Friday, July 30, 2010

College mathematics or math about college

The other day mail came with the most expensive DVD we have ever purchased.  I'll better will not add up because it has not been paid all at once, but it is many thousands of dollars. Have you guessed what it is?
 DVD of my daughter's commencement.

It brought back some of my thoughts from the time when I was full time teaching 120 students. Each student means around $3500-$4000 paid tuition per course. You think I had a fantastic salary?  I cannot blame you for this misconception if your answer is yes.

If you are not a contingent college professor then your image of college professors may be is made up by movies - like a classical one "My Fair Lady" - have you noticed all the staff Professor Higgins had at home?

"My Fair Lady" is my favorite movie but my salary was nothing compared to Prof. Higgins means or any other more recent movies depicting college professors in fashionable home interiors (for example,"Chloe").
In reality my salary for teaching 120 students (do math yourself how much tuition it is) was not enough to cover the tuition for that same semester for one of my children but I had two in college... I felt that something is wrong, very wrong. I am not complaining here that my salary was low (there are many colleges that pay even less per course to contingent faculty), but I thought and think that tuition in American colleges is way too high. If my teaching a class is a product and that is sold to students for 10 times more - the question arises - why are the universities called non-profit organizations? 

I am not alone in my feelings. In a recent public survey report by Squeeze Play 65% of respondents are saying that higher education are rising at a faster rate compared with other things and 74% of them say that this rate is the same or even higher than that of health care costs. In a recent issue of Academe (July-August 2010)  If the costs are rising so rapidly, there should be raised question about the quality of the product that has been sold for such a high price. The whole issue of the current issue of Academe is called "What do faculty owe students?". There is an interesting article by Peter Sacks where he talks that "access to higher education is not determined simply by one's ability to pay for college.Some children are born on the right side of the class divide ...Despite what we may believe about our higher education system's role as a great equalizer for social and economic mobility in the United States, we have created a highly structured system of colleges and universities that, more often than not, actually perpetuates inequality."
May be it would be smart to skip the college? This is one of the questions Rebecca Mead raises in New Yorker because odds of getting job (any job) for class 2010 are lower now than one imagined four years ago. Are colleges responsible for preparing students with degrees that are useless? Why not - parents have paid thousands of dollars or students have taken student loans that have to be paid (even in case a person dies, then this loan passes to the next of kin!!).
There is an interesting discussion about the real cost of college textbooks which also talks about financial burden of college education.When I searched Google for college affordability first jumped out this report - it is from 2003! What has changed except prices are even higher?

Is all what is left is to console ourselves with this Your new college graduate - parents guide?

Saturday, July 24, 2010

The story about the origins of the model of hyperbolic plane

This is something I really wanted to get to the very roots but did not get a chance to do when writing my book Crocheting Adventures with Hyperbolic Planes. It is about the very first model(s) of the hyperbolic plane. When I crocheted my very first hyperbolic plane in 1997 I knew about the existence of paper models and the idea was attributed to Bill Thurston.
I knew also that the very first model was done by Eugenio Beltrami in 1868 but I could not find anything more at that time except that he made a pseudosphere. Later I took a picture of a plaster model which was on an exhibit in Kettle's Yard Gallery, Cambridge University, UK, but it was not an original one. I even did not know whether original Beltrami models still existed!
On July 13 David and I received a message from Italian mathematician and historian of mathematics Piergiorgio Odifreddi who draw our attention to online document by Maurizio Cornalba pointing to the page 15 with pictures of original Beltrami models and asking a question whether this is the same idea as Thurston's.
 I did not know the answer and I always was curious how Bill came up with this idea, so I finally asked him this question. Here what Bill told me:
"I don't recall having seen Beltrami's paper models, although I was aware that Beltrami had shown that the pseudosphere had the intrinsic geometry of the hyperbolic plane. At New College (he graduated from it in 1967 - DT), a handful of math majors held a "senior seminar" in geometry --- purely students --- and we went through large portions of Coxeter's Geometry book. I looked at a few other books about geometry and hyperbolic geometry that were available in the library, but I didn't know very much. I was struggling to get a grip on what the hyperbolic plane looked like, so I started piecing it together in an obvious way (while also doing it for a sphere, for guidance) using concentric circles with shape gotten using trigonometry, length 2 pi sinh(r) and curvature cosh(r)/sinh(r), (radius 2 pi sin(r) and curvature cos(r)/sinh(r) for the sphere). Soon it became obvious that the shapes in the hyperbolic case were converging, so I switched over to using congruent annuli. I was trying to model the hyperbolic plane, not the pseudosphere picture --- but of course it was soon obvious that you could wrap the construction around a pseudosphere. It seems like the kind of thing that could easily have been rediscovered many times. Also, I think people have probably made models for the pseudosphere in lots of ways." 
Here we go - my first correction - I have been saying that Thurston came up with this idea in 1970's but actually he made his first model while still in college! He graduated from college in 1967, that is when I finished 6th grade and had received my first prize in mathematics - Martin Gardner's book (I added a picture of it in my blog about Gardner.) 
This is the picture of the courtyard in the University of Pavia. Eugenio Beltrami started to study mathematics here in 1853 but was expelled in 1856 because of his political opinions according to Wikipedia but according to Mac Tutor biography - he had to stop his studies because of financial hardship and had to take a job...(hmm -  which on of those reasons? or they both go together?). In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him  to take his paper models with him when Beltrami returned to University of Pavia 1876 and taught there for next 15 years. These models are now in the Department of Mathematics of the University of Pavia.  
Prof. Odifredi, who is currently writing a book in history of mathematics, contacted Prof. Cornalba and sent us pictures of the original models, the very first models of the hyperbolic plane, made by Eugenio Beltrami.

