I learned something new today from Myles. I always seem to be learning new things from him. What I’d really like to learn is to cook as well as he does. *grin*

I was never good at spatial reasoning unfortunately. Didn’t fare well in that portion of standardized tests sadly. This information is gacked from wikipedia. But of course =)

The Möbius strip or Möbius band (pronounced /ˈmøbiʊs/) is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. It is also a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.[1][2][3]

A model can easily be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a single strip. In Euclidean space there are in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Möbius strip is therefore chiral, which is to say that it is “handed”.

It is straightforward to find algebraic equations the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above. In particular, the twisted paper model is a developable surface (it has zero Gaussian curvature). A system of differential-algebraic equations that describes models of this type was published in 2007 together with its numerical solution.[4]

Art and Popular Culture
The Möbius strip has provided inspiration both for sculptures and for graphical art. The artist M. C. Escher was especially fond of it and based several of his lithographs on it. One famous example, Möbius Strip II, features ants crawling around the surface of a Möbius strip. It is also a recurrent feature in science fiction stories, such as Arthur C. Clarke’s The Wall of Darkness. Science fiction stories sometimes suggest that our universe might be some kind of generalized Möbius strip. This is especially prominent in the Perry Rhodan-series. In the short story “A Subway Named Moebius”, by A.J. Deutsch, the Boston subway authority builds a new line, but the system becomes so tangled that it turns into a Möbius strip, and trains start to disappear. The Möbius strip also features prominently in Brian Lumley’s Necroscope series of novels.

The mobius strip is even the signature architectural feature of the NASCAR Hall of Fame (www.nascarhalloffame.com), presently under construction in Charlotte, NC.


A popular limerick is often associated with this design which reads [7]

“A mathematician confided
That a Möbius band is one-sided,
And you’ll get quite a laugh,
If you cut one in half,
For it stays in one piece when divided”
In the popular episodic MMORPG “Guild Wars”, there exists an attack called “Moebius Strike”; it is so called, because it gives the player the power to continually use their attack chain (it goes on forever, much like a Möbius Strip).

In “A. Botts and the Moebius Strip”, a short story by William Hazlett Upson first published in 1945 in the Saturday Evening Post, the protagonist secretly restitches a conveyor belt to form a Möbius strip to frustrate a superior’s attempt to “paint the outside, but not the inside” of the belt as a safety measure.

In the seaQuest DSV episode “Playtime” a Möbius strip is used to show a theory involving time travel and the space/time continuum.

“Moebius” is the title of a Stargate SG-1 episode involving time travel where the alternative timelines shown wrap around like a möbius strip.

The videogame F-Zero GX (for the Nintendo Gamecube) features a circuit called “Green Plant”, which is a giant Möbius strip.

In the videogame Ace Combat 04: Shattered Skies (for the PlayStation 2) the call sign of the player’s pilot character is Mobius 1. The emblem of the Mobius squadron is a blue diamond shaped Möbius strip, referred to by several characters as a “ribbon”.

The Nelly Furtado song Hey, Man! contains the following lyrics:

We are a part of a circle
It’s like a Möbius strip
And it goes ’round and ’round
Until it loses a link
The Sonic the Hedgehog comic book series, based on the video game series, takes place on the planet Mobius. This stems from a mistranslation of Yuji Naka stating that a Möbius strip was used in the video game Sonic the Hedgehog 2.

The Japanese band Buck-Tick song Bran New Lover contains the following lyrics:

English Lyrics:

Right now, let’s throw open Pandora’s box
Tear and scatter the Möbius ring, and break free
Japanese Lyrics:

Pandora no hako wo ima akehanateyo
Chigire kakete mebiusu ringu tokihanatou
The They Might Be Giants song Operators Are Standing By contains the following lyrics:

Operators are standing by
Smoking cigarettes and drinking coffee
Pass round the picture of a Möbius strip
The Mountain Goats song Source Decay contains the following lyrics:

I wish the West Texas highway was a Möbius strip
I could ride it out for ever
The Darren Hayes song Love and Attraction contains the following lyrics:

This strip is Mobius
It’s never ending.
Two Orbital tracks contain Mobius references. The first track, from the album Orbital, is titled “The Moebius” and contains a couple lines from an episode of Star Trek: The Next Generation titled Time Squared:

Worf (Michael Dorn):There is the theory of the Möbius — a twist in the fabric of space where time becomes a loop.
Geordi_laforge (LeVar_Burton):When we reach that point, whatever happened will happen again.
The second track, “Time Becomes”, is from the album Orbital 2 and consists of two samples of the “where time becomes a loop” section of the Worf line that are slightly out of phase. The two samples drift further apart and resynchronize.

Iain Banks’ novels contain references. The Wasp Factory: My father once had me believing that the earth was a Möbius strip, not a sphere.(chapter one) The Crow Road The main protagonist Prentice has a white silk Möbius strip scarf which was given to him by his friend who was killed in a road accident. He is devasted when he leaves it on a train.

Mobius Band is an electro-rock band signed to Ghostly Internaional.

The Portland based electro-pop band, Copy, titled their first album “Mobius Beard”.

The logo of the Always brand of sanitary napkins made by Procter & Gamble contains a Möbius strip in the shape of the infinity symbol.
IBM’s Power Architecture logo consists of a Möbius strip in the shape of a ‘P’.
In the Dilbert comic strip from December 14th, 2007, Wally refers to his work list as a Möbius Strip in which each task required him to do another task until it all looped back on itself.[1]
In the xkcd comic strip from February 8th, 2008, named Mobius Battle, two stick figures fight endlessly against each other using a soccer ball as a weapon. Since the pictures are depicted as being on a transparent Möbius strip, each time we complete a circuit, the victim becomes the assailant and vice versa.[8]
In the anime Cardcaptor Sakura the card The Loop visible form looks like a the Möbius ring

Properties
The Möbius strip has several curious properties.

A model of a Möbius strip can be constructed by joining the ends of a strip of paper with a single half-twist. A line drawn starting from the seam down the middle will meet back at the seam but at the “other side”. If continued the line will meet the starting point and will be double the length of the original strip of paper. This single continuous curve demonstrates that the Möbius strip has only one boundary.

If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip – it is the center third of the original strip. The other is a long strip with two full twists in it – this is a neighborhood of the edge of the original strip.

Alternatively, cutting a Möbius strip along the above line, instead of getting two separate strips, it becomes one long strip with two full twists in it, which is not a Möbius strip. This happens because the original strip only has one edge which is twice as long as the original strip of paper. Cutting creates a second independent edge, half of which was on each side of the knife or scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other.

Other analogous strips can be obtained by similarly joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.

Occurrence and use in nature and technology

A demonstration of the mathematical basis of crochet: a scarf designed as a Möbius strip.There have been technical applications. Giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time). Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, as they allow the ribbon to be twice as wide as the print head whilst using both half-edges evenly.

A device called a Möbius resistor is an electronic circuit element which has the property of canceling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s: [9] “Coil for Electro Magnets” was intended for use with his system of global transmission of electricity without wires.

In physics/electro-technology:

as compact resonator with the resonance frequency with half of identically constructed linear coils [10].
as inductionless resistance [11] .
as superconductors with high transition temperature [12]
In chemistry/nano-technology:

as “knot molecules” with special characteristics (Knotane [2], Chirality)
as molecular engines [13]
as Graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism [14]
In a special type of aromaticity: Möbius aromaticity
Charged particles, which were caught in the magnetic field of the earth, can move on a Möbius band [15]
The cyclic protein Kalata B1, active substance of the plant Oldenlandia. O. affinis, as nature cures e.g. for the birth introduction, has a Möbius topology

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