Let's say you're me, and you're in math class,
and you're supposed to be learning about factoring.
Trouble is, your teacher is too busy
trying to convince you that factoring
is a useful skill for the average person to know,
with real-world applications ranging from passing your state
exams all the way to getting a higher SAT score.
And unfortunately, does not have the time
to show you why factoring is actually interesting.
It's perfectly reasonable for you
to get bored in this situation.
So like any reasonable person, you start doodling.
Maybe it's because your teacher's soporific voice
reminds you of a lullaby, but you're drawing stars.
And because you're me, you quickly
get bored of the usual five-pointed star
and get to wondering, why five?
So you start exploring.
It seems obvious that a five-pointed star
is the simplest one, the one that
takes the least number of strokes to draw.
Sure, you can make a start with four points,
but that's not really a star the way you're defining stars.
Then there's a six-pointed star, which
is also pretty familiar, but totally
different from the five-pointed star
because it takes two separate lines to make.
And then you're thinking about how,
much like you can put two triangles together
to make a six-pointed star, you can put two squares together
to make an eight-pointed star.
And any even-numbered star with p points can be made out of two
p/2-gons.
It is at this point that you realize
that if you wanted to avoid thinking about factoring,
maybe drawing stars was not the greatest idea.
But wait, four would be an even number of points,
but that would mean you could make it out of two 2-gons.
Maybe you were taught polygons with only two sides
can't exist.
But for the purposes of drawing stars,
it works out rather well.
Sure, the four-pointed star doesn't look too star-like.
But then you realize you can make the six-pointed star out
of three of these things, and you've
got an asterisk, which is definitely a legitimate star.
In fact, for any star where the number of points
is divisible by 2, you can draw it asterisk style.
But that's not quite what you're looking for.
What you want is a doodle game, and here it is.
Draw p points in a circle, evenly spaced.
Pick a number Q.
Starting at one point, go around the circle
and connect to the point two places over.
Repeat.
If you get to the starting place before you've
covered all the points, jump to a lonely point, and keep going.
That's how you draw stars.
And it's a successful game, in that previously you
were considering running screaming from the room.
Or the window was open, so that's an option, too.
But now, you're not only entertained but beginning
to become curious about the nature of this game.
The interesting thing is that the more points you have,
the more different ways there is to draw the star.
I happen to like seven-pointed stars because there's
two really good ways to draw them, but they're still simple.
I would like to note here that I've never actually left
a math class by the window, not that I
can say the same for other subjects.
Eight is interesting, too, because not only are
there a couple nice ways to draw it,
but one's a composite of two polygons,
while another can be drawn without picking up the pencil.
Then there's nine, which, in addition
to a couple of other nice versions,
you can make out of three triangles.
And because you're me, and you're a nerd,
and you like to amuse yourself, you
decide to call this kind of star a square star
because that's kind of a funny name.
So you start drawing other square stars.
Four 4-gons, two 2-gons, even the completely degenerate case
of one 1-gon.
Unfortunately, five pentagons is already difficult to discern.
And beyond that, it's very hard to see and appreciate
the structure of square stars.
So you get bored and move on to 10 dots
in a circle, which is interesting
because this is the first number where
you can make a star as a composite of smaller
stars-- that is, two boring old five-pointed stars.
Unless you count asterisk stars, in which case 8 was two
4s's or four 2's or two 2's and a 4.
But 10 is interesting because you can make it
as a composite in more than one way
because it's divisible by 5, which
itself can be made in two ways.
Then there's 11, which can't be made out of separate parts
at all because 11 is prime.
Though here you start to wonder how
to predict how many times around the circle
we'll go before getting back to start.
But instead of exploring the exciting world
of modular arithmetic, you move on to 12,
which is a really cool number because it
has a whole bunch of factors.
And then something starts to bother you.
Is a 25-pointed star composite made
of five five-pointed stars a square star?
You had been thinking only of pentagons
because the lower numbers didn't have this question.
How could you have missed that?
Maybe your teacher said something
interesting about prime numbers, and you accidentally
lost focus for a moment.
I don't know.
It gets even worse.
6 squared would be a 36-pointed star made of six hexagons.
But if you allow use of six-pointed stars,
then it's the same as a composite of 12 triangles.
And that doesn't seem in keeping with the spirit
of square stars.
You'll have to define square stars more strictly.
But you do like the idea that there's
three ways to make the seventh square star.
Anyway, the whole theory of what kind of stars
can be made with what numbers is quite interesting.
And I encourage you to explore this during your math class.