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 :
????
( 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 !
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