How many sides has a piece of paper? 1, 2, 3...

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How many sides has a piece of paper? 1, 2, 3...

• The mysterious mathematical world of Topology

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• Topology
• Topology generally is the study of shape, which
  shapes are intrinsically different, and how one
  shape can be transformed into another.

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• Some examples
• You use some examples of topological equivalence
  in everyday life.

• The world isnt really flat, and the countries
  dont really have these areas.

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• Topology shows important interconnections
• We use the London Underground map to understand
  the interconnections between the different lines
  and stations.

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• Planning
• We use topological networks in our planning, to
  show important connections.

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• The most amazing aspects of topology turn out to
  be useful!

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• How many sides has a piece of paper?
• A piece of paper has two sides. If I make it
  into a cylinder, it still has two sides, an
  inside and an outside.

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How many sides has this shape?
It is called a Moebius band. Lets make one and
see!
Cut a rectangle 2cm wide. Join the ends to make a
cylinder.
Now cut another rectangle 2 cm wide, but give it
a twist before you join the ends. You have made
a Moebius band!
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How many sides has a piece of paper? 1, 2, 3 ...

• An experiment
• Draw a line along the centre of your skinny
  cylinder parallel to one of its edges.

• Now do the same on your Moebius band

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• What did you notice?
• - A Moebius band has only one side.

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• The Moebius band
• This was named after August Ferdinand Möbius. It
  isnt just a mathematical curiosity.
• To see why it is useful, colour one of the edges
  of your Moebius band.
• What do you find?

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• Moebius bands are useful!
• You should have found your band only had one
  edge. This has been put to lots of uses. One use
  is in electric shavers.

• Epilady Mini Epishave rechargeable electric
  shaver. Wet / Dry rechargeable lady shaver. Dual
  sided shaver with a single foil for smooth care
  free shave.

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• Moebius band the video!
• http//www.youtube.com/watch?vrYIyXPnWPXc

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• Another experiment
• What do you think would happen if you cut along
  the line youve drawn on your cylinder?
• Will the same thing happen with the Moebius band?
• Try it!

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• More amazement
• Cut a new rectangle, but this time 3cm wide. You
  are going to draw two lines to divide it into
  thirds.
• 3 cm
• Now give it a twist and join the ends to form a
  Moebius band. Cut along one of the lines. What
  happens?

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• You should get a long band and a short band.
• Is the short band an ordinary band or a Moebius
  band? How could you check?

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• A Moebius band with two twists
• Make another Moebius band from a rectangle 2 cm
  wide, but this time give it two twists.
• What do you think will happen if you cut it in
  half parallel to an edge?

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• Are you amazed?
• You should get two intersecting bands. What sort
  are they?
• By now you should see that there is almost no
  limit to the experiments you can perform with
  Moebius bands. What if you chose a 4 cm wide
  rectangle...
• Or 3 twists?

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• Three dimensions
• Up till now we have just looked at flat shapes.
  In order to twist them we need our three
  dimensional world. Mathematicians have wondered
  what would happen if they took a 3D tube and
  twisted it in a fourth dimension before joining
  the ends.

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• Unfortunately we cant do that experiment in our
  world, but mathematicians know what the result
  would be.

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• The Klein Bottle
• By now you should expect the result to be a
  little strange and it is. It is a bottle with
  only one side, which we should probably call the
  outside.
• It cant be made this is just an artists
  impression.

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• Klein bottle videos
• http//www.youtube.com/watch?vE8rifKlq5hc

http//www.youtube.com/watch?vc4vOVA7b4SENR1
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• Is topology useful?
• A question asked by scientists for many years is
  How did our Universe start and how will it end?
  The answer to this question is very much
  dependent on the shape of the Universe is it
  open, closed or flat?
• This is a topological problem.

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• One possibility!

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How many sides has a piece of paper? 1, 2, 3 ...
And now for something completely different! A
piece of paper can have more than two sides.
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• History of the Flexagon
• It all started in 1939 when Arthur H. Stone, a 23
  year-old student, was cutting papers to make them
  fit into a file. Left over were some strips of
  paper that he began to fold. He folded a strip
  into a hexagon and joined the ends and when he
  put three corners together the hexagon turned
  around and showed a hidden side. He had
  discovered the tri-hexaflexagon.
• The next day he managed to make the
  hexa-hexa-flexagon which has six sides. He showed
  some friends what he had discovered and soon the
  whole school was making flexagons.

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• Preparing your strip
• Cut round the outside of your trapezium, and then
  carefully fold along the lines, in both
  directions.
• You are now ready to number the 9 triangles on
  both sides.

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• A net for a flexagon
• Start with your trapezium boat way up. Number
  the triangles to match side A.
• Now turn your trapezium over so that it is roof
  way up. The first 3 on side B will be on the
  back of the first 2 on side A.

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• Folding your net
• The method for folding the nets of flexagons is
  always the same no matter how many faces your
  model will have.

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• Folding instructions
• Start with the highest number on the net (here it
  is 3).
• Fold adjacent matching numbered faces onto one
  another. (Check both sides.) Then fold the next
  highest number. (We wont have to do this for
  our first flexagon.)
• You have a side with five 1s and a 3. Tuck the 3
  under the triangle beneath it and you will have a
  side with six ones. Join the top edge of this
  last 1 with the top edge of the 2 on the other
  side with some tape. Youre ready to flex!

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• Flexing

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• Flexing video

http//www.youtube.com/watch?vSnnvovjcC_o
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• Patterned flexagons
• Once you have gained the skill of flexing your
  hexagon, you might like to put a pattern on each
  of the three faces. Make it symmetrical and be
  prepared to be amazed.

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A visual demonstration of how the dominant faces'
leaves shift position.
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• More nets for larger flexagons

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Here a mathematician with artistic talents has
made a map of the changing faces of a six-sided
flexagon. The web is full of ideas and
information on flexagons. Why not do a search?
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• A health warning from Richard Feynman

This letter was written to Martin Gardener
following an article on flexagons hed
written. Sirs I was quite taken with the article
entitled Flexagons. It took us only 6 or 7
hours to paste the hexahexaflexagon together.
Since then it has been a source of continuing
wonder. But we have a problem. This morning one
of our fellows was sitting flexing idly when the
tip of his tie became caught in one of its folds.
With each successive flex more of his tie
vanished into the flexagons. With the sixth flex
he disappeared entirely. We have been flexing
the thing madly, and can find no trace of him,
but we have located a sixteenth configuration of
the hexahexaflexagon!
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How many sides has a piece of paper? 1, 2, 3 ...

• Now do you know?
• How many sides has a piece of paper?
• The world is your oyster with Topology, another
  branch of Mathematics that I hope you will want
  to explore.
• Not all maths is sums!
• Maths is exciting!