THE MÖBIUS BAND
Here is how to make a Möbius Band.
Take a long strip of paper,and glue the ends together, but
with a twist through 180 degrees.The result might be something like
It is a figure, a surface, with only one edge and only one
Many artists have been fascinated by the Möbius Band and
by one sided figures. Here are two examples by Max Bill.
Möbius Bands also had an important industrial use when
heavy machinery was driven by drive belts from a central power source. These
belts could be made to last longer by making them in the form of a Möbius
Band. Can you explain why?
There are some nice Experiments YOU can do!
Cut your Möbius Band down the middle. What results?
Next cut your new strip down the middle again. What
Make another Möbius Band. Cut it not down the middle but
one third from the edge. What results?
Form another strip of paper, and this time glue the ends together
with a twist of 540 degrees. Again we have a surface with one edge and one
It also is a Möbius Band, but the way it is put into our three dimensional
space is different from the previous one.
This makes an important distinction: between the object itself, and the way
it is a part of a space.
Cut this new object down the middle. What
There is another way of thinking of the structure of the
Möbius Band, which corresponds to the way we made it from a strip. The
Band has a middle circle, which goes round the Band only once. Notice that
there are other circles, seemingly parallel to the middle one, but which
go round the Band twice. Now draw lines on the Band at right angles to the
middle circle. For each point of the middle circle we have a line, and as
this line moves around the middle circle, it twists. This gives a mathematical
model of the Möbius Band which we can realise in a picture. Here are
four views of the Möbius Band. We can also produce a
3D rotating picture (90Kb) which we are
extremely proud of !
There is information explaining how we
did these and the moving pictures, notes for doing the same with
some of John Robinson's sculptures, and examples, such as a
3D rotating DEPENDENT BEINGS
We have also found another interesting picture of the Möbius
Band, and it is in the
Center Graphics Archive (outside link).
Here is another experiment, either in practice or a thought experiment.
Make a Möbius Band out of cloth,
and make a disc of cloth whose edge is the same length as the edge of
your Möbius Band. Now try and sew the
two together along their edges. What happens?
What you are trying to make is called
a Projective Plane.
THE MATHEMATICAL THEMES
Borromean Rings - what they
are and why they don't exist!
Bernard Morin and the Brehm Model
- how Bernard Morin showed John Robinson the Brehm Model of the Möbius
Band and how to make one!
The Projective Plane - how to
create and understand the projective plane when it is not possible physically
to construct it. We also describe the Brehm Model of the Projective Plane
and the Dirac String Trick.
Fibre Bundles - what they
are, how to make them, and examples of them in John Robinson's work.
Knots and Links - Introduction
to the subject of Knot Theory, includes history of the subject, and a rotating
mathematically constructed Immortality (the trefoil being one of the most
Torus Knots - two pages explaining
the basics about torus knots with the help of colourful graphics. There are
also 3D moving images of John Robinson's sculptures of the Gordian Knot and
the Rhythm of Life.
Fractals 3 pages, introducing
Fractals, considering iteration, the Sierpinski Gasket and the applications
of the subject.
"Mathematics and Knots"
John Robinson's Symbolic
Brochure giving a guided
Tour around the Symbolic Sculptures at the University of Wales, Bangor.
About the Centre for the Popularisation
Knots/Edition Limitee 1996
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Last modified 29 November, 1997.