1. Möbius Strip -- From MathWorld html. Escher, M. C. moebius strip I. Wood engraving and woodcutin red, green, gold and black, printed from 4 blocks. 1961. http http://mathworld.wolfram.com/MoebiusStrip.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Surfaces Miscellaneous Surfaces ... cylinder , it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110). Euler characteristic According to Madachy (1979), the B. F. Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts. M. C. Escher was fond of portraying Möbius strips, and they appear in his woodcuts "Möbius Strip I" and "Möbius Strip II (Red Ants)" (Bool et al. 1982, p. 324; Forty 2003, Plate 70). 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

2. Moebius Strip 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.

3. The Shrine Of Escher, Art Gallery 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 http://www.uvm.edu/~mstorer/escher/moebius.html

4. 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.org/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.)

5. Moebius Strip moebius strip, what is it, see an avi movie To obtain a moebius strip, start with a strip of paper 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 Here is an interesting thing to do with 2 Möbius strips. (That is if your stipsare of equal length.) Now do the same again, but this time twist one strip. http://www.cut-the-knot.org/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. 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. http://mathforum.com/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 Home The Math Library Quick Reference ... Contact Us http://mathforum.org/ The Math Forum Webmaster 28 May 1996

10. Graphics Archive - Flat Moebius Strip By Henry Rowley Special TopicsTopology. Flat moebius strip by Henry Rowley. This moebius strip is isometric to a flat rectangle, which differs from the standard parametrization. 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@www.geom.uiuc.edu Created: Sat May 22 23:17:49 CDT 1999 - Last modified: Sat May 22 23:17:49 CDT 1999

11. 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

12. Starting Thru The Moebius Strip The Movie detecting flash player To visit the flash version ofThru The moebius strip, Flash Player 7 is required. http://www.moebiusstrip.net/

detecting flash player To visit the flash version of Thru The Moebius Strip, Flash Player 7 is required

13. Thru The Moebius Strip - The Movie http://www.moebiusstrip.net/start.htm

Enter Flash Version Fit Screen Mode Best View Mode FullScreen View Mode Enter ... Version For BEST viewing effects, make sure you installed MS Internet Explorer 5.5 + and Macromedia Flash Player 7 + Other broswers contain numerous bugs that might impair site functioning.

14. 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

15. Graphics Archive - Flat Moebius Strip By Henry Rowley Flat moebius strip by Henry Rowley. CoArt.com. This moebius strip is isometricto a flat rectangle, which differs from the standard parametrization. http://www.geom.uiuc.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@www.geom.uiuc.edu Created: Sat May 22 23:17:49 CDT 1999 - Last modified: Sat May 22 23:17:49 CDT 1999

16. LookSmart - Directory - Thru The Moebius Strip Thru the moebius strip Learn about the production of the animatedscience-fiction film from the mind of Jean Moebius Giraud. http://search.looksmart.com/p/browse/us1/us317828/us317854/us164424/us270941/us9

@import url(/css/us/style.css); @import url(/css/us/searchResult1.css); Home IN the directory this category YOU ARE HERE Home Entertainment Movies New Releases ... In Production Thru the Moebius Strip - Learn about the production of the animated science-fiction film from the mind of Jean "Moebius" Giraud. Directory Listings About IMDb.com - Through Moebius Strip Credit info about this animated film directed by Frank Foster. Thru the Moebius Strip - 3-D Gate Read a lengthy interview with director Frank Foster and artist Jean Giraud in which they discuss the development of this. Includes images. Thru the Moebius Strip - Upcomingmovies.com Read about the development of this CGI feature based on a story by Jean "Moebius" Giraud and find links to related sites and articles. We're always looking for ways to improve your search experience. Tell us how we're doing. Join the Zeal community and help build the "Thru the Moebius Strip" Directory Category IN the directory this category About Us Advertise with Us Advertiser Log-in Terms of Service

17. Chess In A Moebius Strip Rate this page! Skip to comments Chess in a moebius strip. Take the Cylindrical8 x 14 Variant. The moebius strip is a nonoriented surface. http://www.chessvariants.com/shape.dir/x_moeb.html

Contact Form Ask a Question about Chess Ask a Question about a Chess variant Ask a Question about this Website Ask another question Submit a Link Submit Graphics Submit a Chess variant Submit a piece for the Piececlopedia Submit a Zillions file Submit news for us to post Provide Comments or Information Send in a correction on something document.write ("@ "); More Information on this item Rate this page! Skip to comments 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 chance to a "Black" square (f1-a6-b12).

18. Moebius Chess where you started to draw. A Moebius Chess Board. The chess boardof Moebius Chess is like a moebius strip. The idea is consider http://www.chessvariants.com/shape.dir/moebius.html

Contact Form Ask a Question about Chess Ask a Question about a Chess variant Ask a Question about this Website Ask another question Submit a Link Submit Graphics Submit a Chess variant Submit a piece for the Piececlopedia Submit a Zillions file Submit news for us to post Provide Comments or Information Send in a correction on something document.write ("@ "); More Information on this item La Mecca Chess Encyclopedia Rate this page! Skip to comments Moebius Chess Menno Dekker from Haarlem, the Netherlands, invented Moebius Chess in 1992 or 1993. The rules of this game were sent to me by Rob Nierse. Moebius Chess has not been played often, as far as I know. It is somewhat different from the Moebius Chess which is used sometimes in fairy chess problems: the board of the game described here has 128 squares, whereas the board of fairy chess problems Moebius chess has 64 squares. Moebius Rings Take a strip of paper. Twist it once, and then glue the ends together. Now you have a Moebius Ring, called after the mathematician who invented this concept. Note that the Moebius ring has only one side: when you start to draw a line around the strip, you will first come to the other side of the paper of where you started to draw before you come back to the place where you started to draw. A Moebius Chess Board The chess board of Moebius Chess is like a Moebius strip. The idea is: consider an 8 by 8 board, that is drawn at both sides of a piece of paper. (The paper should either be very elastic, or, better, the squares should be rectangles, e.g., with a size of 10 by 2 centimeters.) Make a Moebius ring of this board, as described above. Rob Nierse wrote me that he actually made a board this way (of paper, with pieces made of pieces of paper): it looks quite interesting, but is hard to use.

19. 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:

20. WorkingForChange-This Modern World: Moebius Strip Foreign Policy Most emailed stories, This Modern World moebius strip Foreign Policy.See more in the Tom Tomorrow archives. Printer-friendly version, http://www.workingforchange.com/article.cfm?ItemID=15540

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