I was really excited seeing these pictures - here they are - the very first models of the hyperbolic plane! I am excited not only as somebody who has taught for many years and loves history of mathematics. My excitement is also for very personal reasons.  I never liked the lines in media once mistakingly put up there and then repeated many times over and over again, against all my wishes - that is a line about this "superwoman who made a first model of hyperbolic plane which men could not do for hundreds of years. I had tried to correct this misquoting many times, only few times I succeeded. 

Thank you Prof. Odifredi and Prof. Cornalba for sending these pictures to me! Thank you also for Beltrami letters to another Italian mathematician Luigi Cremona where he was discussing pseudosphere and why it is a model of the hyperbolic plane.
Odifredi kindly translated some passages from the letter:
"the model is a circular piece of a pseudosphere... the surface is bent into the form a surface of revolution, described by equation n. 16 of my paper of 1868, and its meridien is a transcendent curve, whose equation cannot be given in finite terms. its minimum parallel is one of the diameters of the bent circle: it could also be any geodesic chord, but then the surface would not be symmetrical on both sides" then he says that a few lines are marked, corresponding to some "diameters" and some "horocycles, who, despite having the center at infinity, have a very strong curvature: actually, they're made of little paper arcs with a radius of 25 centimeters, which is the same as the radius of the surface. The total perimeter is more than 6 meters, and its geodesic circumference is made of little paper arcs with a radius of 24 centimeters each". he also says that he was thinking of writing a paper to accompany the model, but apparently he never did. and he adds that he add "some ideas about how to make a better model, with different means and materials". so, the model is actually a piece of pseudosphere, but it's not clear whether he thought that it could be extended to the whole hyperbolic plane. however, beltrami got his two models (the so called Klein and Poincare' models) by extending to the whole circle the geometry of the horocycle corresponding to the pseudosphere."

At the end, of course, there was a question now for all of us: "after all this, what do you think? is it the same model as Thurston's, or is it something different?" Who else than Bill Thurston himself can answer this question:
"The photos look like they're constructed differently, from the region the graph of e^x and e^-x up to some scaling choice on the x and y axes, but the description sounds like it's exactly the same construction with annuli. I wonder if the paper models that Beltrami sent to Cremona are different? In any case, from the translation, it seems clear Beltrami had the idea."

Pictures of Beltrami model and his letter are used here with the kind permission of the Department of Mathematics, University of Pavia. Prof. Odifredi's forthcoming book (volume 1) will be in Italian, in English you can read his book (with a foreword by Freeman Dyson):
The Mathematical Century: The 30 Greatest Problems of the Last 100 Years

This story once more reminded me about the importance of finding the origins of the ideas, going back to the original sources instead of quoting somebody who heard it from somebody else etc.

Tuesday, July 20, 2010

The man who refused a million dollars

since I wrote about this earlier I decided to post this:

from Washington Post, July 19, 2010
by Svetlana Smetanina
Is Grigory Perelman a symbol of the new Russia?
What everyone has been talking about for so long and couldn't quite believe has now finally happened. The Russian mathematician Grigory Perelman has officially refused the Millennium Prize of $1 million awarded him by the Clay Mathematics Institute in Cambridge, Mass., for proving the Poincaré conjecture.

