In 1858, two German mathematicians, August Ferdinand Möbius and Johann Benedict Listing, independently discovered what is popularly known as the Möbius strip. The characteristic feature of Möbius strip is that it is a surface with single side. In its most simplest form a Moebius strip can be constructed out a a strip of paper which is twisted halfway and the ends joined together. If one were to start tracing a surface, by the time they complete one trace they find that they are tracing the opposite side of the paper than the one from which they started. Go another round and you come back to the same side.

A Mobius strip can be expressed mathematically in several diffferent forms. Geometric term for the the form exhibited by a Mobius strip is a *Chiral*. The parametric equations for a Moebius Strip can be expressed as:

x(u,v) = cos(u) + v*cos(u/2)*cos(u)

y(u,v) = sin(u) + v*cos(u/2)*sin(u)

z(u,v) = v * sin(u/2)

Default values for u and v:

u = [0, 2π] for one complete loop;, v = [-0.4, 0.4]

An equation for constructing a Moebius Strip using Matlab can be found at the Univesity of Stutgart’s mathematic department – Matlab code repository.

Among the most famous artwork using a Moebius strip is the one by M.C.Escher, which has a grid in the shape of gigure-8 Moebius strip with ants crawling on different sections along the strip.

M.C.Escher has another piece of art in the form of a Moebius strip. This one has 3/2 twists instead of 1/2 as in the basic strip. By playing around with number of twists, one can create beautiful and complex loops.

A simple, yet intriguing model of a Moebius strip is the one where a 1/2 twist Moebius strip is constructed from a transparent strip of plastic with the word Möbius written on it.

One of the most common example of a Moebius strip one encounters on a day to day basis is the Recycling sysmbol which has three foldeed arrows forming a loop. The folded arrows aren’t identical. One of the arrows folds in the opposite sides from the remaining two arrows. The design of recycling symbol appears somewhat similar to the Escher’s artwork with 3/2 twists.

This comic strip in the form of a infinitely repeating sequence of a stick figure kicking a a football at an unsuspecting stick figure and knocking it down is yet another example of silly, yet creative Moebius stripping.

Origami art, i.e. the art of paper folding, can be used to create some incredible shapes simply by folding a piece of paper. Here is an example of a Moebius strip created with Origami folding. The corrugated pattern on the strip makes it appear quite classy.

Here is an illustration of Moebius strip on a book cover. I haven’t been able to figure out of the illustration is an actual photograph if a jigsaw puzzle or it is merely a creation in Photoshop. Either way, a jigsaw puzzle that would form a jigsaw puzzle in the end would keep anyone occupied for several days – and several more if the structure keeps tumbling every now and then.

Creations that incorporate a Moebius strip are rather intriguing even as still images, but with animations they take on a spectacular form. One is a interlocked gear chain and the other one is an escalator in the form of a half-twist Moebius strip.

Then there are some architectural models of Moebius strip in forms of scuptures that are placed at tourist attractions, and some that are placed in playgrounds.

And then there is a Moebius strip cake – how’s that for sweet!

I was wondering how the Möbius could have been invented/discovered in 1958 when August Möbius was born in born in 1790 and died in 1868. I then realized that it must have something to do with the Klein bottle.

My theory is that it was not Felix Klein who invented this, but in fact, August Möbius invented it in 1858. He was went to a nearby glass factory where a glass blower named Johann Benedict Listing (JBL) attempted to build a Klein bottle under his directions. They tried to avoid the self-intersection by going into the fourth dimension, and got caught in a time warp (the fifth dimension) and emerged in 1958.

Upon further reflection, they realized that rather than trying to build a non-orientable surface with no boundary component in three dimensions they should build a non-orientable surface with a single boundary.

In comparison with the effort involved with constructing a Klein bottle, they were easily able to invent the strip. Möbius was able to go back the 100 years to 1858. However, JBL was not. He found that his initials had been reversed in the process and he was now LBJ. Two years later was elected as Vice-President of the Untied States of America.

Thanks for pointing out the typo error Wes. I changed the date to 1858. Marvellous story development btw

hiya,

im a fashion student at liverpool john moores univercity.Im in my final year and my collection is focusing on mobius! is there any info help advice books leaflets pics samples you could send me? i would be greatful.

many thanks

chio

x

That last one isn’t a cake, it’s a 3D printout done in sugar by the CandyFab3000:

http://www.evilmadscientist.com/article.php/sugarsculpt

http://www.evilmadscientist.com/article.php/candyfab

[...] is used instead of just one side of a two-sided strip. It’s also an attention-getter in art and even [...]

The puzzle, the Wilson sculpture, and the cake are very nice, but they are not Möbius strips, as they are not locally two-dimensional (flat). Of course, in the real world, even a sheet of paper has some actual thickness, but those three pictures aren’t even trying to be representations of a flat object. The puzzle picture is a knotted torus; I’m not sure what I would call the other two figures.

Also note that in an abstract sense a topologist would consider a “1/2 twist Möbius strip” to be the same as a “3/2 twist Möbius strip”, just embedded in space differently. If you were a (two-dimensional) creature living in (not on) either one, you couldn’t tell them apart.

— A Topologist

Your tag for one of the above pics is incorrect, it’s not a cake, it’s a product of a candyfab 3d printing machine> http://www.candyfab.org/

You should credit your image sources better – xkcd (the source of the comic (http://nuclear-imaging.info/site_content/wp-content/uploads/2009/02/mobius_battle.png) uses the Creative Commons BY-NC license, which requires accreditation. And even if it weren’t the law, it’s good practice and just plain polite.

The last image was taken from my web site. It’s not a cake, it’s a 3D printed piece of solid sugar, showing rather uneven caramelization.

cool site!

I am an artist who loves the Möbius strip

This page has (at least) 3 non-Möbius strips on it. M. C. Escher – Moebius Strip I (ribbon), though complicated, definitely has two sides and two edges. It seems Mr. Escher slipped a bit. Additionally, the figure on the cover of The Möbius Strip by Clifford Pickover appears to be a torus that has been cut, tied in a knot, and joined back together. If it is indeed a strip, then one can easily follow the edges to discover that there are two of them, unless there is a twist cleverly (read:stupidly) hidden at one of the overlaps. Finally, the playground climber has two ends! unless they join together underground, it is simply an ordinary strip.

Hello, this may sound weird but in the last 2 weeks, I have seen this symbol in my head and have been doodling it on random pieces of paper, napkins at a resturant…etc.

I had no idea what it was or what it could represent let alone why I have been having the image in my head. It’s quite weird, I know. Just wondering if anyone can shed some light on this for me…

thank you

ashley

@Wes Peacock

Upon closer examination, Mobius Strip I is indeed a Mobius Strip. It simply has 3/2 twists instead of the traditional 1/2.

@Greg

Thank you so much for this very cool post! I’m researching Jose de Rivera’s “Infinity” sculpture for my blog. Very cool to see the various representations of the mobius strip in art, playgrounds and most importantly, CAKE!!

I can’t get the pictures to display (IE 9.0 on Windows Vista. Are the pictures still on the server? Any other ideas?