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.