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Möbius manipulations

Take a strip of paper. Give it a half-twist and tape the ends together to make the shape called a Möbius loop. Run your finger along the inside of the loop. What happens? If you keep going, do you get back to your starting position on the inside? Are the ants in M.C. Escher’s print discovering the same thing?

In topology, the shape of an object is much more important than its size or proportions. One good “shape” question to ask is: How many surfaces are there? For the unjoined paper strip you began with, the answer is two: the upper and lower sides. But your experiment shows that the whole of the Möbius is one continuous surface. Where did the second surface go?

Explore Ø Try cutting your Möbius loop down the middle. What shape is formed?
Also experiment with 1- and 1.5-twist loops.

In diagram form, a Möbius loop can be represented like this:

The arrows show that the top and bottom edges join, but in opposite directions.

What if you join the long edges as well? There are two ways to do this:

The Klein bottle, named after its discoverer, Felix Klein

This geometrical form is called the projective plane. Early arcade
games, played on circular screens, used this geometry.

These shapes don’t really “fit” into three dimensions. But if they did, they’d each have one surface. Why?

Torus topology

Try making this slightly different shape for yourself. How is your shape like a doughnut? Hint: think topologically.

The main observation deck of the CN Tower in Toronto is a torus, as is the shape you just made. Another example of a torus is a doughnut – one with a central hole, that is.

A torus, like a zero-twist loop, does have two surfaces, which is good if you want your structure to have an inside that is separate from the outside!

Some doughnuts, like this honey cruller, have more complex shapes. You can model a cruller doughnut by imagining a strip twisted many times around its long axis. Complete the cruller by joining the ends.

Explore Ø Try predicting how many surfaces a cruller has, based on its total amount of twist. Is there a simple rule? Hint: think about half-twists.

CN Tower, Toronto, Canada

DNA asymmetries

A similar structure to the cruller, but without the ends joined, exists in every cell of your body: DNA.

DNA can be imagined as two separate strips, wound around each other. There are two “asymmetries” (symmetry violations):
• DNA always twists in an anticlockwise-from-above direction
• the two strips leave uneven gaps

Why do you think these asymmetries occur?

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