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# Kruseduller i matematik: Trekantfest

## Video udskrift

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.