Study the decay products produced when a Mobius strip decays via fission.
Hot melt glue the flat side of one clothes pin to the flat side of the other.
Make sure the clothes grabbing ends both face the same way.
To Do and Notice
Make a Mobius strip.
Open both clothes pins and grab the Mobius strip in two places.
Cut across the short direction of the Mobius strip in two places. Tape together the resulting pieces into two loops. Remove the loops from the clothes pins.
Notice that you have a Mobius strip with one half twist (See Counting Half-Twists below) and a loop with no twists.
Some things are preserved after the decay process.
The sum of the lengths of the decay pieces equals the length of the original strip.
The sum of the numbers of half-twists in the decay products, 1 + 0, is equal to the number of half-twists in the original Mobius strip, 1.
The number of half-twists is a conserved quantity.
Is the number of sides a conserved quantity?
No it is not, the original Mobius strip had one side, the decay product had 1 + 2 sides.
Make two identical twisted loops of paper with 2 half-twists.
Investigate the decay products.
Notice that the two half-twist strip can decay in two ways, into a strip with two half-twists and a strip with no twists, and also into two strips each with one half-twist.
The number of twists is conserved, the length of the decay products equals the original length. But the number of sides does not remain the same in both decays.
Make three identical loops with three half-twists.
Study the possible decays.
What's Going On?
In a real radioactive decay the energy is conserved, just as the length of the strips remains the same in a Mobius decay.
The spin of particles before and after a radioactive decay remains the same just as the number of half-twists in the decay products equals the number in the original particle.
If you start with a no twist loop you can hold it at the top and bottom in the clothes pins.
Then remove the top from the clothes pins and give it a half-twist.
Cut the loop at the top and bottom and tape it together as two loops.
What does conservation of twists say about these resulting Mobius strips?
If the original loop had 0 twists, what must be the sum of the half-twists in the two decay particles? Answer 0.
But each of the two decay products is a Mobius loop with a half-twist, how can they combine to make zero twists?
Answer one is a Mobius strip with +1 half-twist and the other is an anti-Mobius strip with -1 half-twists. the sum is zero.
Collide these two Mobius strips and see what happens. They annihilate!
How to Count the Half-twists
Counting the number of half twists can be a difficult problem.
But you need to know the number.
Try the following, Hold the twisted strip with one hand pulling down on the bottom and the other hand on the top.
Hold it so that on one side of your two hands there are no twists. All the twists will be on the other side.
Twist the top of the Mobius strip to unwind the twists on one side of your hands and move them to the originally untwisted side.
Count the number of half twists it takes to completely remove all of the twists from one side of the strip.
This is the number of half-twists originally in the strip.
Scientific Explorations with Paul Doherty
20 October 2000