Number Bracelets: Clock Arithmetic
What's going on here? (Clock arithmetic, Fibonacci sequences,
and discrete dynamical systems)
[This page under construction. Check back for more details
and a cool Java applet.]
By dropping all but the ones digit of the numbers, you are really
doing arithmetic on a clock with 10 hours instead of on a number
line. Draw yourself a 10-hour clock and add by counting around
(clockwise, of course).
The Fibonacci sequence is the sequence of whole numbers
you get starting with 1 and 1 and adding the last two numbers
to get the next number of the sequence. This is the rule
for Number Bracelets, except that (a) you can use any two
starting numbers in Number Bracelets and (b) the
Fibonacci sequence uses whole numbers on the number line,
not on a clock.
Fibonacci, also known as Leonardo of Pisa, was a medieval
mathematician who worked in the field of algebra
(that's high school algebra, which was hot stuff then,
not abstract algebra). The Fibonacci numbers arose in finding a pattern
in the way a rabbit population grows. Fibonacci numbers
are found all over the place in nature; there is lots
of interesting material to read about them.
Discrete Dynamical Systems
A dynamical system is sort of like a treasure hunt: every location
has instructions telling you how to get to the next location.
In the number bracelets game, think of the ordered pairs of starting
beads as locations, or points. The rule for getting to the
next pair is:
Example: Start with the pair (2,6).
- The first number in the new pair is the second number of the
- The second number of the new pair is the sum of the two
numbers of the old pair.
The next pair is (6,8).
The next pair is (8,4).
The next pair is (4,2).
The next pair is (2,6): we're back to the starting pair.
This is just another way to think about the bracelet 2 6 8 4.
The ordered list of all the points that are visited on
any one trip is called an orbit, in analogy with the orbit
of a planet (the route a planet takes around the sun).
In the number bracelets game, there are 6 orbits, having lengths
1, 3, 4, 12, 20, and 60, through the "solar system" of
Discrete vs. Continuous
The word discrete means that a situation can
be described by whole numbers, without using fractions or
irrational numbers like the square root of 2 or pi.
The opposite of discrete is continuous, which means
that a situation is more appropriately described by real
numbers: all the numbers on the number line.
The number bracelets game is a discrete dynamical system:
there are 100 points; there are no other points between them.
Continuous dynamical systems are very important in the
physical sciences (and other fields, like biology and economics).
An example is a vector field describing fluid flow:
at every point of a region (such as a pipe or a wind
tunnel) there is an arrow
showing how fast and in what direction the fluid is flowing.
By following ("integrating") the arrows, you can trace
the path of a molecule of the fluid.
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