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.
Basic Moebius strip (twisted ribbon)
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.
Rendering of Moebius strip using Matlab
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 - Moebius Strip II (insects)
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.
M. C. Escher - Moebius Strip I (ribbon)
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.
Moebius Strip written onto a transparent Moebius Strip
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.
Commonplace use of Moebius Strip - Recycling Symbol
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.
Cartoon depiction of a continuous reel Moebius Strip
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.
Origami paper folding - Moebius Strip
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.
Moebius Strip depicted as a jigsaw puzzle on a book cover
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.
Moebius strip in the form of a gear chain animation
Animation of Moebius Strip as an escalator
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.
Robert R. Wilson's sculpture of Moebius Strip at Ramsey Auditorium Fountain
Moebius Strip structure at a cildren's playground
Moebius Strip - 12-panel Moebius Climber at a playground
Moebius Strip Jungle Gym near the Temple of Honor
And then there is a Moebius strip cake – how’s that for sweet!
Cake baked in the shape of a Moebius Strip