
.--------------------.
/ \
/ ,.:::::::.. \
: ( ) :
| .\_________/. |
| . . |
Klein bottle | . . |
============ | . . |
| . . |
- wfct | . . |
| . . |
| . . |
| . `. . |
| . `/^---------.
| `. ( : \
| ` . . . .\________. \
| | \ :
`. .' | |
\ / | |
\ / ' ;
\ ( / /
\ \ / /
\ \_______' /
\ /
`-------------------------'
--------------------------------------------------------------------------
* * *---* N.Z. graph theory is... SELF-COMPLEMENTARY !
|\ | / <---------------------------~~~~~~~~~~~~~~~~~~
| \ | / (Who else would compliment us...?)
| \| /
* * *---* Bill Taylor W.T...@math.canterbury.ac.nz
--------------------------------------------------------------------------
|> As far as I can tell, you actually *can* store stuff in this
|> 3-dimensional version, but not very easily: you need to pour it in the
|> bottom, then tip it round slowly so it goes throught the spout and ends
|> up in the `inside' of the bottom.
Zackly so. In fact, I have done it with the glass Klein bottle my colleague
has in his office. It's fun getting the last few bubbles out! Mind you,
it's a bummer if you want to get the "inside" dry again in a hurry!
Of course the 3-dimensional glass KB is only a physical model, not the
real thing, which cannot be embedded in E^3, though it *can* be immersed,
(allows self-intersection). The glass version isn't even that - there is
a slab of glass missing: the little circle where the small part of the tube
passes through the larger part. But it'll do.
|> Why you would want to do that, I have absolutely no idea.
For the same basic reason you'd want to do ANY math:- IT'S FUuuunnnn...
|> If you feel like making a Klein bottle, you can do it by taping two
|> Moebius strips to each other, all the way along both sides.
If you casn mange this with paper ones, you're a better man than I am,
Gunga Din! You might succeed with some cloth ones, by sewing them together,
if you've carefully arranged for the edges to be equal lengths. But a cloth
one won't show the essential features.
It's got to be glass!
My colleague even has a glass model of a Boy's surface! (Immersion of the
projective plane into E^3 without singular points.) Now that is *seriously*
bizarre! It was also the hardest job the University glassblower ever had
to do, he said. My colleague also has a home-made wire-netting version.
Neato. But I'll leave someone else to try the ascii version of that!
But an ascii sphere-with-crosscap shouldn't be *too* difficult.
(Same surface but differently immersed - it has two "pinch points").
Anyone want to give that a try?
_____ ____________________________________________________________ _____
|X| . | Bill Taylor. W.T...@math.canterbury.ac.nz |X| . |
|~ . .| ----------------------------------------------------------- |~ . .|
|___._| Entropy - where energy goes to die. |___._|
===========================================================
/|
/ |
______| |_____
| | | |
| __| |___ |
| / |/ / |
|__/ /____|
/_______/
|/
This is the central core of the Boy's surface... the easy part!
=============================================================================
Matthew Thomas <mpt...@spamfree.land> wrote: >Any picture you see of a Klein bottle is a 2-dimensional picture of a >3-dimensional simulation of a 4-dimensional shape, so it's bound to be a >bit screwy. Here's my version ... > ______________ > _,--' "`-._ > ,' `--.__ > | `--. >, \ `-. >| .`........._ `. >| .:: `""--.___ \ >| .::_::::::__ / )"-. ______ \ >| .;' `""-.\__; `..-'mt-2 "`. | >` / `--_ `-. ) | > | ,-' `: `"-----"' / > `._ _;-' `:_. _;' > `--._____;--' `--:___,-' That looks more like a stomach reflux to me. :-) .-------. !_______! (((`\\\ ))) (((_o___o_))) ((| |)' |_______| l l ==================================================================== .----. ===(_)== THIS WONT HURT A BIT... // 6 6 \\ / ( 7 ) \ '--' / \_ ._/ __) (__ /"`/`\`V/`\`\ / \ `Y _/_ \ / [DR]\_ |/ / /\ | ( \/ / / / \ \ \ / \ `-/` _.` jgs `=. `=./ `"` ______________ _,--' "`-._ ,' `--.__ | `--. , \ `-. | .`........._ `. | .:: `""--.___ \ | .::_::::::__ / )"-. ______ \ | .;' `""-.\__; `..-'mt-2 "`. | ` / `--_ `-. ) | | ,-' `: `"-----"' / `._ _;-' `:_. _;' `--._____;--' `--:___,-' -Matthew Thomas- |________|___________________|_ | | | | | | | | | | | | |________________ |________|___________________|_| , | | | , -Marcia Chabert- ==================================================================== Bye,