 Perelman's explanation for this refusal was as surprising as his actual refusal. He disagrees with the decision of the mathematics community: "I do not like their decision, I consider it unfair," he said. "I consider that the American mathematician [Richard] Hamilton's contribution to the solution of the problem is no less than mine." What Hamilton thinks about this is not yet known.

Since March, when the prize winner was announced, people had been guessing whether Perelman would take the money or not (he had earlier refused a prize of $10,000), and if he did take it, then what he would spend it on. Any number of people wanted to help him in this less-than-simple matter. The most forthright turned out to be the communists of St. Petersburg, Perelman's native city. They wrote him a letter containing a detailed plan of action. First, said the communists, you need to quickly take the money, preferably with interest. Second, invest it in the construction of a scientific center to educate children from low-income families. Third (our personal request), contribute $100,000 to the Lenin's Tomb Fund.

The Russian government also had something to say about "Perelman's problem." Vladimir Putin, in a speech to Russia's academicians who are always asking him for money for science, suggested that they follow Grigory Perelman's example. "We try to help him at least in some way, but he won't take even our money," the prime minister said with pride. 

This hitherto unknown mathematician has suddenly become incredibly popular. The public has asked that he be made an honorary resident of Petersburg. Viktor Vekselberg, head of the high-tech Skolkovo Project, has asked Perelman to be on the innovation city's advisory board. Some people are even trying to befriend the reclusive mathematician. For example, fellow Petersburger Sergei Mironov, Speaker of the Federation Council. Like the communists, he wrote Perelman a letter full of compliments and flattery and asked to meet with him to discuss problems of science. Who knows, Perelman and Mironov may indeed have something to talk about. Mironov, too, has refused money. Judging by his income declarations, he is the "poorest" of all of Russia's governors. He literally lives from paycheck to paycheck. Even so, Perelman did not write him back. 

Perelman's widespread popularity is easy to explain. His having proved the PoincarĂ© conjecture clearly has nothing to do with it. Most Russians have no idea what that is. What they like is Perelman's stubborn refusal to take the money. The money is being pressed on him, but he still won't take it. Even now, in this most materialistic of post-Soviet times when everyone seems to be screaming in your ear: earn money, get rich, spend it, get all you can out of life! "I don't need anything, I have everything I need," Perelman explained to journalists through the closed door of his apartment. Yet the neighbors say Perelman lives not just modestly, but poorly. 

One could, of course, chalk all this up to the eccentricity of a mathematical genius. But a former colleague at the institute where Perelman worked until 2005 put it this way: "He is exceedingly punctilious. Sometimes he would see violations of moral codes where, in fact, there were none." The mathematical community considers that Perelman solved the problem, but he says that Hamilton deserves the prize. We, his contemporaries, feel sure that a million dollars is the equivalent of happiness in life and that one shouldn't refuse such presents. But Perelman has a different opinion. Do you catch his meaning? He simply has other criteria concerning what is moral and what isn't, what is correct and what isn't. It seems that he doesn't just see "violations of moral codes where there are none," but sees more than all of us put together. Perhaps that is what helped him solve the "unsolvable" problem. 

Modern-day Russia has a complicated relationship with money. Life is bad when you don't have any. On the other hand, when you do have money and especially when you have a lot of it, life, for some reason, is still not good. Take Russia's beloved and wildly popular sports: soccer and hockey. The government invested huge amounts of money in both. As a result, the coaches got a million euros, our soccer players became millionaires, to say nothing of our hockey players. And what happened? Our soccer team didn't make it to the World Championship, while our hockey team lost the World Championship and the Olympics. It turns out that in addition to top salaries, our players need to have something else as well--in their hearts or in their heads. I don't know what it's called, but you obviously can't buy it. 

So what does Grigory Perelman have to do with this? Everything. He seems to have that mysterious "something," and it's worth a lot more than a million dollars. He knows why he's here and what he needs. What are all the temptations of the world to him compared with that knowledge?

The mathematician William Thurston had this to say about Perelman's refusal of this valuable prize: "I am filled with a deep sympathy and admiration for his inner strength and purity, for his ability to be true to himself. We learned from Perelman the mathematician. Perhaps we would also do well to think about ourselves and learn from his attitude towards life."

