1. Moebius Strip moebius strip. Sphere has two sides. A bug may be trapped Escher (18981972).To obtain a moebius strip, start with a strip of paper. Twist one http://www.cut-the-knot.com/do_you_know/moebius.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Moebius Strip Sphere has two sides . A bug may be trapped inside a spherical shape or crawl freely on its visible surface. A thin sheet of paper lying on a desk also have two sides. Pages in a book are usually numbered two per a sheet of paper. The first one-sided surface was discovered by A. F. Moebius (1790-1868) and bears his name. Sometimes it's alternatively called a Moebius band. (In truth, the surface was described independently and earlier by two months by another German mathematician J. B. Listing .) The strip was immortalized by M. C. Escher To obtain a Moebius strip, start with a strip of paper Twist one end 180 o (half turn) and glue the ends together (the avi file takes 267264 bytes). For comparison, if you glue the ends without twisting the result would look like a cylinder or a ring depending on the width of the strip. Try cutting the strip along the middle line. People unacquainted with Topology seldom guess correctly what would be the result. It's also interesting cutting the strip 1/3 of the way to one edge. Try it. I have put together a short (155648 bytes) avi movie of a twisting Moebius strip. (When you get to the movie page click on the frame to start the movie.)

2. Graphics Archive - Flat Moebius Strip By Henry Rowley Flat moebius strip by Henry Rowley This moebius strip is isometric to a flat rectangle, which differs from the standard parametrization. The steps involved in its creation are found in Rowley's Summer Institute 1991 report. http://www.geom.umn.edu/graphics/pix/Special_Topics/Topology/moebius_strip.html

Graphics Archive Up Comments Special Topics ... Topology Flat Moebius Strip by Henry Rowley This moebius strip is isometric to a flat rectangle, which differs from the standard parametrization. The steps involved in its creation are found in Rowley's Summer Institute 1991 report. How to make it: Mathematica was used to obtain the parametrization, and MinneView (the precursor to Geomview) was used to view it. Image created: summer, 1991 The Geometry Center For permission to use this image, contact permission@geom.umn.edu External viewing: small (100x100 2k gif), medium (500x500 20k gif), or original size (400x400 15k tiff). The Geometry Center Home Page Comments to: webmaster@geom.umn.edu Created: Sat May 22 23:17:49 CDT 1999 - Last modified: Sat May 22 23:17:49 CDT 1999

3. Moebius Strip A moebius strip is a onesided surface that can be constructed by affixing the ends of a rectangular strip after first having given one of the ends a one-half twist. http://zebu.uoregon.edu/~js/glossary/moebius_strip.html

Moebius strip A moebius strip is a one-sided surface that can be constructed by affixing the ends of a rectangular strip after first having given one of the ends a one-half twist. This space exhibits interesting properties, such as having only one side and remaining in one piece when split down the middle. The properties of the strip were discovered independently and almost simultaneously by two German mathematicians, August Ferdinand Moebius and Johann Benedict Listing, in 1858.

4. The Shrine Of Escher, Art Gallery moebius strip II, woodcut, printed from 3 blocks. 1963, 45x20 cm. A moebius strip is a surface that has only one surface, as it is attached to itself and twisted halfway around. Thus all nine red ants are on the same side. http://www.uvm.edu/~mstorer/escher/moebius.html

Moebius Strip MOEBIUS STRIP II , woodcut, printed from 3 blocks. 1963, 45x20 cm. A Moebius Strip is a surface that has only one surface, as it is attached to itself and twisted halfway around. Thus all nine red ants are on the same side. Go Back to The Art Gallery.

5. Moebius Strip Find out how to make a moebius strip or view images of models created by Mathcad. Includes the math behind the shape. Creating a moebius strip is a 3dimensional affair. http://www.cut-the-knot.com/do_you_know/moebius.html

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Moebius Strip Sphere has two sides . A bug may be trapped inside a spherical shape or crawl freely on its visible surface. A thin sheet of paper lying on a desk also have two sides. Pages in a book are usually numbered two per a sheet of paper. The first one-sided surface was discovered by A. F. Moebius (1790-1868) and bears his name. Sometimes it's alternatively called a Moebius band. (In truth, the surface was described independently and earlier by two months by another German mathematician J. B. Listing .) The strip was immortalized by M. C. Escher To obtain a Moebius strip, start with a strip of paper Twist one end 180 o (half turn) and glue the ends together (the avi file takes 267264 bytes). For comparison, if you glue the ends without twisting the result would look like a cylinder or a ring depending on the width of the strip. Try cutting the strip along the middle line. People unacquainted with Topology seldom guess correctly what would be the result. It's also interesting cutting the strip 1/3 of the way to one edge. Try it. I have put together a short (155648 bytes) avi movie of a twisting Moebius strip. (When you get to the movie page click on the frame to start the movie.)

