So now I want to show you another
type of turtle program we can send to
the to the plotter machine. - (You're)
(there, waiting for your next job) - Yeah,
and we're going to continue with this
idea of just giving it a list of angles
to turn through. And I just need to find
some source of
angles, and I thought maybe let's look at
the digits of numbers like pi and that
kind of thing. Here we have our original
thing where we started exploring the
Euler spiral; like what if I just start
sending the digits of pi?
...8,4,6.
(Wow you know that many off by heart?) - I know
more than that but-
The next one's a 2. You always
need to know one more.
If I sent this this would just be like
3 radians, 1 radian and stuff it- I
think what would make more sense would
be to do- well what I've marked up on on
this here,
I've split this into ten directions.
So
if I do- for the first three if I do one
two three, I could- that might make more
sense. So I'll be turning three tenths of
a full turn,
and then I'll turn
you know then I'll turn one tenth, then
four tenths, one tenth, five, nine; and then
we'll see what- see what picture gets
drawn. - (So does that mean 1 will be)
(36 degrees?) - Yeah, yeah we're doing
multiples of 36 degrees. So yeah let's
put in that, 36 degrees.
It would have been pointing this way and
it turned back three tenths to go up
this way.
And then it here it turned just 36
degrees, the smallest amount it can turn,
and so on.
Here when it got to 5 you'll notice that
it just did a
180 degree turn
because 5
tenths is a half. Any time you see a 5
that means it goes back;
so if I see two fives basically it
doesn't do anything right?
Yeah so
let's draw more. We're going to stick to
base 10 to start with, and and then it
will do this picture, it'll plot the
turtle drawing for that number.
So here we have pi
in base 10, and let's do
3,000 steps of pi. I got this picture here,
kind of like a fractal island or a
sort of wild walk
going around everywhere. Let's do more
digits, see what it looks like, so here's
10,000.
Our original picture, I think was up in
this corner here, now it's branched off
and gone over here.
20,000
100,000
So like here is some
way of visualising the digits of pi;
let's just sort of come out from playing
around with turtle programs.
(I've got so many questions coming to my)
(mind like)
(where will it ultimately go? Will it ever)
(come back to the start? Will every)
(possible shape be formed at some point?)
(Like, you know you know how they)
(say the monkeys writing Shakespeare?)
(Will you eventually draw the Mona Lisa)
(just by luck?) - Right, yeah yeah. I have
those same questions as well. One
conjecture about pi is that it's normal,
the digits of pi are normal, meaning that
every possible sequence that you could
think of:
0,1,2,3 up to 9 would be
in the digits of pi.
That- I guess that means if you had some
shape that you could draw with 36
degrees turn you would expect that
shape to eventually show up in this
drawing.
I guess you don't know exactly where in
the paper it would be
right? So I'm- so I
I don't know if you would expect after
drawing for infinitely long, would you
just have like a black sheet of paper
that's fully filled in with ink?
(When you get a very long string of)
(0s you'll get like really long)
(straight lines?) - Long straight line yeah.
Another question I think is natural to
ask is like, if I gave you this picture
could you have worked out the digits of
pi?
Right? Or if I sent someone this as like
a secret message or I broadcast this
across the universe as like a sign that
there's some intelligence on Earth; would
it be a good way to show like, hey we
know about pi? Let's do 3,000 so
we get this kind of like
dancing off away from the main kind of
hub.
This is pi base 10,
almost 3,000 digits. Base 10 is arbitrary
right? So I mean you want to try a
different base? Got one in mind?
(Well let's go base 3.) - Base 3 is cool.
So let's let's change the base to 3
and let's boost up to 10,000
points. So one cool thing about base
3 is that our our turns are going to
be uh 120 degrees,
that means that we're going to be
constrained to this triangular grid,
triangles that tessellate the plane.
Whereas with base 10
we have this weird 36 degree angle, it's
kind of like pentagons right? And
pentagons don't tile the plane so yeah
you can
get a real mess. You're always going to
have sort of collections of triangles if
you use base 3. - (So will base 4)
(give us squares?)
Yeah that's right!
So now we've got squares. I like this as
maybe
if I needed to generate some kind of
like roads for a zoomed out city in a
video game or something, you might
you know create a network like this. - (They)
(look like world maps to me) - Yeah
some people say they're kind of like-
like ways to generate an island.
It looks like Japan maybe. We're could do
more uh-
let's do 100,000 in base 4.
So I'm- I chose pi because I want like
an infinite list of turns that is just
going to spew out the Mona Lisa at some
point.
A less interesting digit sequence would
just be a rational number. So let's change
pi to
1/7th. I like 1/7th because it's
0.142857
142857... If you do
that enough you're just gonna end up
back where you started in the same
orientation and you're gonna create a
rotationally symmetric picture like this.
Could make it like one over some bigger
prime;
like I guess 119 is a prime
probably? Then you get a like a a more
detailed but still rotationally
symmetric picture. It's kind of
a nice way to see a stark difference
between rational numbers and irrational
numbers.
Do more irrational numbers, will we try e?
So let's do e,
base 10,
100,000.
There you go, that's e.
(What's going on here? That's a cool)
(little place,)
(there's a lagoon) - Yeah this seems
like an unlikely configuration. - (That)
(looks like Britain, up there) - Yeah yeah
Oh yeah! We're here right now. - (Yep)
(Scotland, yeah)
I like
golden ratio?
So what? You got square root of 5 plus
1. In some sense the most irrational
number.
I don't know if it gives a
sort of more squiggly map but it looks
cool to me.
(Imagine if you just put some arbitrary)
(number in one day, like randomly, and)
(pressed it and your face appeared!) - Yeah!
Or
conversely, could somebody send me the
number that draws my face?
This is the Feynman point, so six 9s,
99999. So that
won't be beaten- there is a string of
seven 3s
but that takes another seven hundred and
ten thousand digits, so we'll be nearly
three quarters of the way down the mile of
pi before we find a longer string of the
same digit and the first one is within
the first two metres.