Monday, July 5, 2010

My (mathematical) trip to Ireland

I was invited to give a talk in Dublin, Science Gallery on June 2. It was in connection with Hyperbolic Crochet Coral Reef exhibit there.
The invitation to go to Dublin actually first time came already in January but then some funny things happened with scheduling for the reasons not known to me. Finally I offered to stop in Dublin on my way to Riga where I was to give an opening keynote for the European Conference Textures. There were two reasons I wanted to go to Dublin - I learned that there are some of my works on display there (not mentioned in official exhibition booklet), but most of all I wanted to meet with Irish Reef enthusiasts - and I am very happy that we all met and connected. My mathematical models that are in collection of The IFF were on display in so called Math Chapel (strange name must say for the place where people could learn about hyperbolic geometry and make their own models but of course - titles are up to organizers of the exhibit according to their vision).
There was also my piece "The Land and the Sea" exhibited - (yes, the same one which is at the top of the blog).That was not credited at all, at least I could not find anything that mentioned it. The day I arrived in the Science Gallery the piece was missing from its display place. I was told that the director has it in his office since he took it with him travelling to Helsinki where he received a prestigious award for the Science Gallery. May be I should have felt flattered that my piece had traveled to Finland, but I doubt that it was credited properly there. May be that was one of the reasons why the director was too busy to talk to me, even so busy that had no time to say simple hello.
I guess the missed display time on the wall was made up for my piece to be the last one taken off when the whole show was over on June 11.(Thanks, Irene, for the picture, I borrowed it from your blog.). Thanks to all who came to see me and thanks for the chat before and after my talk - it was such a great energy and joy around us!

We had great time in Dublin. We had nice talks with the gallery volunteers, were well taken care by Beth and Lynn who had organized my talk.

I was happy to reconnect with my models that were on display and use them for my talk adding a nice model of the hyperbolic soccer ball made in the gallery by one of the guides and I had some of my own stuff with me.

The hotel David and I were staying was close to Trinity College and Science Gallery. It was a little surprise first to find out that the hotel is actually in the fire station - fire trucks and ambulances were stationed right outside our windows. Amazingly - we could sleep well (no, no, Guinness has nothing to do with that!).
The day we arrived first we had to have a nap because of time difference between NYC and Dublin, then wee walked to the Gallery - the exhibit looked wonderful, I could see how much love Irish crocheters had put in their work. Also the gallery itself is a  great place for exhibits.
From the gallery we walked to the Old Library to see the Book of Kells. I have always been fascinated with old books and manuscripts but this is the most beautiful piece of art and is really amazing. It was also great to have an opportunity to visit the Long Hall. (It is not allowed to take pictures in there, so I posted an image from website.) David and I tried to imagine David's great grandfather coming here and reading around the turn of 20th century.

The other place in Dublin connected with David's great grandfather is Christ Church Cathedral. From here he was forced to leave because of the disagreement with the church, his writings were banned, and he emigrated to Montreal, Canada, where he is commemorated for his service behind  the altar of St. Patrick's Basilica.  (coincidence - we visited that place when I was giving my previous public lecture last November.)
By the way Jonathan Swift was once a dean here.

Christ Church Cathedral was an interesting place to visit also from the mathematical point of view - it has a lot of interesting tiles.

When I was teaching history of mathematics class for years I was telling my students that William Rowan Hamilton discovered quaternions while walking across the bridge in Dublin (according to the sources I had). Now, of course, Wikipedia gives more precise description. Anyway, finally I was in Dublin and had to find this magic bridge that gave such a divine inspiration. I liked Ha'penny bridge, but it would not fit because it is an iron bridge and Hamilton could not inscribe the famous equation on it. On the Internet I found the name of the bridge - Broom Bridge but none of the bridges in the center of the city (close to trinity College) would have such name.
While walking around we found some anchor rings - during my talk I mentioned them when showing my model of crocheted hyperbolic pants - both are the examples of double hole torus.

The Millenium sculpture picture can be used talking about Euclid's Parallel Postulate :-)

The other nice use of the mathematical form was this pyramid as a memorial to Irish soldiers in Merrion Square Park.

So the day after my talk we were determined to find the Broom Bridge. Google maps gave us a location - it has to be on Broom bridge Road. make sense. We found the bus that goes to the right direction. A bus driver did not know what we were talking about, so we kept looking for Royal Canal.

