Doodling in Math Class: Stars
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Doodling in Math Class: Stars : More videos/info: http://vihart.com/doodling Check out this cool star-making applet Ruurtjan sent me: http://stars.ruurtjan.com Doodling Infinity Elephants: http://www.youtube.com/watch?v=DK5Z709J2eo Doodling Snakes + Graphs: http://www.youtube.com/watch?v=heKK95DAKms Doodling Binary Trees: http://www.youtube.com/watch?v=e4MSN6IImpI http://vihart.com
- 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.
- 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.
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