hyperbolic crochet: July 2012

I was invited to give TEDxRīga talk in May and had one month to prepare it while having a very stressful time due to my motherš illness. I crocheted the big yellow model and also the one I took in hands at the beginning of the talk because all my other models were in US. Lucky my daughter came to visit us in Riga and brought some teaching models. The night before the talk I was expecting to rehearse it on the stage but I had only 5 minutes to be on it since everything had to be rushed. My TedxRīga talk has appeared online with minimal editing so there are several places where I am refering to slides which are not seen in online version of the talk. So I decided to post here the full version of the talk.

Does this look scary? Would you touch it? Of course, you would. Would you think of hyperbolic geometry with the same ease? Why not?

Today I want to tell you my story what happened when I mixed crocheting with hyperbolic geometry.

It was spring of 1977. I am a senior at the University of Latvia and counting the days to graduation in order to fulfill all those great dreams each of us have at that age. Still there are couple of classes to finish and one of them is Hyperbolic geometry. Although I was a good student in my other classes, I struggled with this one. By the end, I really disliked it.

Clearly, something has changed for me, but why was this the case? At the time, it seemed like hyperbolic geometry required a little too much imagination for me to make sense of it.

The instructor drew two pictures on the chalkboard and claimed that in Euclidean geometry through the point outside the line you can draw only one line parallel to the given one but in Lobachevsky geometry (she used this name for hyperbolic geometry) those three intersecting lines are parallel to the one line below. „Why?” was this picture on the chalkboard showing what the instructor said it was supposed to show? She said, „You have to use your imagination or make an assumption.” How could I make an assumption if my previous experience told that only one of these lines would be parallel? , or how can I imagine something if I do not know what to imagine? Fortunately this was a pass/fail class. I passed – otherwise I would not be here – but I hoped I would never have to think about hyperbolic geometry again.

It was not in my dreams that 20 years later, I would be, on the other side of the world, faced with the task of teaching Fall semester hyperbolic geometry at Cornell University. There was no way I could use my previous „experie nce”, so I looked to what others were doing. In June 1997, I attended a week-long workshop led by David Henderson, who explained and demonstrated hyperbolic geometry showing a paper model that he had learned to make from Bill Thurston. My fingers itched to touch it, but I was not allowed to touch, because this fragile little thing would be easily destroyed. At this point, it became clear to me that I had to figure out how to make something more durable. It looked like, once again, that I would have to turn to my imagination. The problem I faced was how to present exponential growth in tactile way.

I remembered how as a child I was first introduced to powers of two with an example how rumors spread – first I tell you something, then the next day each of us tells just one more person, the next day there are four of us who tell just one more person and so on. Nowadays of course it is much simplier to explain – it is enough to mention Ponzi scheme, and everybody gets the point. Right there during the workshop I graphed powers of two and then my „Aha!” Moment happened – the graph looked like a crochet pattern!

The same night I began to crochet it. Powers of two grow very fast, so I decided to slow it down a little. The first crocheted hyperbolic plane was born with a very simple „recipe”. Instead of crocheting back and forth with each row as I might for something flat, like a scarf, I would add an additional stitch into every third stitch that I came across from the previous row. So instead of „crocheting” y = 2^x I crocheted y=(3/2)^x.

This model came out too ruffled, so I figured the exponential growth of the number of stitches should be slowed down more, making the ratio even closer to one and I tried others, such as 9/8, 12/11, ...

The next models became something that I could use to finally figure out how lines on a hyperbolic surface look.

So, to go back to the original question. How do you get a straight line on a hyperbolic plane? Well, how do you do it on a flat sheet of paper? If you can fold it – that gives a perfect straight line.

It turns out, that I can do the same thing with my crocheted hyperbolic plane and voila! – here is a straight line on it! ...

And finally there it is! I have in my hands a tangible picture that I had so much trouble imaginining as a student!

