As you enter my house my wife and I
wanted to have some kind of artwork that
was personal to the house and also
related to the work that we do; and so
you see two things: one
of them is the dragon curve, I'm gonna explain
to you more about in a minute, and
another one is this work here which is a
grook by Piet Hein. 'Road to wisdom? Well
it's plain and simple to express. Err and
err and err again but less and less and
less.' It's one of my favorite- it sort of tells
the story of my life is making mistakes
and trying to learn from that.
They designed it in the shape of a superellipse
which is Piet Hein's great
great discovery of a- of the ellipse that
that is is more full and has been used
in in many architectural products. So
what's the idea of the dragon curve? This
is something invented by a couple of
physicists, late 60s, and you start with a
piece of paper and you fold it, fold it
again, and you fold it again. And you do
this n times, where n is nine in the
case of the dragon curve to our left
but here n is four or something. But wh-
when you look at it you see that some of
the folds go in mountain folds
some are valley folds. If you imagine
walking here you're turning left here,
you turning left here, but then you're
turning right; and another right turn and a
left turn and so on. So- so there's a
pattern of lefts and rights. The amazing
thing is that if you continue this like
nine times and then you open all these
folds up to ninety degree then you get a
path starts to fill the space and makes
a- quite a nice fractal pattern. So
start here with- at this tile for example;
if you if you zoom in on it you'd
that my wife and I had signed our names
here, but we can
talk about that story later. But it
starts out and makes a left turn, and a
left turn, and a right turn, and a left turn,
left and so on; so this is exactly the-
the pattern that you get from folding.
Since I folded nine times there's 512
steps to this - well
511 probably - to this pattern. So it ends-
it ends over here; and it keeps going but
in the middle it gets very intricate. And
the great thing is though from a
standpoint of geek art, you know, is that
you only need three kinds of tiles; a tile
that's completely blank, or a tile that
has one loop in it like this one, or a
tile that has two loops in it like this
one. And of course I could take the tile
with two loops and I could tip it this
way, then it wouldn't be joining from
here to here but it would be- that
would make this into an isolated circle.
All right, yeah, I had to count how many
blanks- how many blanks I needed, how many
single loops and so on. And so we
made a mock-up in in brown paper and
cardboard and had it in the apartment
where we were living while our house was
being designed. And we said, hey, this is a
cool design, it'd be kind of nice to have this
in our house when when the house was finally
built. So then we we both loved the tiles
that are made by Heath ceramics in
Sausalito, California; and we had a
mock-up of the design so we went up to
see the Heath ceramics and talked to the
people there; and we wondered if
they could make custom tiles for us,
certain number blank, certain number of single.
And the answer was, well, we could do it
except that we all- we only make tiles
that are three and a half inches square.
And what- my design called for two inches
square. If this- the mathematics could
not be changed even if I went down to
8 instead of 9 it wouldn't have
been- it wouldn't have been as interesting
a pattern, but also it would be huge and
wouldn't look right at scale. This-
this kind of pattern is very appealing
at this size but not at a bigger size. So
we were down, we looked very disheartened
evidently because the person we were
talking to saw that we were very
disappointed, that we had, you know, driven
up here and had this nice idea. But they said, wait!
One week in June we're going on vacation, we
close our factory, and you- but but
we have workers, somebody here is a security
person, you could come in and use our
equipment. We let you use our clay and
extrude tiles and if you cut them to the
size, two inches by two inches, then
we'll also let you put on our glazes and
we'll fire the tiles for you afterwards.
And so that's- that's what we did, we we
went up there and spent a day making all
these tiles, and a few spares, and put our
initials in this one for the starting
point. Then they fired it and- there are
some of the accidents in the fire and
you can see, sometimes it cracked a
little bit during the during the firing.
So we had- we had nine of these actually
so I could- so I put those in so that
after- after 8 steps
it's here. And then after 16 you're here,
32 I'm here. So years went by and and I
finally put a chapter in my book on fun
and games that talks about this. And I
reread all the papers I had written about
the dragon curve and come to a shocking
discovery; that is that this is wrong. You
get this by making nine folds but the
ninth fold I made in the wrong direction,
instead of going this way I went that way.
And so actually this is- the last half of this
has a different bend, and there's something-
mathematically I've been living with
this for thirty years and didn't realize
that I had a mathematical error in the
design. - (Brady: How do you feel about that now?)
Well I feel the
moral is: err and err and err again but less and less and less.
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We'll have more from Don Knuth coming
soon so stay tuned to Numberphile, and
our sister channel Computerphile, where
those videos will be appearing. And if
you'd like to watch even more about the
dragon curve we've done some previous
videos on that which go into more detail.
Links are on the screen and in the video description.