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Is it art or is it geometry? It's both, of course. This stainless-steel Möbius strip casts a reflection in a circular pool on top of the Ramsey Auditorium at the Fermilab in Batavia, Illinois. The sculpture measures 8 feet in diameter. [Courtesy of the Fermi National Accelerator Laboratory]
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Where and when does an appreciation of the beauty and uniqueness of mathematics begin? The Möbius strip, or band--this wondrous shape--makes a great place to start. This paper with a twist gets its name from August F. Möbius (born in 1790), whose birthday we celebrate today.
The shape is simple and complex: a rectangular strip of paper that has been given a 180 degree twist and then is joined at the two ends. Tracing a path on the strip to demonstrate that this configuration has a single, continuous surface makes for a commonsense-defying act that will intrigue and engage your audience.
And now for a historical twist. The Möbius strip is named after August F. Möbius--and rightly so, or at least not wrongly so, since Möbius did discover the shape that bears his name. The truth is, however, that he was not the first to discover it. Another German mathematician, Johann Benedict Listing, published information about this unusual surface four years before Möbius. In 1861, Listing published a paper in which he described the Möbius strip and explored components of surfaces and connectivity of surfaces (Listing is also credited with the first use of the word topology). Unaware of Listing's work, Möbius published a description of the Möbius strip in 1865.*
*You may have noticed that the name Mobius (no umlaut over the "o") in the title differs slightly from the way it is spelled in the rest of the text ("o" with the umlaut). We've done this to make out title easier to index--to prevent odd-looking little symbols from appearing instead of letters. However, a Möbius strip is a Mobius strip is a Moebius strip. Any way you spell it, it's still the same shape. If you want to use the spelling that Möbius used but don't know how to type "ö," hold down the Alt key and then type 148 on the number pad.
The Möbius strip can tie into a study of algebra or geometry and make use of literature and art in the exploration. For example, Flatland (listed below in the Web Resources section) by Edwin Abbott is a classic story set in two dimensions. "Paul Bunyan versus the Conveyor Belt," a short story by William Hazlett Upson (from Clifton Fadiman, The Mathematical Magpie, Springer Verlag, New York, 1997, 2nd ed.), is lots of fun to read. Visit "Math That Makes You Go Wow" (listed in the "Ready-to-Go" Activities section) to find an online exploration with activities that connect the Möbius band and other non-orientable surfaces with literature, music, and toys.
If you can't convince your students that non-orientable surfaces are worth knowing because of the history, literature, and music surrounding them--in other words, for those students who always seem to ask, "But how does this apply to real life?"--try a more tangible approach. Inform them that the Möbius strip, according to the book Möbius and His Band, was the basis for a number of patents, including a patent for an endless sound record, filed in 1920 by the well-known inventor and "father of the radio" Lee de Forest. Some sources also report that the design is sometimes used for conveyor belts. It helps the belts to wear more slowly and more evenly.
Finally, for those who like to engage in philosophical discussions, note the resemblance between an infinity sign (which first made its appearance in the 1600s) and a Möbius strip. Have students discuss why a Möbius strip seems appropriate to symbolize infinity.
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Flatland
(ENC#020810) The full text of Edwin Abbott's classic story set in two dimensions is available online.
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Symmetry and the Shape of Space
(An ENC Digital Dozen selection, ENC#010300) This site contains a comprehensive unit on symmetry and properties of three-dimensional space. Designed for the secondary level and above, the site can also be used as a source of patterns and designs for elementary students. The first part of the unit deals with transformations in the plane, which are studied through definitions, theorems, images, and problems. Real-life applications of patterns and symmetry are shown in artists' paintings and Escher motifs as well as historical artifacts, such as a ceremonial digging board from the Inca Valley of Peru.
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Search Dr. Math
Visit Dr. Math and search the archives for answers to frequently asked questions about the Möbius strip.
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Neat Stuff
Offers pictures of topological surfaces.
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A Gallery of Topological Surfaces
Offers pictures of topological surfaces.
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Topology
Here you will find background information on the mathematical meaning of surface. If you dig deeper into the site, you'll find intriguing, fairly advanced activities involving knots.
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Möbius Concepts in Architecture
A Möbius strip-shaped preschool building? Playground? Museum? This paper explores the practical possibilities of twisting the Möbius strip into a variety of architectural structures. The drawings and photos that accompany the article are fascinating.
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Math That Makes You Go Wow
Developed by undergraduates at Yale University, this site encourages even the least mathematically inclined student to find beauty in a Klein bottle or be fascinated by the idea of living on a Möbius band. It offers a multidisciplinary exploration of non-orientable surfaces, of which the Möbius band is an example.
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The Mobius Strip and the Annulus
This activity takes students through nine questions that explore why a Möbius strip has only one side.
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Chasing Arrows
This site links the Möbius strip to the familiar symbol for recycling.
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The Möbius Strip
This web page contains a discussion about and directions for knitting a Möbius strip, and crocheting one too.
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Mathematrix: Mobius Strip
This page demonstrates what can be done with a Möbius strip and contains an illustration of its one-sidedness.
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Famous problems and their mathematicians
  Featured in ENC Focus
Date: 1999 Grade(s): 5 - 12 Cost: $24.00 ENC#: 017540
This teacher's resource book contains 61 chapters, each presenting the historical background of a mathematician or a famous problem along with a student activity based on the problem and its solution. The chapters, in reproducible format, are designed to be integrated into the curriculum.
(For more details see ENC Record)
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G is for googol: a math alphabet book
Featured in ENC Focus
Date: 1998 Grade(s): 6 - 12 Cost: $15.95 ENC#: 016027
This illustrated book explains mathematical terms, including the abacus, Möbius, and zillion.
(For more details see ENC Record)
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Critical thinking puzzles
Date: 1996 Grade(s): 4 - 8 Cost: $6.95 ENC#: 013991
Here you will find thought-provoking challenges from a variety of mathematical fields including topology.
(For more details see ENC Record)
Find more materials by searching Curriculum Resources on ENC Online using these terms: Knots or Topology.
Realizing that there are many ways to use this Classroom Calendar entry, we chose these standards because they relate to the entry, in full or in part. If you want to explore the standards further, please use the link(s) provided.
Connections to NCTM Standards
Geometry: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
Geometry: Specify locations and describe spatial relationships using coordinate geometry and other representational systems
Geometry: Use visualization, spatial reasoning, and geometric modeling to solve problems
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Judy Spicer
Instructional Resources
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First posted: 9/25/2001
Last modified: 4/30/2004 |
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