Transcriber: Ilze Garda Reviewer: Cristina Bufi-Pöcksteiner
Does this look scary?
Is it? No.
You would touch it, would you?
Of course you would.
Would you think with the same ease about hyperbolic geometry?
(Audience) No. Daina Taimiņa: No. Why?
(Audience) Yes.
DT: That's good, good, good.
So, well, why not?
OK, well here's my story.
It's 1977.
I am a senior at the University of Latvia, just across the channel here.
Up in those three tiny windows, there was an auditorium.
I'm sitting there, senior, having great dreams -
what am I going to do after graduation? -
that all of us have at that age.
Well, but there are a couple classes I need to finish,
and one of them is hyperbolic geometry.
Well, not my favorite one, but a required one,
and I have trouble with it.
I was a good student, generally, but I had trouble with that.
And, well, what's the case? What happened?
At that time,
it seemed like this hyperbolic geometry asked from me too much imagination.
The instructor drew these two pictures on a chalkboard and added,
"Well, so this on the left side, that one is in Euclidean geometry;
the right one is in Lobachevskian geometry.
All of them are correct."
I had the question, "Why?"
Well, the answer was
"You have to make an assumption, or you have to imagine that."
How could I imagine something which I don't know what to imagine?
That was pre-Google era.
And then my previous experience told me that I couldn't make an assumption,
because my previous experience is that only one of them can be correct.
Anyway, I had this puzzling question, but I finished the class.
Disliked it.
It was pass/fail, so I passed.
Otherwise, I wouldn't be here.
But I truly believed, well, that's it.
I will never have to think about hyperbolic geometry again.
It wasn't in my dreams that 20 years later
I'll be on the other side of the world at Cornell University,
facing the fact that for a whole semester I'd have to teach hyperbolic geometry.
(Laughter)
I couldn't use my previous experience -
I didn't want my students to think about that course
like I was thinking about it -
so I had to do something.
I attended a workshop,
and that was where I first saw this model of a hyperbolic plane.
My fingers itched to touch it.
I wasn't allowed because that thing would fall apart.
I tried to make something, a replication, well, something like this.
So I had to do something; I had to make some tactile model.
What I faced was the problem
that I had to visualize - or make in a tactile way -
exponential growth.
So I remembered how, as a child, I was taught about powers of two
and how the exponential growth comes through rumors.
You know - I have something.
I tell Pāvils.
"And just don't tell anybody else."
But next day, all of us, just one more person,
and so on, and so it goes.
Nowadays, you mention Ponzi scheme, and that's it - everybody gets the point.
It's easier.
At the same workshop, I just drew this pattern, and then I thought, "Oh!"
And then my "Aha!" moment happened:
"This looks like a crochet pattern."
The same night I started to crochet.
So, the stitches grow exponentially:
instead of crocheting flat, I started to add stitches,
to increase the number of stitches.
They grew exponentially, so I had to slow it down,
and the first model became too [crumpled],
and I couldn't do anything.
So I tried - okay, I need to slow down this exponential growth
and make this ratio closer to one.
So I tried the next one, which was closer to one,
and now I could do something that I can show.
So how can I get a straight line on a hyperbolic plane?
Well, if I have a plain sheet of paper,
then I can get a perfect straight line just by folding it - here we go.
OK?
So I can do the same thing if I have a hyperbolic plane.
Now I can use this, and now you will see that this is it.
Okay, well.
Yeah, so I can fold it the same way, but it doesn't look nice.
That's how it goes.
But if I have a crocheted one, I can fold it nicely,
and here I have a straight line.
So I kept folding, and voila!
Here was a picture
which I had so much trouble imagining as a student.
So it appears at a -
(Applause)
Thank you.
So here's another version.
That was the original one.
So here it is:
here is a straight line,
and here are the lines which are intersecting,
but all of these three lines here -
but that means if there are three, there can be infinitely many -
wouldn't cross the third one.
What does it come from?
