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  • So you're me and you're in math class

  • and-- triangles, triangles, triangles-- I don't know.

  • The teacher keeps saying words, and you're

  • supposed to be doing something with trigons,

  • whatever those are.

  • But you're bored and-- triangles, triangles,

  • triangles.

  • Sure you could draw your triangles separately,

  • but then they get lonely.

  • They're happiest when snuggled up together

  • into a triangle party.

  • Everybody knows triangles love parties.

  • Sometimes they get together and do these triangle congo lines.

  • If you keep adding new triangles on the same side,

  • it gets all curvy and spirally.

  • Or you can alternate and it goes pretty straight.

  • In fact, since all the sides of the triangle

  • are supposed to be straight lines,

  • and since they're all lying on top

  • of some previous straight line, then this whole line

  • would have to be straight if these were actually triangles.

  • Since it's not, it's proof that these aren't quite triangles.

  • Maybe they've been partying a little too hard,

  • but hey, at least you're not doing math.

  • Speaking of which, the teacher is still

  • going on about types of trigons, and you're

  • supposed to be taking notes.

  • But you're more interesting in types

  • of triangles, which you already know all about.

  • There are fat triangles, and pointy triangles,

  • and perfect triangles, and cheese

  • slice triangles which are a kind of pointy triangle

  • but are symmetric like a slice of cheese or cake.

  • Super pointy triangles are fun to stack into triangle stacks.

  • You can put all the points facing one direction,

  • but the stack starts to wobble too much

  • towards that direction.

  • So it's good to put some facing the other direction before you

  • go too far.

  • You'll notice pretty quickly, that the skinnier the triangle,

  • the less wobble it adds to the stack.

  • To compensate for a big wobble, you

  • can put just one not so skinny triangle

  • that's pointing the other way.

  • Or maybe you want to wobble, because you

  • have to navigate your triangle stack around your notes.

  • In which case you can even alternate back and forth

  • as long as you make the triangles point towards where

  • you want to go, a little less skinny.

  • The easy part about triangle stacks,

  • is that there's really only one part of the triangle that's

  • important, as far as the stack is concerned.

  • And that's the pointy point.

  • The other two angles can be fat and skinny, or skinny and fat,

  • or both the same if it's a cheese slice,

  • and it doesn't change the rest of the stack.

  • Unless the top angle is really wide,

  • because then you'll get two skinny points,

  • and which side should you continue to stack on?

  • Both?

  • Also, instead of thinking fat and skinny,

  • you should probably create code words

  • that won't set off your teacher's mind reading

  • alarms for non-math related thoughts.

  • So you pick two words off the board, obtuse and acute,

  • which by sheer coincidence I'm sure, just

  • happen to mean fat angle and skinny angle.

  • Of course, those are also kinds of triangles.

  • Which doesn't make much sense, because the obtuser one

  • angle of a triangle is, the more acute the other two get.

  • Yet, if you make an acute triangle

  • with the same perimeter, it has more area,

  • which seems like an obtuse quality.

  • And then, can you still call an obtuse equilateral

  • triangle a cheese or cake-slice triangle,

  • because these look more like [INAUDIBLE].

  • Point is, triangle terminology is tricky,

  • but at least you're not paying attention to the stuff

  • the teacher is saying about trigonometric functions.

  • You'd rather think about the functions of triangles.

  • And you already know some of those.

  • There are sines, and cosines, but enough of this tangent.

  • The thing to pay attention to is what affects your triangle how.

  • If you start drawing the next triangle on your triangle

  • stack this way, by this point, you already

  • know what the full triangle would have to be.

  • Because you just continue this edge until it

  • meets this invisible line, and then fill the rest in.

  • In fact, you can make an entire triangle stack just

  • by piling on triangle parts and adding the points later

  • to see what happens.

  • There's some possible problems though.

  • If you start a triangle like this,

  • you can see that it's never going to close, no matter

  • how far you extend the lines.

  • Since this triangle isn't real, let's

  • call it a Bermuda Triangle.

  • This happens when two angles together

  • are already more than 180 degrees.

  • And since all the angles in a triangle

  • add up to 180-- which, by the way,

  • you can test by ripping one up and putting the three

  • points into a line-- that means that if these are

  • two 120 degree angles of a triangle,

  • the third angle is off somewhere being negative 60.

  • Of course, you have no problem being a Bermuda Triangle

  • on a sphere, were angles always add up to more than 180,

  • just the third point might be off in Australia.

  • Which is fine, unless you're afraid your triangle might

  • get eaten by sea monsters.

  • Anyway, stacking triangles into a curve is nice,

  • and you probably want to make a spiral.

  • But if you're not careful, it'll crash into itself.

  • So you'd better think about your angles.

  • Though, if you do it just right, instead of a crash disaster,

  • you'll get a wreath thing.

  • Or you can get a different triangle circle

  • by starting with a polygon, extending the sides in one

  • direction, and then triangling around it,

  • to get this sort of aperture shape.

  • And then you should probably add more triangles,

  • triangles, triangles--

  • One last game.

  • Start with some sort of asterisk,

  • then go around a triangle it up.

  • Shade out from the obtusest angles,

  • and it'll look pretty neat.

  • You can then extend it with another layer of triangles,

  • and another, and if you shade the inner parts

  • of these triangles, it's guaranteed

  • to be an awesome triangle party.

  • And there's lots of other kinds of triangle parties just

  • waiting to happen.

  • Ah, the triangle.

  • So simple, yet so beautiful.

  • The essence of two-dimensionality,

  • the fundamental object of Euclidean geometry,

  • the three points that define a plane.

  • You can have your fancy complex shapes,

  • they're just made up of triangles.

  • Triangles.

  • Dissect a square into triangles, make symmetric arrangements,

  • some reminiscent of spherical and hyperbolic geometry.

  • Triangles branches into binary fractal trees.

  • Numbers increasing exponentially with each iteration

  • to infinity.

  • Triangles, with the right proportion

  • being a golden spiral of perfect right isosceles triangles.

  • Put equilateral triangles on the middle third

  • of the outside edges of equilateral triangles

  • to infinity, and get a snow flake

  • with the boundary of the Koch Curve.

  • An infinitely long perimeter, continuous yet nowhere

  • differentiable.

  • With a fractional dimension of log four over log three,

  • and-- uh oh, teacher's walking around,

  • better pretend to be doing math.

So you're me and you're in math class

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B1 中級

算数の授業で落書きトライアングルパーティー (Doodling in Math Class: Triangle Party)

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    林宜悉 に公開 2021 年 01 月 14 日
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