6. Moebius Strip The resulting shape will be a square. (That is if your stips are of equallength.) Now do the same again, but this time twist one strip. http://www.cut-the-knot.com/do_you_know/moebius2.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Date: Tue, 21 Oct 1997 17:00:34 +0200 From: To: Alexander Bogomolny Hello Alexander Here is an interesting thing to do with 2 Möbius strips. Firstly, some background : Years ago I saw a problem posed by Martin Gardener. It went something like this : Take two strips of paper, glue them together without twisting them so that you have two cylinder like structures. Now glue the outsides together at a 90 degree angle. Now you cut the strips down the middle. The resulting shape will be a square. (That is if your stips are of equal length.) Now do the same again, but this time twist one strip. In other words one strip becomes a Möbius of "order" 1. What do you get if you cut the strips down the middle? The answer of course is another square. Now to the interesting part: Do the same as above but this time twist both strips. What do you get if you cut the strips down the middle? The answer is not obvious, it depends on the direction of the turns. if both strips are turned in the same direction then you will get a two separate pieces, one of which is completely twisted and the other is a "biangle", if I may call it that!

7. Moebius The moebius strip A moebius strip is a loop of paper with a half twist in it. How to make a moebius strip. 1. Take a strip of paper. 2. Give it a half twist (turn one end over). 3. Tape the ends together. What to do with a moebius strip. http://forum.swarthmore.edu/sum95/math_and/moebius/moebius.html

The Moebius Strip A moebius strip is a loop of paper with a half twist in it. How to make a moebius strip. What to do with a moebius strip. Find it on the Web or at the Library. How to make a moebius strip. 1. Take a strip of paper. 2. Give it a half twist (turn one end over). 3. Tape the ends together. What to do with a moebius strip. After you make your moebius strip, take a pencil and draw a line along the length. How many sides does the moebius strip have? Take a pair of scissors and cut the moebius strip along the line you have drawn. What happened? What do you think will happen if you cut it down the middle again? Try it. Find it on the Web... Moebius was a mathematician and astronomer in the 1800s.  Take a look at the path of the ants on Moebius Strip II , a woodcut by M.C. Escher . Which side of the strip are the ants walking on? or at the Library. If you have trouble accessing the Escher graphics over the Web, check with your school or public library for a book of Escher's work, which includes a lot of interesting mathematical figures. Return to main Mathematics and...

8. Moebius The moebius strip. A moebius strip is a loop of paper with a half twistin it. How to make a moebius strip. 1. Take a strip of paper. http://mathforum.org/sum95/math_and/moebius/moebius.html

The Moebius Strip A moebius strip is a loop of paper with a half twist in it. How to make a moebius strip. What to do with a moebius strip. Find it on the Web or at the Library. How to make a moebius strip. 1. Take a strip of paper. 2. Give it a half twist (turn one end over). 3. Tape the ends together. What to do with a moebius strip. After you make your moebius strip, take a pencil and draw a line along the length. How many sides does the moebius strip have? Take a pair of scissors and cut the moebius strip along the line you have drawn. What happened? What do you think will happen if you cut it down the middle again? Try it. Find it on the Web... Moebius was a mathematician and astronomer in the 1800s.  Take a look at the path of the ants on Moebius Strip II , a woodcut by M.C. Escher . Which side of the strip are the ants walking on? or at the Library. If you have trouble accessing the Escher graphics over the Web, check with your school or public library for a book of Escher's work, which includes a lot of interesting mathematical figures. Return to main Mathematics and...