It comes - just my understanding came and that of my students, luckily -
came from the simple fact that if I fold it,
then in a tactile way, through my fingers,
I can really experience,
yes, how these two lines come closer only at one point
and then they diverge away on both sides.
And that is that phenomena on this hyperbolic plane,
which is actually hard to imagine if you are not having this experience.
Well, now you have a question -
why do we care about hyperbolic planes - don't you?
(Laughter)
Well, I can tell you: it's actually very rich and beautiful.
You won't believe it, but that's true.
Also, this gives us a way to think about the shapes in nature -
I'll show you some examples a little later.
It also makes us think about the shape of our universe,
which has been a puzzling question for thousands of years,
and actually it's still an open question -
what is the real shape of our universe?
Well, I'm not going that far,
but I can show you
some interesting properties from the hyperbolic plane.
First, to understand that, we have to understand
how we are characterizing a hyperbolic plane.
So what's happening is
if we have a surface which has constant curvature,
then we can define a geometry on it.
And if we can define a geometry on it, then we can describe it.
Well, what is a curvature?
We all know there are positive numbers, negative numbers, and zero in between.
We can make the same analogy with surfaces,
and that was suggested by Carl Friedrich Gauss,
early 19th century.
So he said,
"Why don't we characterize surfaces also with a curvature?"
In two dimensions, it can also be flat -
this is more or less like the floor I'm standing on.
It's flat, and so its curvature is zero.
Or we can have a constant positive or a constant negative [curvature].
How to get it? Let's use some tilings.
If we are having regular hexagons, we can nicely tile a flat plane.
Now, if at each vertex three hexagons come together,
that gives us a perfect 360 degree angle, and it becomes flat.
It's no more in fashion, but it will come back.
Well, if at each vertex, I remove one of these hexagons,
then I get a pentagon,
and eventually this surface will close.
So that is an example of constant positive curvature,
or it is a way of how to make soccer balls.
Nowadays, they've changed this design,
but that's deliberately done to destroy my talk.
(Laughter)
If I remove a hexagon and instead use a heptagon, or seven-sided polygon,
then you can see that this surface starts to bend away:
it bends again, but now it's infinite.
So this is an example of constant negative curvature.
We can explore these things,
and we can explore the properties of a hyperbolic plane
by using these paper models,
but, of course, I have crocheted ones.
Well, one of the things we can explore
is what is the sum of the interior angles of a triangle?
We all learned in school
that a Euclidean plane is counted 180 degrees
and it's constant.
If we have a sphere, it's more than 180 degrees -
actually, you can cut an apple or something just to check -
and that really works like that.
If we are on a hyperbolic plane, we can have a triangle
where some of the interior angles are going to zero -
it's getting very small -
or it can be almost zero, like in this case.
Now, so let's go to this model,
and I can demonstrate what happens with this.
Because as soon as I learned how to crochet hyperbolic planes,
my husband, who is a topologist, said,
"You have to make this model."
He knows it from the theory, but he never held it in his hands.
In a hyperbolic plane, I can make a regular octagon with 45 degree angles.
That would be, in a Euclidean plane, an equivalent of a stop sign.
And now, what I can do is that I can identify or glue these sides.
This way, from this hyperbolic stop sign,
I can get something which has a very fancy name -
in topology, it's absolutely official -
and you can easily guess what it is.
So this is a hyperbolic pair of pants.
Well, so if I continue,
there are two more points which I can identify like this -
you can better see it in the blue one on the screen.
And then - something that I forgot to order from the organizers, sorry -
at this point, I need the fourth dimension;
can you provide it?
Sorry, you have to imagine;
now you have to turn to your imagination.
So we can have this hole glued here, and these holes glued together -
or this hole here and this one here -
and we are getting a two-holed torus.
That's a proof that two-holed torus has hyperbolic geometry in it.
Here we go.
OK, there is something else we can do.
When I was in school,
I was taught to write the numbers on a nice and neat square paper,
and you had to do it very purposefully.
