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.


  1. Very lovely! Thank you for the pictures!

  2. The model made by Beltrami can be easily reproduced today using flexible Fiberglass or Composite plastic. It is cut out from the lateral surface of pseudosphere, not central Tracticoid variety but hyperbolic pseudosphere.
    Differential parametrization can be expressed using geodesic polar coordinates i.e., as involutes. (I post on Math Stackexchange.. maybe I had posted earlier also as Lanther? ..) Narasimham G.L. 26 Sept 2019