Sunday, October 21, 2012

Happy birthday, Martin Gardner (1914-2010) !

Today is Hexaflexagon Day - around the world math enthusiasts are celebrating birthday of Martin Gardner. You can see and listen Martin Gardner on Nature of Things. or read an interview with Martin Gardner. Of course, I cannot believe if you are reading this blog and have not joined those many millions on YouTube who has become fans of hexaflexagons watching Vi Hart's videos on hexaflexagons.
My first introduction into "recreational mathematics" was in 1966 when I as a sixth grader won my first math olympiad. My teacher gave me a little book by Martin Gardner. It was translation in Latvian of Martin Gardner's 1956 book:
There are still  only two books by Martin Gardner that are translated in Latvian (the other one is Relativity Theory for everyone). The other books by Martin Gardner on my bookshelf in Riga were all in Russian. All those books are partly responsible for me becoming a mathematician (like for many others). Unfortunately university mathematics was nothing like the exiting topics from Gardner's books, so I was discouraged to become a research mathematician. Instead I decided to become a teacher so that I can teach mathematics in fun and accessible ways. My teaching carrier in school was not too long - about 5 years as part time job. Most of my teaching has happened in universities  - University of Latvia and Cornell University. And I did end up with PhD in Mathematics but that is another story.
Martin Gardner is certainly responsible that I always tried to make math visual and tangible.
So my tribute for Martin Gardner's birthday is this online exhibit.
This is how many people see mathematics - "I have no idea what is about".
The following paintings are all having something in common. Can you figure out what it is?

If so far you have not figure it out - here is a clue:

Sunday, October 14, 2012

Sleepless night's musings: What is Veech surface?

Every fall there is a night which comes with cruel cold and kills flowers in my garden. It happened Friday night. It was clear and still with stars looking particularly bright through the skylight above my bed. I could not sleep so I reached for my iPad to catch up on Twitter I neglected yesterday.
Evelyne Lamb's tweet pointed out on some strange label in Science post. This post is interesting because it eatures interesting competition - Dance your PhD. There are videos of finalists of this competition, and of course, as a mathematician you would look up is there any about mathematics. Yes, there is one - Cutting sequences on Veech surfaces - except it is labeled as ...physics!:
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May be this whole thing is somehow connected to physics in actual PhD thesis but I could not find any reference, except this on Wikipedia which says that Diana Davis is a fictional character  with PhD in physics. However the real Diana Davis is a mathematician whose research interests about Veech surfaces arose from her interest in symbolic dynamics.

I never heard about Veech surfaces before and I do not know what symbolic dynamics is about. So I do search and come to Wikipedia definition of Veech surface. If you are as ignorant in this area as I am - clicking on provided link won't help. The next step logically is to read An Introduction to Veech surfaces by Pascal Hubert and Thomas Schmidt. They promise "a gentle introduction to the basics of Veech surfaces" . I start reading until I get to this place:
The 1-form dz on the complex plane induces a 1-form on our surface.
There is a unique complex structure on the surface such that this 1-form is holomorphic. The process thus results in a Riemann surface with a distinguished abelian differential (that is, holomorphic 1-form). There is a close relationship between the flows on the flat surface and various properties of the 1-form.
 It is 2:30 am, and I can understand each word separately but cannot put all this together. I am relieved that next follows an example of unfolding.
The first one is easy to understand because it is about flat torus. Animation how to transform rectangle into flat torus is here. Jeff Weeks has developed Torus games.

Torus has even inspired piano piece:
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By now it is 3 am, and I  am still thinking of Veech surface. That dance gives a very nice illustration how this surface can be otained by identifying sides but I cannot visualize. I am trying to get back to sleep but I am still puzzled with what I read in Hubert's and Schmidt's paper: " An easy Euler characteristics calculation shows that it has a genus two. As a genus two surface it has a hyperbolic covering."

I crocheted genus two surface - those are "hyperbolic pair of pants" - identifying every other side of the octagon on a hyperbolic plane with interior angle 45 degrees we will end up with a two hole torus.
Hyperbolic octagon with 45 degree angles and velcro strips on the sides that will be identifyed (glued together)

In order to make a two hole torus the two points at the bottom have to be identified. This will result of having four equal holes which could be glued together in 4th dimension to keep hyperbolic geometry on this surface.

Hyperbolic pair of pants were mentioned in very good Erica Klareich's paper Getting into Shapes: from Hyperbolic Geometry to Cube Complexes and back celebrating 30 years since Bill Thurston's original paper.

3:30 am - Veech surface can be covered with hyperbolic plane? I know what the hyperbolic plane is but how could I possibly make this Veech surface out of it? If flat torus "opens up" in a square, will Veech surface can be "opened" in rectangular pentagon on a hyperbolic plane?
it is impossible to have a square grid on a hyperbolic plane but we can have rectangular pentagonal grid, also hexagonal, heptagonal and so forth
Trying in my mind to navigate on this rectangular pentagonal grid I am dowsing off into sleep.
In the morning I draw two joined pentagons and color their sides the same way I saw in the movie.
Next step is to glue together (identify) two sides with the same color. The black strip in the middle is already two sides joined, so I join green ones.

