Toffee Maths

 The Background

 Through the dusty window of a wooden shack on a wooden pier off the coast of California, a group of children are watching a silent but remorselessly rotating band of sticky white material alternately stretching and folding back onto itself …

This is the Monterey Taffy Company and this famous salt-water toffee is unknowingly home to an example of one of Mathematics’ most exciting new studies – ‘Chaos’.

 The Taffy

 The length of toffee hangs between two arms, which are about 1 foot apart ( we’re in California, remember ). Each time these arms rotate, the toffee is stretched to double its original length, and the excess is then folded back onto itself, so that it is once again 1 foot in length.

As you can see, the section from 0 to 0.5 foot is stretched onto the whole 0 to 1 foot section, while the 0.5 to 1 foot section is first stretched between 1 and 2 feet, and then folded back on top of the 1 to 0 foot section. 

The piece of toffee ( or the ‘point’ )  originally at 0.3 foot is ‘mapped’ onto the point 0.6, while the point at 0.9 is mapped onto 0.2

The Problems :

A         We can write this ‘mapping’ as :   


  While              for   0.5< x <1        x   ????     ( This is your bit ! )


B         As the silver arms steadily rotate, some points in the toffee don’t move at all, some move in regular cycles, and some points seem to move all over the place… 

Can you find any points which stay unchanged by this stretch-and-fold process ?

( There are more than one … )


C         What happens to the point x = 0.5 after a few repetitions of the process ?

What other points end up at this ‘attractor’ point ?

How many of these points are there ?


D         Now examine the behaviour of the points 0.1, 0.2, 0.3, …, 0.9

What’s different about this compared to the behaviour of the previous points ?

Is there an ‘odd-man-out’ within this group ?

Which other points will eventually end up in this same repeating cycle ?


E         Now examine the behaviour of points like 0.01, 0.02, 0.03, …, 0.09

What happens to these points ?

How long is the ‘period’ of these cycles? (ie how many terms are there in the cycle?)

Again, is there an ‘odd-man-out’ ? Can you see a pattern here ?


So far, we have only looked at points in their ‘decimal’ form.

F          What will happen to the points that starts off at     or     etc ?

How many of these regular cycles do you think there can be ?

What about the vast majority of points that are known as ‘irrational’ and consist of infinitely long, irregular decimals ? What would you expect to happen to them ?


G         Finally, consider the ( almost ‘twin’ ) points  0.44  and  0.441

These begin their Toffee Journey only 0.001 foot ( less than half a millimetre ) apart, so you might expect them to have similar Life Histories … Take a look !


What you have done is dip your toe into the beautiful, but unpredictably turbulent waters of what we now describe, mathematically, as ‘Chaos’.


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Last modified: June 18, 2007