of the MÖBIUS band

In 1990 Ronnie Brown visited John Robinson with the French geometer
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

**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. *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 3-dimensional 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, The three interior triangles With respect to edges, this means: we add to the three horses
heads the following edges: On the one hand, the boundary of the first triangle mentioned
above, namely

and

A
_{-1} B_{-1} C_{-1}

A
_{1} A_{-1} C_{-1},

B
_{1} B_{-1} A_{-1},

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.

C
_{1} C_{-1} B_{-1}

A
_{1}B_{0} C_{-1},

B
_{1}C_{0} A_{-1},

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.

C
_{1}A_{0} B_{-1},

as well as

A
_{-1} B_{-1}, B_{-1} C_{-1}, C_{-1}A_{-1}

A
_{1}C_{-1}, B_{1}A_{-1} , C_{1}B_{-1}.

**Reference**: Brehm, Ulrich: *How to Build Minimal Polyhedral Models of the Boy Surface*, **The Mathematical Intelligencer **Vol. 12, 51-55 (1990)

**© Mathematics and Knots/Edition Limitee 1996 - 2002**

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