Bill Taylor wrote: >... > Of course the 3-dimensional glass KB is only a physical model, not the > real thing, which cannot be embedded in E^3, though it *can* be > immersed, (allows self-intersection). The glass version isn't even > that - there is a slab of glass missing: the little circle where the > small part of the tube passes through the larger part. But it'll do. Just what I said originally -- the 3-dimensional version is only an approximation. > |> Why you would want to do that, I have absolutely no idea. > > For the same basic reason you'd want to do ANY math:- IT'S > FUuuunnnn... <Uni. of Canterbury only> Er, you're talking to someone who failed MATH 107 here ... </Uni. of Canterbury only> > |> If you feel like making a Klein bottle, you can do it by taping two > |> Moebius strips to each other, all the way along both sides. > > If you casn mange this with paper ones, you're a better man than I am, > Gunga Din! You might succeed with some cloth ones, by sewing them > together, if you've carefully arranged for the edges to be equal > lengths. Um, reread what I said originally: < If you feel like making a Klein bottle, you can do it by taping two < Moebius strips to each other, all the way along both sides. Just < remember, you need to be in at least 4 spacial dimensions at the time. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ < Extra dimensions can be bought from your local hardware store. Joke, dang it, joke! (Sigh.) You can't make one in 3-D, otherwise we wouldn't have the problem of having to immerse it ... > My colleague even has a glass model of a Boy's surface! (Immersion of > the projective plane into E^3 without singular points.) Now that is > *seriously* bizarre! It was also the hardest job the University > glassblower ever had to do, he said. My colleague also has a > home-made wire-netting version. > > Neato. But I'll leave someone else to try the ascii version of that! <Uni. of Canterbury only> Wow! We have our own *glassblower*?!? </Uni. of Canterbury only> Well, I found a VRML version of a Boy's surface here: http://www.mhri.edu.au/~pdb/geometry/boy/ and churned out the following two pics. First, viewed straight on: ,--. | .:`. J .::`. L . .::\ J .:`. \ . .::\ \ .::\ \ . .,-`----..._____ \ L. . . `""--._ \ . `.__....:..:..:.......) \ /:. `>::::::::::::_,-' ,'`-':. / `""""""""""' /:. . /mt-2 /:. . ,' /:. . / /:. . ,' /. . ,' L ,' \_.' Secondly, on an angle which lets you see a bit more of the geometry: _ / `. /: | /:. | |:. . | L:. | ______________ J:. . | ,-'"""mt-2 """"`-----;;. L J . |.. J L . . . . J:. . J `-._ . `.::__. . . L:. L `-._ "`--.._J:. . L `"--..__ J `"""-+----...___ L J .:;` L \ . .:' L \ .. | \ . / `-..__.-' How's that? > But an ascii sphere-with-crosscap shouldn't be *too* difficult. > (Same surface but differently immersed - it has two "pinch points"). > > Anyone want to give that a try?> Well, the only picture of a `crosscap' that I can find is http://www.cs.msstate.edu/staff/banks/public_html/pics/thesis/crosscap.sig.gif and it isn't terribly enlightening. Is that a sphere-with-crosscap? What is a `crosscap', anyhow? Any pointers? > /| > / | > ______| |_____ > | | | | > | __| |___ | > | / |/ / | > |__/ /____| > /_______/ > |/ > > This is the central core of the Boy's surface... the easy part! Really? (Scratches head, looks confused ...) -- .._o Matthew Thomas (mpt26 @ student . canterbury . ac . nz) ..://\/ http://cantua.canterbury.ac.nz/~mpt26/ (Site for sore eyes) ..:_/\ ASCII FAQ: http://cantua.canterbury.ac.nz/~mpt26/art/ascii/ ..:/ Banana custard!
Matthew Thomas <mpt...@spamfree.land> writes:
|> Just what I said originally -- the 3-dimensional version is only an
|> approximation.
Yep; just emphasizing old son, just emphasizing.
|> > For the same basic reason you'd want to do ANY math:- IT'S
|> > FUuuunnnn...
|>
|> <Uni. of Canterbury only>
Hah! Didn't notice that before. Hi!
|> Er, you're talking to someone who failed MATH 107 here ...
Bad luck.
|> Um, reread what I said originally:
|> < remember, you need to be in at least 4 spacial dimensions at the time.
|> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Yep, reply as above. Stay loose, old buddy!
|> < Extra dimensions can be bought from your local hardware store.
DO YOU HAVE A REFERENCE FOR THIS?!
|> <Uni. of Canterbury only> Wow! We have our own *glassblower*?!?
Well, we *did* back then. He was a very skilled artisan indeed.
That particular one retired, but I suppose we still have one.
They must be witty fellows - they make very good retorts!
|> First, viewed straight on:
|>
|> Secondly, on an angle which lets you see a bit more of the geometry
Very neat pics! Thanks, though they aren't really enough to show what's
axshully going on. You need to hold one in your hands. Nice to see the
tri-partite symmetry is pictured...
,--.
| .:`.
J .::`. (Hmmmm... looks like it could be used as a boomerang...)
L . .::\
J .:`.
\ . .::\
\ .::\
\ . .,-`----..._____
\ L. . . `""--._
\ . `.__....:..:..:.......)
\ /:. `>::::::::::::_,-'
,'`-':. / `""""""""""'
/:. . /mt-2
/:. . ,'
/:. . /
/:. . ,'
/. . ,'
L ,'
\_.'
My colleague, who has the glass and wire-netting models, also has a
home-made clay model, which is trisected into 3 congruent parts. Neato.
