View Full Version : Möbius Curve


Rollingthunder
16th Mar 2009, 03:31
A Mobius curve is a single closed curve. It has only one plane. If you draw a line around the the curve you will discover that you travel on both the "outside" and "inside" yet meet your starting place. The Mobius curve where place and time turns back on itself again. Here in a painting by M. C. Escher we see ants traveling on a Mobius curve:
http://www.dr-mcginnis.com/MoebiusAntsSmall.gif

An endless ring-shaped band usually has two distinct surfaces, one inside and one outside. Yet on this strip nine red ants crawl after each other and travel the front side as well as the reverse side. Therefore the strip has only one surface.

In making a Mobius curve, you take a long thin rectangular piece of paper, give one of the narrow ends a 180 degree twist and glue or tape the ends together. What happens when you cut a mobius band all around its length?

This is so much fun that you *must* try this yourself. But the result is...you get just one band of paper, not two, amazingly enough.

Next, my candle trick.



Loose rivets
16th Mar 2009, 05:20
Yeh, right.

I can't pass an IQ test that has things like this in it. Today, I couldn't even work out how to put a puppy's harness on. (mind you, I had three of them climbing on my head at the time)

Just a mental failing. All the IQ questions about unraveling string. Nill Points.

Mobius...and Escher. Total pain in the Rse.

Bushfiva
16th Mar 2009, 05:29
Rollingthunder, two other things you'll enjoy: starting with the result of cutting the mobius strip in half, cut that in half, too.

Starting with the original mobius strip, start cutting at one-third of the width.

Whirlygig
16th Mar 2009, 07:31
When you've had enough fun with the Mobius Strip, have a go at making one of these. :E It can be done but only with some lateral thinking.

http://pirate.shu.edu/%7Ewachsmut/Workshops/Escher/worlds/box.jpg

Cheers

Whirls

B Fraser
16th Mar 2009, 07:41
mmmmh, it looks like an old MFI coffee table to me. :suspect:


A circle has two sides, an inside and an outside.

SoundBarrier
16th Mar 2009, 07:41
Whirls,

That is one messed up beer crate. Lends one to believe that one should stay drinking at the pub instead of trying to take some home. :)

SB

Blacksheep
16th Mar 2009, 08:09
If you want to have some real fun, try making two mobius strips and then glue them together... :hmm:

Bern Oulli
16th Mar 2009, 08:29
And when you have damaged your brain sufficiently, try a Klein Bottle (http://www.kleinbottle.com/whats_a_klein_bottle.htm) which is, of course, a single sided bottle with no inside or outside, has zero volume and yet can contain a liquid. Unfortunately it can only truly exist in a 4-dimensional universe. However, a 3-D representation (that is the equivalent of a 2-D photo of a 3-D object) can be made and bought. I bought my scientist father one as a prezzy. :ugh:

oxenos
16th Mar 2009, 08:42
Try taking a strip of paper, giving it a 360 twist, sticking the ends together, and then cutting it in half along its length.

Blacksheep
16th Mar 2009, 10:48
The mobius strip is made by turning one end through 180 degrees and joining it to the other. A Klein Bottle results from stitching two mobius strips together, effectively joining one end twisted through 180 and the other directly. In mathematical terms, by multiplying the y-axis of a radian function by -1 and the x-axis by +1.

There is no technical objection to multiplying both axes of a radian function by -1. This would create an extremely interesting model, which although I have yet to see an acceptable 2D illustration of this 5 dimensional construction, can be imagined as long as you try not to look at the whole model in one view. (The model may appear impossible, but only when you consider the time element of the equation (time = 't') as being constant throughout the body of the model. )

Blacksheep
16th Mar 2009, 10:52
http://www.andrews.edu/~calkins/math/webtexts/klein.gif

tony draper
16th Mar 2009, 11:01
I was just going to say that Mr Blacksheep.:rolleyes:

Storminnorm
16th Mar 2009, 11:10
If one was to dwell on this sort of thing, one would rapidly
lose one's power of reason, and hit the bottle in a big way.

goudie
16th Mar 2009, 11:15
Me 'ead 'urts!

Storminnorm
16th Mar 2009, 11:28
Told you so goudie!!!

Been Accounting
16th Mar 2009, 11:52
This would be fun to fly with a smoke trail!

Blacksheep
16th Mar 2009, 12:25
For those who are interested in following the Blacksheep on an exploration of topology, this (http://www.geometrygames.org/TorusGames) is an excellent starting point. Yes goudie, it may make your head hurt at first, but it is more mind expanding than any of the drugs people experimented with in the sixties.

Once you master the spatial possibilities, advance to the possibilities of the fourth dimension being different at all points in the structure. Once you get this picture, mere weirdness becomes normality and quantum mechanics becomes questionable. Perhaps I'm slightly mad?