If I would be in a hyperbolic plane, I wouldn't be able to do it,
because if I tried to fold
and get a right angle on this hyperbolic plane -
so whenever I try -
and you're welcome to try that afterwards -
I can fold it and get a right angle, but I end up with regular pentagons.
There are no right angle squares on a hyperbolic plane.
So let's go back there.
Here are my trials, and this is a close-up
of these right angle pentagons on a hyperbolic plane.
Well, here are some things in nature.
And now, next time you see it in nature,
you will recognize where the negative curvature is.
So this is what we are seeing.
Well, of course,
there are some stereotypes about mathematics.
Because when I started to do these presentations 15 years ago -
actually started to use these planes -
there were people who said, "Oh, yeah. Now, that's great!"
Students liked it; college professors adopted that.
But then there were people who said,
"What are you doing?
Mathematicians do not crochet; they do mathematics."
There is still this stereotype
that mathematics is something to stare at, with some confusing rules,
and accessible only to some very chosen people -
I'm not the one.
Well, the same thing [happens] by the way, with crochet, too.
Crochet is something which women do when they have nothing else to do.
So when we tried to publish an article about crocheting hyperbolic planes,
in a magazine,
the editor's first reaction was,
"What? Crochet instructions in the math journal?"
So he really didn't believe that would be something useful,
but I won, I convinced him, and he published it eventually.
But I was very happy three years later
to see another crochet model on the cover of the same magazine.
Hinke Osinga and Bernd Krauskopf crocheted a Lorentzian manifold,
which they crocheted using a computer-generated pattern,
to show a model of chaos.
Well, this is very important.
I always use this to warn my audience:
"For God's sake, please give it up" - before you've started.
"Fear it no less than the sensual passion, because it, too, may take up all your time
and deprive you of your health, peace of mind and happiness in life."
That was a warning
the father Wolfgang Bolyai said to his son Janos Bolyai,
one of the discoverers of non-Euclidean geometry,
when he said that he wanted to do it.
Well, it has become viral.
Crocheting hyperbolic planes,
this idea of explaining hyperbolic geometry through them,
was adopted by Margaret Wertheim, and she has done it now worldwide,
popularizing the project about hyperbolic crochet coral reef.
This is "Latvian Reef."
It was in 2009, curated by Tija Vīksna.
And so, well, it's even made it to a TED talk in 2009.
"When I was a child ... "
When somebody hears that I am a mathematician,
always I hear,
"Oh, I was so bad at math!"
When I was in school -
and this goes back to what [Aldis Kalniņš] said about these negative remarks -
I was told by the art teacher,
"You might be the best in the maths, but you are horrible in arts."
Therefore, it was like a lightning
when I got an invitation to participate in art shows.
My first art show was at 1111 Pennsylvania Avenue,
the same one that connects the White House with the Capitol in Washington DC.
So I said, "Yes, you can have my works,"
but then I thought, well, "What on earth am I going to do?"
Because I'm a mathematician, so I had to learn how to be an artist.
Mathematics is not scary when you can touch it.
The editor of my book said,
"It's the best if you can make it yourself."
The editor of my book, Charlotte Henderson,
after editing my book, decided to learn to crochet and crocheted one for herself.
And then she said, "Once I did it,
the properties of hyperbolic geometry, which previously I accepted in my head,
now I can accept with my heart."
So she even wrote a poem, using the crocheting.
Crocheting is this simple method to connect people,
and it's like really giving inspiration in music and poetry,
psychotherapy and biology and physics, and just a lot of friends like that.
The great mathematician Bill Thurston - he wrote in the foreword of my book.
He wrote,
"Mathematics is an art of human understanding.
Mathematics sings if you can feel it with all your brain.
People like music, but they are afraid to sing.
You can learn to sing only by singing."
So, do dare to make unexpected connections.
Be ready for tears and joy, for praise and rejection.
So, now, would you touch it?
Could we have some adventures with hyperbolic planes?
Thank you.
(Applause)