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

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