Module Trifold

The Hairy Ball Theorem

There really is a mathematical theorem called "the hairy ball theorem"! It says that if you have a hairy ball, you can't comb the hair on it so that it all lies smoothly. You will always have a cowlick. That's not necessarily true for other surfaces--investigate with the hairy ball and the hairy doughnut!

Topology in Clay
Use the clay to make a shape which is topologically the same as the ball or the doughnut in the pictures. Then make a shape which is not topologically the same as either of those.

A Topological Trouble
Help with my sewing project! I've nearly finished the ball and doughnut I am making--I just need to turn them right-side-out so that none of the seams show. Will you do it for me?

Knot Mathematics
Make one or more of the sailing knots. See if you can match the knot to the pictures in the Knot Catalogue.

OR

Sort the collection of knots--which knots are the same? Can you match these to the knots in the Knot Catalogue?

TOPOLOGY TRICKS

What is topology?
It's the geometry of position: whether or not things have holes, how things cross or connect. It is sometimes called "rubber sheet geometry". Two shapes are called "topologically the same" if you can stretch or squeeze or shrink one of them to look like the other, as if you were working with clay. You can stretch or squeeze your shape as much as you want to but if you poke any extra holes or close up any of your holes then you have a "topologically different" shape.

Topology is also the study of knots: which knots are different, and how to classify knots. Mathematical knots are the same as ordinary knots, except that the ends are joined together. Two knots are the same if you can tangle or untangle one to look like the other without cutting the knot. Knot studies have applications in the study of DNA.

Moebius Bands

Make a Moebius band. Take a strip of paper and twist it once and tape the ends together.

Things to do with your Moebius band:
1. Draw a line down the center of the Moebius band.

2. Cut your Moebius band down the center.

3. Draw a line down the center again.

4. Cut along your line once again.

OR
Write a Moebius band-inspired poem or story around your Moebius band.

Klein Bottles
A Klein bottle is a sort of 3 dimensional version of a Moebius band. It's a bottle with no inside or outside. The bottle doesn't really cut through itself. The "twist" is supposed to occur in a fourth dimension, but we can't visualize or make a model of that! The best we can do is to make a model with the bottle intersecting itself.

Three jolly sailors from Swindon-on-Tyne
Went to sea in a bottle by Klein
Since the sea was entirely inside the hull,
The scenery seen was exceedingly dull.
from The Space Child's Mother Goose by Frederick Windsor.