JOURNEY AND THE BREHM MODEL
OF THE MÖBIUS BAND

In 1990 Ronnie Brown visited John Robinson with the French geometer Bernard
Morin who has been blind since the age of 5. John laid out a range of
maquettes for Bernard to see with his hands.
One of the perceptive comments Bernard made was that John was a modest person,
as shown by the way he can obtain an effect without dazzling technical means.
An example is Eclipse (not shown here) which is made from two offset
hemispheres of bronze, with differing surfaces.


Bernard Morin showed us the Brehm model of the Möbius Band, and
JOURNEY is John Robinson's version of this.
BUT WHAT IS THE BREHM MODEL ?
There is a space known to Mathematicians as the "
Projective Plane". It has relations to the notion, used in technical
drawing of projection, which concerns looking at a model or object from different
viewpoints.
However there is no way of building this projective plane in our 3dimensional
space. This is a mathematical fact, a theorem, but a little experiment will
convince you that a disc of cloth cannot be sewn onto the edge of a möbius
band. The way in which an attempt at this process gets tangled up shows that
there might be a model which crosses itself.
The first of these models was produced by Boy, a student of Hilbert, at the
end of the last century. A remarkable model with flat faces has recently
been discovered by U Brehm. First one makes three "horses heads".
The crucial feature of these is that the lengths of the parts AB and CD are
to be the same. Also there is a right angle at BCD. These three horses heads
are glued together so that the part AB of one is attached to CD of another.The
result is a Möbius Band. You can make this model for your self.
Now to form the projective plane, seven more triangles have to be added.
Four of them,
A1 B1 C1 ,
A1 A1 C1,
B1 B1 A1,
C1 C1 B1
are added on the outside and cause no problems, but it is a good idea to
cut a hole in the middle of the first triangle so that you can see inside.
The three interior triangles
A1 B1 C1 ,
B1 A1 C1,
C1 A1 B1
intersect each other and the three horses, so to make the model you have
to cut holes in the triangles. Detailed instructions for making this model
are in Brehm's article.
THE MATHEMATICAL THEMES

Borromean Rings  what
they are and why they don't exist!

The Möbius Band  what
one looks like, experiments to try, and a beautiful rotating golden one enabling
you to really see what one looks like in 3D (this is optional as 90Kb).

The Projective Plane
(88Kb), (95Kb)  two pages explaining how to create and understand the
projective plane although it is not possible physically to construct it.
Also, the relation to the Brehm Model of the Möbius Band, 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 basic knots).

Torus Knots  two pages
explaining the basics about torus knots with the help of excellent 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.

Exhibition
"Mathematics and Knots"

Ronnie
Brown's Homepage

John Robinson's Symbolic
Sculpture

Brochure giving a guided
Tour around the Symbolic Sculptures at the University of Wales, Bangor.

About the Centre for the Popularisation
of Mathematics
©Mathematics and
Knots/Edition Limitee 1996
This material may be used freely for educational, artistic and scientific
purposes, but may not be used for commercial purposes, for profit or in texts
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