If you want to fly the smoke trail Been Accounting, a figure of eight knot would make a visualisation of a trail on the surface of the simple five dimensional structure that I mentioned earlier.

bnt
16th Mar 2009, 12:39
You sometime hear people attributing magical qualities to the Möbius Curve or Klein Bottle, as if they do something physically impossible. If you bend a surface in the 3rd dimension, it's no longer a purely 2-dimensional surface, is it? So you haven't really gained anything "new" in the process. :rolleyes:

Blacksheep
16th Mar 2009, 12:45
To bend a surface (i.e. two dimensions) in the third dimension, you might try manipulating the fourth dimension, bnt. As you say, you gain nothing for of course, a klein bottle encloses no volume. But once you accept that the fourth dimension is not fixed, the concept of volume is no longer necessary or relevant. Being three-dimensional creatures we find that difficult to visualise, but there's really no need to invoke magical qualities.

ORAC
16th Mar 2009, 13:06
http://www.mcescher.com/Gallery/recogn-bmp/LW439.jpg

Whirlygig
16th Mar 2009, 13:15
Unfortunately it can only truly exist in a 4-dimensional universe.
Would this be a different 4-dimensional universe to the normal one that you and I live in? :confused:

Cheers

Whirls


width, length, depth ...... and time.

unstable load
16th Mar 2009, 17:09
But Whirls,
Surely the length= depth? Time is just a matter of practice and concentration...:E

BlooMoo
16th Mar 2009, 19:22
Orac's Waterfall image is a good analogy of current Government economic 'logic'.

pigboat
17th Mar 2009, 02:49
Ok, time for a topic hijack sorta. With all this stuff about Mobius strips, has anybody read any of Larry Niven's Ringworld series?

ORAC
17th Mar 2009, 07:01
Ringworld? Poor man's Dyson sphere.....

Much preferred A Mote in God's Eye.

MadsDad
17th Mar 2009, 11:39
Read all the Ringworld but prefer Iain M Banks these days. Lots of mini ringworld 'Orbitals' in the Culture.

Blacksheep
17th Mar 2009, 13:34
Why mess with a Ringworld when there's a Discworld to play with?

SoundBarrier
11th Aug 2010, 00:58
Thinking of these curvey things....

Have a look at this guy, a weird speech but dynamic and fun nonetheless. My question, how the hell would you create a mobuis curve from glass?

I like his *jug* too :}

Clifford Stoll: 18 minutes with an agile mind | Spoken Gems: Speeches, Lectures, Monologues (http://spoken-gems.com/2008/08/06/clifford-stoll-18-minutes-with-an-agile-mind/)

SoundBarrier
11th Aug 2010, 01:26
I found his website...
Acme's Top Mouth Erlenmeyer Klein Bottle (http://www.kleinbottle.com/top_mouth_erlen_klein.htm)

What a cool decanter or wine bottle that would be!

OllyBeak
11th Aug 2010, 01:44
If you cut along the middle of a Mobius strip, you get...

And this is pretty, even if it's not much to do with the lad:

Thingy. (http://www.finecutfilms.com/images/Frame.gif)

SoundBarrier
11th Aug 2010, 02:42
I'm going to over-think your question:

If you cut along the middle of a Mobius strip, you get...

1. You would cut air if t'was the *middle*
OR
2. You would get a flat strip since you cut the strip and it now has truly defined edges, i.e. two surfaces
OR
3. TWO mobius curves

:)

Caboclo
11th Aug 2010, 04:30
If I wanted to be dizzy and confused, I'd get drunk.

crippen
11th Aug 2010, 04:41
or read jetblast.:\

chuks
11th Aug 2010, 05:51
He is the artist who did that work on the first page, here. He did some amazing woodcuts that play with perspective.

Escher reproductions used to be a given, along with one of those Mateus wine bottles made into a candle-holder and an avocado pit suspended in a festering bowl of water by three toothpicks, in any given college dorm room of a certain sort of freshman girl. With the right name-checks sexual congress was also a given. Allegedly!

If you can remember the Sixties you weren't there...

Captain Stable
11th Aug 2010, 07:44
If you cut along the middle of a Möbius loop you get one loop, but with a double twist in it.

If you then cut along the middle again, you get two strips, but linked together.

chuks
11th Aug 2010, 08:31
It is much easier to construct a Möbius strip and see all this for yourself, when "seeing is believing." Paper, glue and scissors is all you need.

There are all sorts of surprises lurking if you just take the time to experience them, such as the blind spot in our vision. For that one you just need one eye and a piece of paper with a dot on it.

Another fun thing is to play with is the logic of guessing what is behind "Door Number Three": Take three doors, One, Two and Three. Behind one of the three there is a bag of gold, behind another a goat. You guess "One" for example, when I then tell you that the goat is behind "Two." Does it pay to make a different guess for the bag of gold, now that you know that? (If you don't have ready access to a bag of gold and a goat you can play this game with three cups and two different objects.)

jcbmack
11th Aug 2010, 11:11
is cut into fourths:confused:

tony draper
11th Aug 2010, 11:18
Better than being halved into three.:uhoh: