16: Doodling

    I'm sure you've had the experience of being in a boring meeting or
class, and starting to draw random pictures in your notebook instead of
paying attention.  If you don't have any drawing skills, as in my
case, chances are you make abstract patterns instead.  I sometimes
distract myself by trying to figure out how many patterns I can make
with a fixed number of dots.
    For example, suppose you have five dots.  You can draw them in a
row in a line, which is kind of boring.  Another familiar image is to
put the dots in the arrangement you would see on the '5' side of
dice.  If you arrange them in a pentagon shape, depending on the lines
you draw, you can end up with all sorts of pentagon and pentagram
variants, which is a bit more interesting.  Distorting the pentagon a
little to make two right angles at the bottom, you can get a sort of
house shape.  And, if you think in three-dimensions, you can actually
connect the five dots to form a small square-based pyramid.  As you
try this with more dots, you are able to come up with more interesting
shapes.
    Another fun thing to try is to relate the numerical properties of
what you are drawing to the pictures you end up with.  For example, if
the number of dots you have is a cube, or third power, of some other
number, you can literally draw a cube-- with 8 dots, you can draw a
2x2x2 cube, since 8 = 2 to the third power.  To get a more interesting
example, let's look again at the square-based pyramid we made with 5
dots.  The reason this formed a nice pyramid like that is that 5 is
the sum of two squares: 2 squared, the four dots on the bottom layer,
and 1 squared, the single dot on the upper layer.  Extending this
another step, if we have 3 squared + 2 squared + 1 squared dots, that
makes 14-- so with 14 dots, we can make a nice three-layer
square-based pyramid.  This is pretty surprising-- initially staring
at 14 dots on the page, you might not think they would make such a
regular, visually pleasing pattern.
    In his book "Mind Tools", mathematician and cyberpunk Rudy Rucker
took this idea further, attempting to come up with patterns for all
the numbers from 1 to 100.  Some of his efforts were a little
half-hearted though-- for example, he gave up on the numbers 23 and
29.  On the other hand, some numbers led to an unexpected set of
diverse images, such as 91, which can make a square-based pyramid
(being the sum of the first 6 squares), a regular triangle with a base
of 13 dots, a nice hexagonal pattern, or a pair of cubes.   Perhaps
you can come up with more interesting patterns.

    And this has been your Math Mutation for today.
 
Rudy Rucker on Wikipedia