Doodling in math: Triangle party Triangles!
Doodling in math: Triangle party
- So you're me and you're in math class
- --triangles, triangles, triangles--
- I don't know--the teacher keeps saying words?
- And you're supposed to be doing something with trigon
- whatever those are.
- But you're bored and--triangles, 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 into these triangle conga lines.
- And if you keep adding new triangles on the same side
- it gets all curvy and spirally
- or you can alternate and it goes sorta straight
- In fact, since all the sides of the triangle are supposed to be straight lines
- and since they are all lying on top of the previous straight line
- then this whole line would have to be straight
- if we were actually triangles
- since it's not
- it's proved 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 trigots,
- and you're supposed to be taking notes,
- but you're more interested in types of triangles
- which you already know about.
- There are fat triangles
- pointy triangles
- perfect triangles
- 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 could pull all the points facing one direction,
- but the stacks start 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 could put just one not so skinny triangle that's pointing the other way.
- Or maybe you want to wobble
- because you have to navigate you 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?
- 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'd 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 obtuseish quality
- and then, can you still call an obtuse equilateral triangle a cheese or cake slice triangle?
- because these look more like common <u><u></u><u></u></u>.
- 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 you 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 will 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 are some possible problems though.
- If you start a triangle like this,
- you can see that's 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
- this means that if these are two 120 degree angles of a triangle.
- The third angle is off somewhere being negative 60.
- Of course you'd have no problem being a bermuda triangle on a sphere
- where 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 will get eaten by sea monsters.
- Anyway, second triangles into a curve is nice
- and you probably want to make your spiral
- but if you're not careful, it will crash into itself
- so you'd better think about your angles.
- Though, if you do just right instead of a crash disaster
- you'd 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 triangleing around it to get this sort of apeture shape
- and then you should probably add more
- triangles, triangles, triangles, triangles.
- One last game.
- Start with some sort of asterisk
- then go around and 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 each triangle
- 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.
- dissect a square into triangles
- make symmetric arrangements
- some reminiscent of spherical and hyperbolic geometry
- triangles branch into binary fractal tree
- numbers increasing exponential with each iteration to infinity.
- with the right proportion being a golden spiral of perfect right, isoceles triangles
- put equilateral triangles on the middle third on the outside edges of equilateral triangles
- to infinity and get a snowflake with a boundary of a koch curve
- an infinitely long perimeter
- continuous yet nowhere differentiable
- with a fractional dimension of log 4 over log 3
- And uh-oh,
- teacher's walking around
- better pretend to be doing math!
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