On a hyperbolic plane I can have infinitely many lines through this point and they all will be parallel to the given line. The solution to the puzzle of why this is possible comes through fingers touching it: by folding, I can really see that any of these pair of parallel lines is closest at one point then on both sides they diverge away.

Why do we care about hyperbolic plane? It is rich and beautiful. It helps us to better understand Euclidean geometry. It also helps to understand shapes in nature, which we will see later, and helps us to think about the shape of our Universe, the problem that has puzzled people for thousands of years and still is an open question.

If we have a surface or space that is the same everywhere, then we can define geometry that allows to describe it. The simpliest ones - Euclidean plane, sphere, and hyperbolic plane are all two dimensional surfaces with the constant curvature therefore they each can have their own geometry.

What is a curvature? We know that numbers can be positive, negative, and zero. In early 19th century Carl Friedrich Gauss suggested to use similar characteristic for surfaces. Let us use some tilings.

We can cover floor or walls with hexagonal tiles. Notice that - at every vertex 3 regular hexagons come together and the result is flat surface. So we say that Euclidean plane has zero curvature.

If at every vertex I remove one hexagon and replace it with pentagon, then surface starts to bend and eventually closes) This is one way how soccer balls can be made. This is the example of constant positive curvature or sphere.

If instead of pentagons I use regular heptagons – 7-sided polygons – the surface bends again but this time it is not closing- this is another model of the hyperbolic plane which has constant negative curvature. Keith Henderson, a mathematics teacher, was explaining a notion of curvature to his class this way, and I really liked it.

If you do not want to crochet, you can use these paper models but I will use crochet ones to show some more interesting properties of hyperbolic geometry.

We all learned in school that the sum of interior angles of a triangle is 180 degrees. That is true only on a Euclidean plane or zero curvature surface. On a sphere triangle will have angles which add up to more than 180 degrees. You can check this by cutting an apple.

On a hyperbolic plane, a triangle will have interior angles which add up to less than 180 degrees; and it means that we can have a triangle which has the sum of interior angles almost zero!

My husband is a topologist and as soon as I figured out how to crochet hyperbolic plane, he said that I have to make this model which he knew from a theory but never had in his hands. On a hyperbolic plane it is possible to construct a regular octagon with all interior angles 45 degrees. (In Euclidean plane it would be a stop sign.)

If we identify or „glue” its sides accordingly we get so called „hyperbolic pair of pants” which in 4th dimension would be possible to turn into a two hole torus. If organizers would have provided me with it here, of course.

In first grade when I learned how to write numbers I had to do that neatly in a notebook with square graph paper.

It is impossible to have a square grid in hyperbolic plane – if we fold a regular polygon so that all angles are right angles we end up with right angled pentagons! And we can go even further and have right angle hexagons, heptagons and so on, but no squares on the hyperbolic plane!

We are familiar with the flat plane and sphere but surfaces with negative curvature are no stranger to us. Look at nature! These forms have been there long before mathematicians learned how to describe them!

Without any theory nature has figured out how to fit maximum surface area into minimal volume.

When I started these demonstrations with crocheted models of hyperbolic planes 15 years ago I received mixed reactions. My students were excited about having hands-on experience, and many college professors started to use it. But there were people who said that I am not serious enough – mathematicians do mathematics not crochet! There is still alive the stereotype that mathematics is an austere and formal subject concerned with complicated and confusing rules and therefore it is accessible only to very few chosen people. Perhaps that is because we are teaching mathematical facts and results but keeping it almost secret how actually they are discovered. In school we are rarely taught mathematical thinking.

It is not only mathematics, crochet suffers from a steoreotype as well like – crochet is what women do when they have nothing else to do. When in 2001 we tried to convince the editor of The Mathematical Intelligencer to accept a paper about crocheting hyperbolic planes, he was very hesitant to do so. He said – how it will look – crochet instructions in a math journal? Eventually he agreed. At that time he did not really believe my argument that crochet is useful for making mathematical models and somebody else can be inspired to use it also. But it did!