He also tried to make a TWO-LAYERED model out of pipe cleaners. He has
done it, in fact, but its WAY to small to be any use, alas. Two-layered
is what you get if you "split" a paper model of one, all the way round the
paper; i.e. separate the inside and outside faces of the paper. As there
IS only one side, this means you finish up with a single sheet of paper.
It's like a higher-dimensional version of the standard cutting of a Mobius.
The neat thing about this two-layered version is, that it is topologically
equivalent to a sphere! Or i.e. a sphere can be doubly-immersed onto
a Boys surface. And this in turn means one can EVERT a sphere, by mapping
it onto a doubled Boys surface, pass the whole surface through itself(!),
i.e. "exchange the two layers", and re-map it back to the sphere the same
way as before, only now it's inside-out. The waaaaaay kooooool thing
about this particular eversion of the sphere is, EVERY DAMN POINT is
self-intersected through the surface SIMULATANEOUSLY with every other!
Not just a single "equator" of them, like in the usual eversions.
|> Well, the only picture of a `crosscap' that I can find is
|> http://www.cs.msstate.edu/staff/banks/public_html/pics/thesis/crosscap.sig.gif
|> and it isn't terribly enlightening. Is that a sphere-with-crosscap? What
|> is a `crosscap', anyhow?
Well I've got a polystyrene one in my office, if you want to see it!
It's a Mobuius strip self-intersectedly folded up so that the single
edge is in the shape of a circle (almost). Then a hemisphere can be glued on.
|> > /|
|> > / |
|> > ______| |_____
|> > | | | |
|> > | __| |___ |
|> > | / |/ / |
|> > |__/ /____|
|> > /_______/
|> > |/
|> >
|> > This is the central core of the Boy's surface... the easy part!
|>
|> Really? (Scratches head, looks confused ...)
Yes; if you look at the very centre of the diagram above, you'll see
a dot where three parts of the surface intersect at right angles.
Of course you can only see the outside of the three, there.
Here's a slightly better version of my pic...
,
/|
/ |
/ | (watch out for those sharp corners!
______| ._|____ you could get paper cuts off your VDU...)
| | | |
| __| !___ |
| / |/ / |
|__/ ' /____|
/_______/|
| !
| /
| /
|/
'
-------------------------------------------------------------------------------
Bill Taylor W.T...@math.canterbury.ac.nz
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The unrefined and sluggish mind Of Homo Javanensis,
Could only treat of things concrete And present to the senses.
-------------------------------------------------------------------------------
Khall (jpr...@a.saner.side.of.never.land.gte.net) wrote: : I know this is a 'dumb' question...but what's a klein bottle? I've seen one : before...but what are they used for? Klein bottle for sale, see inside. Mobieus strip for sale, please turn over. They're not used for anything...outside euclidean gemoetry.
In article <6o2b28$t3...@hpwin055.uksr.hp.com>, den...@bri.hp.com (Contractor - Dennis Yau) wrote: > Khall (jpr...@a.saner.side.of.never.land.gte.net) wrote: > : I know this is a 'dumb' question...but what's a klein bottle? I've seen one > : before...but what are they used for? > > Klein bottle for sale, see inside. Mobieus strip for sale, please turn over. > > They're not used for anything...outside euclidean gemoetry. > Well ---not exactly ---large machinery which uses long flat belts often have Mobius strip belts for even wear --one side is all there is! -----== Posted via Deja News, The Leader in Internet Discussion ==----- http://www.dejanews.com/rg_mkgrp.xp Create Your Own Free Member Forum
Thanks to everybody, but as I said, I have seen one, on the back of my high school mathmatics book, as a computer generated graphic. However it's been...uhmmm *counting on his fingers* like 9 yrs so I couldn't remember where it was. Or what it was. Much appreciation for all the answers but this was resolved a week ago. K. who likes to keep the exposure of his ignorance to a minimum:P On Thu, 9 Jul 1998 17:17:19 -0400, "C.C." <cco...@fas.harvard.edu> wrote: > >>: I know this is a 'dumb' question...but what's a klein bottle? I've seen >one >>: before...but what are they used for? > > >A Mobius strip is a piece of paper with one side. A klein bottle is a >bottle in which the inside and outside are the same thing. > >Odds are you haven't seen one. Just like it take us three dimensions to >visualize a Mobius strip, it takes 4 dimensions to visualize a klein bottle >(even though it is 3 dimensions. > >Someone correct me (nicely) if I'm wrong . > > >CC >
C.C. (cco...@fas.harvard.edu) wrote:
: Odds are you haven't seen one. Just like it take us three dimensions to
: visualize a Mobius strip, it takes 4 dimensions to visualize a klein bottle
: (even though it is 3 dimensions.
Well, I'd say we can "see" 4-dimensional objects just as well as we "see"
3-dimensional ones. Of course what our eyes give us are 2d projections of
the objects, from which our brains can construct a model of the actual
multidimensional shape. (significantly harder in 4d than in 3d, but not
impossible)
--
Ilmari Karonen (ilt...@sci.fi) http://www.sci.fi/~iltzu/
"Moebius Strip: See instructions on other side."
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