*This post is of general interest.*

I was reading Prof. Alexei Sossinsky ‘s coffee table book on** Knots** – *Knots: Mathematics with a Twist** *, *and it mentioned a couple of interesting cases of blind mathematicians. These couple of cases ignited enough interest to publish an old draft on blind mathematicians albeit now with a different flavor.

*(Note that the book has poor reviews on Amazon which I honestly don’t relate to. I think the errors reported in the reviews have been corrected plus the book is extremely short ~ 100 pages and hence actually readable on a few coffee breaks)

Sossinsky’s book gives an example of Antoine’s Necklace:

Antoine’s Necklace is a *Wild Knot* that can be constructed as follows:

1. Start with a solid torus say .

2. Place inside it four smaller tori linked two by two to make a chain. Let’s call this chain .

3. Inside each of the tori in step 2, construct a similar chain. This would be a set of 16 tori. Let’s call this

4. Repeat this process *ad-infinitum*. The set obtained by the infinite set of Tori will be Antoine’s necklace.

Antoine’s Necklace is not a mere curiosity and has very interesting properties. One would suppose that constructing such a structure would require considerable visualization, which is indeed true. However one of the most interesting things about this knot is that ** it was formulated and studied by Louis Antoine, who was blind**. After he lost his eyesight, the famous mathematician Henri Lebesgue suggested to him that he study topology.

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I have noticed (it is a common observation) that it is almost a rule that mathematicians who are blind are usually geometers/topologists. Such a correlation can not be mere coincidence.

Before reading Sossinsky’s book which also mentions G. Ya. Zuev as another influential blind topologist, the two best examples that I was aware of were L. S. Pontryagin and the great Leonhard Euler. Pontryagin is perhaps the first blind mathematician that I had heard of who made seminal contributions to numerous areas of mathematics (Algebraic Topology, Control Theory and Optimization to name a few). Some of his contributions are very abstract while some such as those in control theory are also covered in advanced undergrad textbooks (that is how I heard of him).

Pontryagin lost his eyesight at the age of 14 and thus made all of his illustrious contributions (and learnt most of his mathematics) while blind. The case was a little different for Euler. He *learnt* most of his earlier mathematics while not blind. Born in 1707, he almost lost eyesight in the right eye in 1735. After that his eyesight worsened, losing it completely in 1766 to cataract.

His mathematical productivity however actually increased, publishing more than half of his work after losing eyesight. Remarkably he published one paper each week in 1775 aided by students who doubled up as scribes. It is noteworthy that he is the most prolific mathematician to have ever lived in terms of number of pages published (Paul Erdős produced more papers), becoming one of the most influential mathematicians to have ever lived.

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This excellent (as usual) *Notices of the AMS *article lists a few more famous blind mathematicians. Bernard Morin and Nicholas Suanderson to name a couple. Bernard Morin is famous for his work on sphere eversion (i.r homotopy, many youtube videos on this theme are available, video below).

It is difficult to imagine for *ordinary* people that such work could be done by somebody who has been blind since age six. What could be the explanation for what I atleast consider an extraordinary and counter intuitive case?

Sossinsky in his book talks briefly of what he thinks about it and of some research in the area (though he doesn’t point out specific papers, it turns out there is a lot of interesting work on this aspect on spatial representation in blind people). He writes:

“It is not surprising at all that almost all blind mathematicians are geometers. The spatial intuition that sighted people have is based on the image of the world that is projected on their retinas; thus it is a two (and not three) dimensional image that is analysed in the brain of a sighted person. A blind person’s spatial intuition on the other hand, is primarily the result of tile and operational experience. It is also deeper – in the literal as well as the metaphorical sense. […]

recent biomathematical studies have shown that the deepest mathematical structures, such as topological structures, are innate, whereas finer structures, such as linear structures are acquired. Thus, at first, the blind person who regains his sight does not distinguish a square from a circle: He only sees their topological equivalence. In contrast, he immediately sees that a torus is not a sphere […]”

The *Notices article* has a line: “In such a study the eyes of the spirit and the habit of concentration will replace the lost vision”, referring to what is called as the *Mind’s Eye* commonly (i.e it is commonly believed that people with disabilities have some other senses magnified). Some of the work of the celebrated neuroscientist Oliver Sacks (who I also consider as one of my role models. Movie buffs would recognize him from Dr Malcolm Sayer’s character in the fantastic movie Awakenings) talks of individuals in which this was indeed the case. He documents some of such cases in his book, * The Mind’s Eye*. He also notes that such

*magnification*ofcourse does not happen in all of his patients but only in some fascinating cases.

