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# Doodling in math: Stars

Video transcript

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.