9. Math Forum: Mathematics And... - Jan Garner Mathematics and by Jan Garner. Back to sum95 Projects Page Forum Web Units.Art. Perspective Drawing. Things to make. The moebius strip. Polyhedra. Spreadsheets. http://mathforum.org/sum95/math_and/

Mathematics and... by Jan Garner Back to sum95 Projects Page Forum Web Units Art Perspective Drawing Things to make The Moebius Strip Polyhedra Spreadsheets Introduction to Math and Spreadsheets Spreadsheet Basics and Graphing Examples Download notes or examples. jgarner@halcyon.com Visit SUNY Potsdam Suggestion Box Home The Math Library ... Search http://mathforum.org/ The Math Forum Webmaster 28 May 1996

10. The Moebius Strip The moebius strip Arnita Newton Kenwood Academy 4800 Chicago Beach Drive 5015 BlackstoneAvenue Chicago IL 60615 Chicago IL 60615 312548-1446 312-535-1350 http://www.iit.edu/~smile/ma9113.html

The Moebius Strip Arnita Newton Kenwood Academy 4800 Chicago Beach Drive 5015 Blackstone Avenue Chicago IL 60615 Chicago IL 60615 312-548-1446 312-535-1350 Objective : To investigate mathematical patterns using the Moebius Strip. Materials Needed : Strips of paper 10 inches long and 2 inches wide. Adding machine tape, construction paper or graph paper. Allow 10 strips for each student. Markers or colored pencils optional. Students will also need Scotch "Magic" tape and scissors. Strategy : Students are asked to examine their strips of paper to determine that each strip Conclusion : The Moebius Strip is an interesting topological figure. The investigation also provides a good exercise in having students derive a generalization from their empirical observations. Return to Mathematics Index

11. :: 3Dgate.com - Your Source For All That Is 3D :: Thru the moebius strip Preproduction. Jean Moebius Giraud and Frank Foster. DVWhere did the idea for the story of Thru the moebius strip come from? http://www.3dgate.com/techniques/2001/010205/0205moebius.html

document.write(""); document.write(""); Editorial Policy Contact Information Advertising Info Publicist Info ... Contact 3Dgate Thru the Moebius Strip : Preproduction Jean "Moebius" Giraud and Frank Foster by Dominic Milano Less than a quarter hour after leaving the rental car lot at Los Angeles International Airport, I pull into a tiny industrial complex. No. "Industrial complex" is too grandiose a way to describe the low-profile building situated off a narrow yet busy side street near the Marina Del Rey harbor. The place could be Any Auto Repair Shop, but for the artful logo painted on its double glass door entrance. Like so many structures that house film production companies, this one is cleverly disguised. The logo on the door isn't even the name of the company residing within, all the better to hide it from star-struck autograph hounds with nothing better to do than disrupt folks who are trying to put in a hard day-and-a-half's work. A test render that Coproducer and Director Frank Foster describes as "very rough, but it gives an idea of how well Moebius's drawings translate to 3D." Not that the star struck are likely to be looking for the L.A. offices of Global Productions, Ltd. That will change if things go as planned. Headquartered in Kowloon, Hong Kong, Global's parent company, Global Digital Creations Group, hopes to grow into an animation industry powerhouse. To get things started, they're animating a Saturday morning 3D animated show called

12. Chess In A Moebius Strip Chess in a moebius strip. Take the Cylindrical 8 x 14 Variant. Why should theboard be glued so nicely? The moebius strip is a nonoriented surface. http://www.geocities.com/Area51/Corridor/8611/x_moeb.htm

Chess in a Moebius Strip Take the Cylindrical 8 x 14 Variant . Why should the board be glued so nicely? Why not twist it before the glue? The board would still look like the picture below, however, along the 11th/12th (or -3rd/-2nd) junction, there would be a "inversion", so that a11 would join h12, b11 would join g12, and so on: All other rules (castling, en-passant, promotion, etc) apply An interesting aspect of this board is that, while on most standard variants, the King is facing the other King, here, along the 8-9-10-11-12-13-14-15 board, each King is facing the opponent's Queen. The Moebius Strip is a non-oriented surface. While this concept can hardly apply to chess, a minor consequence is that the Bishops don't have a fixed colour: a bishop that starts in a "White" square (say, f1) can change to a "Black" square (f1-a6-b12) A sample game could be: 1- d12 e11 [actually in front of the d12 pawn] 2- c12 [White is playing a Queen's Gambit, while Black is playing a King's Gambit!] 2- ... exc12 3- e12 g11 [defending the c12 pawn] This variant can lead to subvariants, or similar games:

13. The City Of Absurdity: David Lynch, Paper On the Lost Highway Lynch and Lacan, Cinema and Cultural Pathology.DIGRESSION 2 HOW TO MAKE A moebius strip Ill 1 The moebius strip. http://www.geocities.com/~mikehartmann/papers/herzogenrath3.html

On the Lost Highway : Lynch and Lacan, Cinema and Cultural Pathology DIGRESSION 2: HOW TO MAKE A MOEBIUS STRIP Take a strip of paper Make sure that it has two sides Take one end of the strip, make a 180 degree twist, and put it to the other end. Tape - or, better, with respect to suture, which is important, as we will see - stitch the two ends together. As a result, you now have a one-sided figure instead of a two- sided figure. Ill 1: The Moebius Strip The Moebius Strip subverts the normal, i.e. Euclidean way of spatial (and, ultimately: temporal) representation, seemingly having two sides, but in fact having only one. At one point the two sides can be clearly distinguished, but when you traverse the strip as a whole, the two sides are experienced as being continuous. This figure is one of the topological figures studied and put to use by Lacan. On the one hand, Lacan employs the Moebius Strip as a model to conceptualize the "return of the repressed," an issue important in Lost Highway as well. On the other hand, it can illustrate the way psychoanalysis conceptualizes certain binary oppositions, such as inside/outside, before/after, signifier/signified etc. - and can, with respect to

14. Möbius Strip -- From MathWorld Classic Surfaces from Differential Geometry moebius strip. http//wwwsfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_MoebiusStrip.html. http://mathworld.wolfram.com/MoebiusStrip.html

Geometry Surfaces Miscellaneous Surfaces Recreational Mathematics ... Trott A one-sided nonorientable surface w with midcircle of radius R and at height z = can be represented parametrically by for and The coefficients of the first fundamental form for this surface are the second fundamental form coefficients are the area element is and the Gaussian and mean curvatures are from to , which can unfortunately not be done in closed form. Note that although the surface closes at , this corresponds to the bottom edge connecting with the top edge, as illustrated above, so an additional must be traversed to comprise the entire arc length of the bounding edge. paradromic rings (Listing and Tait 1847, Ball and Coxeter 1987) which are summarized in the table below. half-twists cuts divs. result 1 band, length 2 1 band, length 2 2 bands, length 2 2 bands, length 2 3 bands, length 2 3 bands, length 2 2 bands, length 1 3 bands, length 1 4 bands, length 1 A torus even number of half-twists, and a Klein bottle There are three possible surfaces disk : the Boy surface cross-cap , and Roman surface Euler characteristic (or genus ), so the

15. Mobius Strip A moebius strip is a twisted loop, normally made of paper. MathematicalIdea. A twisted it. This is a moebius strip. (see image.). The http://www.questacon.edu.au/html/mobius_strip.html

Outreach Contents: NRMA RoadZone Photonics Questacon Balloon Questacon Maths Centre ... Teacher Workshops Maths Centre Activities What is Topology The Handcuffs Puzzle Handcuff Puzzle Images Moebius Strip ... Bubble Mix Recipe A Moebius Strip A Moebius Strip is a twisted loop, normally made of paper. Mathematical Idea A twisted loop is very different from a normal loop. Materials Needed Strips of paper, sticky tape, scissors and a pen. Demonstration Take a strip of paper and some sticky tape. Turn the paper into a loop, but before you stick it down, flip one end of the paper over. This should give you a piece of paper with a half-twist in it. This is a Moebius strip. (see image.) The Moebius strip has several strange properties. The Moebius strip has only got one side. If you draw a line down the middle of the strip until you get back to your starting point, you will find that you draw on both sides of the paper. The twist in the paper makes you change sides as you draw around. The Moebius strip used to be common in belt drives (like a car fan belt). With an ordinary belt only the inside of the belt was in contact with the wheels, so it would wear out before the outside did. Since a Moebius strip has only one side, the wear and tear on the belt was spread out more evenly and they would last longer. However, modern belts are made from several layers of different materials, with a definite inside and outside, and do not have a twist.