Three years later on the cover of the same journal there was a crocheted model again.

Hinke Osinga and Berndt Krauskopf were not shy to use crochet to create a Lorentz manifold using computer generated crochet instructions. This beautiful surface illustrates how chaos arises and is organised in various systems. They were inspired to use crochet because of seeing crocheted hyperbolic planes.

I made my first model, then next, then the whole classroom set, then other people wanted to use these models in their classrooms. I kept continuing my experiments – changing ratios, forms, colors. It was interesting because before I started a piece I had no idea how it will look at the end. Was it becoming an obsession?

In early 19th century Wolfgang Bolyai warned his son Janos who became obsessed with the idea of creating a new geometry:

For God's sake, please give it up. 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.

I used this quote to warn my audience, but I should’ve listened to it myself! Somehow once people start crocheting hyperbolic planes, they cannot stop. Exploring hyperbolic geometry using crochet models became very popular and got out of mathematical circles - it has become quite viral.

Margaret and Christine Wertheim learned from me the idea of explaining hyperbolic geometry through crocheted models and have widely used it in connection with their project Hyperbolic Crochet Coral Reef spreading these ideas worldwide, including a TED talk "The Beautiful Math of Coral" in 2009. By now their project has involved more than five thousand crocheters around the globe and the numbers just keep growing.

This is a fragment of Latvian Project "From coral reefs to Baltic Sea" on exhibit here in Riga in 2009. Fortunately crocheting hyperbolic planes is not depriving people from their health and peace of mind but making them happy to become connected through the simple crochet pattern used in many creative ways – in physics, biology, in design, even psychotherapy, music, and poetry.

Often after somebody has learned that I am a mathematician, I hear – oh, I was so bad at math in school! In fourth grade I did my best to draw „Memories of my summer”. Art teacher glanced at it and told me that I might be the best student in math, but I have no artistic eye and my drawing is horrible. I liked to draw and was sad about such teacher’s prejudgment, so I grew up convinced that I am really bad at art.

It was like a lightening from a clear sky when in March 2005 while gardening I received a phone call from Washington. Binnie Fry asked me if she could have some of my works for an art show Not the Knitting You Know in 1111 gallery on Pennsylvania Avenue. Yes, the same one that connects the United States Capitol to the White House. I authomatically said – yes, of course, but afterwards started to think – what should I do now? Before I used craft yarn for it was cheap and durable, and my models were made for use in geometry class not the art gallery. I did feel very scared but I had to keep my promise. First I went to the local yarn store to find some nice yarn, then I had to learn how to be an artist.

Since then I have been invited to participate in more than 20 art shows. One of my pieces even is included in Textile Collection of Cooper-Hewitt National Design Museum in New York City. I wish I could talk to my art teacher now. I wish no child would be told – you are no good at math. That is why I keep crocheting and talking about these hyperbolic planes.

Mathematics is not scary when you can touch it, think about it in fun ways, or even better – make it yourself. Charlotte Henderson after editing my book Crocheting Adventures with Hyperbolic Planes decided to crochet one herself. She said: „Once I had made one for myself, I was able to believe hyperbolic geometry properties that I had previously accepted with my head but not my heart.”

At the opening of Latvian Crocheted Reef exhibit I met several young people with learning disabilities who also had participated in the project. They wanted to know what is hyperbolic geometry. I showed them what I showed you and they understood. And afterwards one of them said the best praise I have ever heard about my work: „Thank you for making us feel as part of others.”

Bill Thurston wrote in a forward to my book: „Mathematics is art of human understanding...Mathematics sings when we feel it in our whole brain. People like music but they are afraid to sing. You only learn to sing by singing.”

Dare to make unexpected connections and be ready for joy and tears, praise and rejection!

How about some adventures with the hyperbolic planes?