But now I cannot get any further in three dimensional world we live in - I need another dimension. While I am pondering on how to get over this obstacle my husband passes by and asks what am I trying to create. I tell him that I want to see how Veech surface can have a hyperbolic plane as a covering. David looks at me and says - you are trying to think about it as geometric manifold but it is not - this is topology. Hyperbolic plane is its covering in the same sense as a sphere is a covering of the cube. That is something I grasp easy - imagine your cube (tetrahedron, dodacahedron, etc.) made from a rubber, then you can blow it up like a baloon and it will result in a sphere.

For Veech surface it should be opposite to the sphere - instead of "blow up" it should rather be called "suck in". I sigh and think that I should have woken David up at night to resolve this.
Still I do not have an image how Veech surface would look. If it is topologically like a two hole torus then I should be able to make it using play-dough.

This far I got with my paper model. Now I should stretch it and should be able to join more sides.

Well this is one step further. It reminds me of something....

Here it is! My play-dough sculpture looks like a hyperbolic plane I crocheted years ago and then folded. My wish to see how Veech surface could be connected to the hyperbolic plane is satisfied, and I go outside to assess a damage the freeze has done to the garden.
Mums have survived, so I will have flowers for a while.
In the afternoon I pass by David's study and notice a lot of post-its on his desk.
Now it is my turn to ask - what are you doing? He smiles - you infected me. Get that play dough.
So we continue together until we do find the way how these four holes can be connected to get a two hole torus. I must say - nothing elegant but at least I can get away to cook a dinner.

I wrote in my book about some uses and connections  of the hyperbolic plane. I am still very interested to find more, so mention of the hyperbolic plane in connection with Veech surface caught my interest and I wanted to understand what is this connection. I am happy to find that the hyperbolic plane is getting more and more popular. The latest use of it in popular culture was this nice cartoon strip by Bill Amend from

I am going now to make hyperbolic pancakes - may be that will be easier than crochet...
Greetings from Ithaca where after Friday night's freeze today was almost summer...

Sunday, October 7, 2012

Hyperbolic crochet for Math Fair

Last Thursday I visited River Valley Waldorf school in Upper Black Eddy, PA. You know that it is not a usual school when you are first greeted by these nice chicken :-)
The school is planning Math Fair in November, so the purpose of my visit was to tell students, teachers, and parents about my crocheted hyperbolic planes and gve some ideas what would work for Math Fair.
page 198
Students were impressed with my inaugural Guiness World Record in "hyperbolic crochet" but for now it is not enough time to break the record by the time of their Math Fair. So I suggested to make a large hyperbolic soccer ball using paper templates or to crochet. 
To understand hyperbolic plane it is important to understand what is curvature.
When I was in elementary school in fall we had to draw leaves. I always had trouble because I was drawing them like they are flat. But they are not!
Let us carefully cut a thin strip around the edge of the leaf. If I let it freely lay on the table then it is springing up.
If I am trying to flatten the edge s that it would all be on the table, then I notice that the edge is "too long" - it does not come together at the point but is starting to overlap. 
Try this experiment with other leaves! You will find that there will always be some overlap. It is because leaves are not flat but they are curved.

Let us look at a crochet pattern. It is called African Flower (pattern/tutorial). It lays nicely flat on the table because it is forming a hexagon. If I make in this flower one petal less, I have a pentagon which does not lay flat on a table - it curves up or down. 

 You can imagine how by adding more these five petal flowers I would get something very much like a ball. If you do not believe me - make 12 such motifs and then join them together at the edges. This will make your crocheted dodecahedron. Your surface will be closed like a ball or sphere. This type of curvature is called positive curvature. Since sphere is the same everywhere - you can see it has constant positive curvature. Stitching together pentagons and hexagons will end in a shape of traditional soccer ball. Now the pattern has chnaged but I hope you still remember this:
Now let us use the same African flower pattern but this time instead of six petals let us make seven.
This flower also does not lay flat on a table but is curved differently than the flower with five petals - there are some "ups and downs". It is more like curly parsley or cale. This type of curvature is called negative curvature. If this negative is the same everywhere (or we say - constant) then we have hyperbolic plane. In the sense of curvature we can say that hyperbolic plane is something opposite to sphere. You can try to make your own hyperbolic plane using template of "hyperbolic soccer ball". (here is another version , you can see the animation how it will grow here.)
This time I crocheted one - middle motif is with seven petals but I used only one color yarn to stress that there are seven hexagons around the heptagon.

And this is my crocheted version of "hyperbolic soccer ball".
Another crochet version could be made using just seven petal flowers alone - this way curvature will show off faster.
In class students made their own hyperbolic soccer balls from paper templates. We did not have time to join them all together but hopefully that will be done for Math Fair. 

photo Derek Smith
photo Derek Smith
Waldorf school is ver hands-on oriented and of course children wanted to find a practical applications of hyperbolic planes. Some said they will be making hats in shape of pseudosphere, some will make scarves. I am very much looking forward to see what ideas of our meeting will be used in Math Fair. 
photo Derek Smith
 My thanks to Derek Smith for organizing this trip and to all my very nice audience in River Valley Waldorf school.
If you would like to see some more ideas -  Crocheting Adventures with the Hyperbolic Planes are now available in Kindle version. I have not seen it myself and not very happy about not knowing that the book will be made in Kindle version - it was not written as e-book, so I wish I would be given an opportunity to edit it accordingly.
On my way back to Ithaca I stopped at Delaware Water Gap where I took this picture.