16. UM-VRL: Moebius Strip In VRML A virtual reality model (VRML) of the moebius strip with an animated processionof balls moving in an endless loop along this unique onesided surface. http://www-vrl.umich.edu/project2/moebius/

The Moebius Strip in VRML The mathematician A. F. Moebius (1790-1886) discovered this one-sided surface that became known as the Moebius Strip or Moebius Band. The artists M. C. Escher (1898-1972) was intrigued by the strip's puzzling geometry and included a procession of ants crawling along a Moebius Band in his collection of drawings depicting spatial illusions. The Moebius Strip, however, is not an illusion. It can be modeled in 3D and its topological features can be explored using an interactive computer model. Inspired by Escher's work, Bert Schoenwaelder created this unique VRML application with an animated procession of balls during his internship at the University of Michigan Virtual Reality Laboratory. Escher Schoenwaelder Main Features A simple endless band in the form of belt-shaped loop (below left) has two distinct surfaces and two edges. Moving from one surface to the opposite site requires crossing one of the edges. The Moebius Strip is an endless band that includes a half twist (below right). Amazingly, the band has only one surface and only one edge. Moving along the surface (like Escher's ants) will bring you to the opposite site without crossing the edge. simple endless band load VRML endless band with half twist load VRML Explore the Animated Model Load the VRML Animation (11.0K) of the Moebius Strip

17. UM-VRL: Moebius Strip 1 click to enlarge to screen size http://www-vrl.umich.edu/project2/moebius/scr.html

click to enlarge to screen size click to enlarge to screen size

18. Artsadmin/Gilles Jobin/Moebius Strip http://www.artsadmin.co.uk/artists/gj/moebiusstrip.html

19. Artsadmin/Gilles Jobin/Moebius Strip/Text Printable page. The moebius strip (2001). The moebius strip (2001) Photo ManuelVason. A moebius strip is a two dimensional surface with only one face. http://www.artsadmin.co.uk/artists/gj/moebiusstriptext.html

Top of page Artists: Gilles Jobin Introduction A+B=X Macrocosm Braindance ... General Information Printable page The Moebius Strip (2001) The Moebius Strip Photo: Manuel Vason A Moebius Strip is a two dimensional surface with only one face. A Moebius Strip represents infinity. When you cut a moebius strip in half, lengthways, you get a double moebius strip. If you cut it again you get a triple strip, and so on into infinity. When you ride a moebius strip, it is like going back to the beginning, in space, on a one-dimensional surface. If you cut it lengthways, then you must continue, and continue, and you will never cross your starting point again. Like life itself, it is a perpetual restart, never from the same point. The sound surrounding the 5 dancers is full. The space is vibrating. Their bodies are lines in space. The movements are continuous and fluid. The tension comes from the density of the movement, from the density of the music. The piece is evolving, in a circular way. Always close to the starting point, always different. The Moebius Strip is a journey that starts long before the spectators enter the auditorium. Top of page Choreography Gilles Jobin Music Franz Treichler Lighting design Daniel Demont Dancers Gilles Jobin, Christine Bombal, Lola Rubio, Jean-Pierre Bonomo, Vinciane Gombrowicz

20. Moebius Strip moebius strip. By Andrew Looney. Allen Murphey, President of NapoleonIndustries, one of the country's largest manufacturers of useful http://www.wunderland.com/WTS/Andy/SecretWorld/MoebiusStrip.html

Moebius Strip By Andrew Looney Allen Murphey, President of Napoleon Industries, one of the country's largest manufacturers of useful household appliances, sits at his desk. It is the largest desk in the largest office of the largest building owned by the company. Allen Murphey is pleased to be at this desk, in this position of enormous power. The fact that he never had to work very hard to get this position, it having been passed on to him by his father, does not bother him. The fact that he does the least work and makes the most money of anyone in the corporation does not bother him. The fact that his father was killed by a faulty electric can opener manufactured by their company does not bother him. What does bother him is a round stain on the varnished walnut surface of his desk. The stain was made by a coffee cup, which was not placed upon a coaster, as it should have been. Allen Murphey thinks, "Perhaps I should get a new desk." Allen Murphey rubs at the stain with his thumb for a span of about thirty seconds, and then looks at his very expensive wristwatch. He must go to a meeting in twenty eight minutes and nineteen seconds. He sets the alarm to remind him when it is time to leave. He calls up The Time, to see if his very expensive wristwatch is correct. It is. Allen Murphey looks at a picture on one of the walls of his very large office. It is a photograph of his father with Scott Carpenter, the astronaut. They were good friends.

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