After crossing it several times we decided it is time to get out. There was a wide open space with new developments nearby and some bridge straight ahead.  We walked there.

From the pictures of the plaque I knew it had to be on the side but this bridge did not have any, also the road crossing it was not the right one. Few passerby there to interview, and the few we asked did not know who Hamilton was. Of course, when we explained that he was famous physicist and we are mathematicians, people  felt safer to leave us alone.

At least we figured out that we have gone too far and started to look for the bus stop. This time we were lucky - the bus driver knew where the Broom Bridge is.

He was a little suspicious about us wanting to visit the place, and we enthusiastically started to tell him a reason why we were interested to find the place.
The driver said nothing, stopped the bus between bus stops to point out exact direction of the bridge. His politeness seemed to me bordered with certain degree of sympathy.
When we reached the place I could clearly see why - for sure I would not be comfortable walking there alone and I was happy that two of us went there in bright daylight.

To get to the canal, we crossed train tracks but there was nothing on a bridge indicating wonderful mathematical discovery.
We used the overpass to get on the other side of the canal - and here it was! At that point I realized how important it is to go back to the sources - Hamilton was not walking ACROSS the bridge but Along the canal UNDER the bridge! Of course, now it is possible to find more tuned story on Web but - sorry my former math history students!

Somebody might hope that in the place like this it is possible also to come up with some brilliant idea yourself. I looked on both sides of the bridge, and the only thought I could have was - why there are well kept up memorial places for writers (in Dublin particularly - Oscar Wilde, James Joice, Jonathan Swift, George Bernard Shaw, WB Yeats,...) but you really cannot find such places for scientists? In Dublin tourist map there is only one mention of Sir William Rowan Hamilton - the house where he was born. We were not the only ones who tried to look for the Broome Bridge. I just found this account about a group of mathematicians searching for it in 2004. Comparing pictures I have to say that there has been an improvement in this place - at least there was no graffiti this time. 

But may be I am unfair wishing that place would be better kept up. One of the most interesting things to see in highly recommended Dublin City Gallery The Hugh Lane is Francis Bacon's studio. He was quoted to say that he cannot be creative in a neat and tidy place. 

We did not have enough time in Ireland because we had to go to Riga. We hoped to have more time to spend exploring Ireland on our way coming back but things turned out such that we had only one more day - June 29th.

That was a day we went to see Newgrange, the World Heritage site. I learned about this site once researching on spirals, when they first appeared as human symbols. And this is one of the oldest sites - built about 5000 years ago. It is not allowed to take pictures inside the passage. But it is indeed amazing to see how well built this place is. We were walking around it and thinking what amazing skills and knowledge those early humans had. Life expectancy at that time was around 30 years of age. So the place was built during several generations.  

The passage and chamber of Newgrange are illuminated by the winter solstice sunrise. A shaft of sunlight shines through the roof box over the entrance and penetrates the passage to light up the chamber. The dramatic event lasts for 17 minutes at dawn on the Winter Solstice and for a few mornings either side of the Winter Solstice.

To have such precision there had to be previous observations. Also stones are gathered from different places, some even as far as 80 km, so they had to know where those stones are and how to transport them.

Here I am next to the mysterious spirals that nobody really knows what they mean. Our tour guide gave different possible versions people have come up. I would vote for the one that said that this stone represented a map of the area where three interlocking spirals mean three mounds -  Newgrange, Knowth, and Dowth, diamond shapes are fields, but wavy line on the bottom is Boyne river.

There are more of these signs on stones that are around the hill. They certainly felt to me not as ornaments but containing some important information those early humans wanted next generations to know.

  May be we all have gone too far in our technological world that we do not know anymore how to understand and translate basics? 

In one of the talks in Riga conference, a presenter quoted a line from the autism manifesto:

If I cannot speak that does not mean I have nothing to say     

I was remembering this phrase walking around the mound of Newgrange.

We also went to see some of high crosses of Ireland. This is one of crosses in Monasterboice. This particular cross tells the story in the Bible. Instead of writing the manuscript they just carved it in stone!

The place was built late late 5th century. At that time there were small communities that lived together and built high towers to protect themselves. At that time most of he Europe was burning in war flames, badly damaged by Romans. Many important ancient manuscripts were saved by Irish monks. I always wondered why, and I think here in Monasterboice I saw the answer.
I was looking again on the geometric signs on the crosses and tombstones. 
 There was so much more I wanted to see in Ireland. Good bye - I hope to be back.