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 sophorific voice reminds you of a lullaby
but you're drawing stars.
And because you're me, you quickly get bored of the usual 5-pointed star
and get to wondering: why five?
So you start exploring.
It seems obvious that a 5-pointed star is the simplest one-
the one that takes the least number of strokes to draw.
Sure you can make a star with 4 points but that's not really a star
the way you're defining stars.
Then there's the 6-pointed star which is also familiar
but totally different from the 5-pointed star because
it takes 2 seperate lines to make.
And then you're thinking about how
much like you can put 2 triangles together to make a 6-pointed star,
you can put two squares together to make an 8-pointed star.
And any even numbered star with "p" points can be made of 2 "p over 2" gons.
It is at this point that you realize if you wanted to avoid thinking about factoring
maybe drawing stars was not the brightest idea.
But wait! 4 would be an even number of points
but that would mean you could make it out of 2 "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 4-pointed star doesn't look too star-like
But then you realize that you can make a 6-pointed star out of 3 of these things
and you've got an asterisk, which is definitely a legitimate star.
In fact, for any star with a number of points that 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 q 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 successfull game in that previously you were considering
running, screaming, from the room
or the window is 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 7-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 have never actually left a math class via the window,
not that I can say the same for other subjects.
8 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 9,
which, in addition to a couple of other nice versions, you can make out of 3 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.
4 4-gons,
2 2-gons,
even the completely degenerate case of 1 1-gon.
Unfortunately 5 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 and 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, 2 boring old 5-pointed stars.
Unless you count asterisk stars, in which case 8 was 2 4s, or 4 2's. or 2 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 five which itself can be made in 2 ways.
Then there's 11, which can't be made out of seperate parts at all, because 11 is prime.
Though here you start to wonder how to predict how many times around the circle
it will go before getting back to the 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 5 5-pointed stars a Square Star?
You've 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.
And, oh no.
It gets even worse.
6 squared would be a 36 pointed star, made of 6 hexagons.
but if you allow use of 6 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 7th 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.