Here in the video below (many more available on youtube) Dr Sacks describes some of such cases:

I wonder when we’d know enough. For such cases tell us something interesting about the brain, it’s adaptability, vision and spatial representation.

The *Notices* article also cites some examples of famous blind mathematicians who were not geometers, perhaps the more interesting cases if I could loosely put it that way.

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**Translation of the Article in Romanian:**

Geometri Blind by Alexander Ovsov

**Recommendations**

1. The World of Blind Mathematicians – Notices of the AMS, Nov 2002 (pdf)

2. The Mind’s Eye – Oliver Sacks (Amazon)

3. Knots Mathematics with a Twist – Alexiei Sossinsky (Amazon)

4. Biography of Lev Pontryagin

5. Mathematical Reasoning and External Symbolic Systems – Catarina Dulith Novaes

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on July 31, 2011 at 4:51 pm |David CurranI love the story where the guys use a 3d printer to show Morin a new shape related to one he developed

http://emsh.calarts.edu/~mathart/Tactile_Optiverse.html

on July 31, 2011 at 5:03 pm |David CurranActually the really touching photo of Morin feeling the new shape seems to be gone. A small version si still on the shapeways page though

http://www.shapeways.com/blog/archives/297-Touching-3D-printing-story.html

on August 18, 2011 at 4:21 pmShubhendu TrivediI think you were referring to

thislink! These photos are really good. I have written to the author for permission to put one of them here.on July 31, 2011 at 5:17 pm |Shubhendu TrivediThank you very much for the two links!

As an aside: Is the photo on page 3 of the

Noticesarticle the one that you are talking about?I almost wrote a blog post on Morin a couple of years back. But then I did not as I thought I did not know enough of his mathematics. I think now that I know more, the more I am in awe of the man.

As an undergrad the first blind mathematician that I came to know was Pontryagin. The story of how he learnt math through his mother is also very interesting. Very inspiring stories, almost bordering on being unbelievable.

on July 31, 2011 at 9:21 pm |David CurranIt is not that photo but its from the same occasion and set.

While I am talking to you. Have you seen the kinetic sculpture that BMW used to advertise? It is on youtube here http://www.youtube.com/watch?v=81jd9c8WeSw they have balls at the end of a piece of fishing line. the line moves up and down on motors to make the shape of a car. I thought it would be really cool to use a similar system to make functions tactile for the blind.

Not to outstay my welcome but I made an all black tactile rubiks cube that is all black, link here http://liveatthewitchtrials.blogspot.com/2009/04/blind-rubiks-cube-2.html . Each side feels different. If you want it I will post it to you.

David

on August 1, 2011 at 3:06 amShubhendu TrivediThank you for both the links!

I wasn’t aware of the Kinetic Sculpture video nor had thought of a Rubik’s cube for the blind. Excellent stuff!

on August 1, 2011 at 12:31 am |Catarina Dutilh NovaesI was directed to this post by a friend who had read two recent blog posts of mine on blind mathematicians. Since they are on the same topic, I hope it’s ok if I do a bit of self-promotion here:

http://m-phi.blogspot.com/2011/07/mathematical-reasoning-and-external.html

http://m-phi.blogspot.com/2011/07/what-is-it-like-to-be-blind.html

on August 1, 2011 at 3:07 am |Shubhendu TrivediHi Catarina,

Both of your writings are really good. I am linking one of it on the recommendations.

Thanks for sharing! :)

S.

on August 1, 2011 at 7:33 pmCatarina Dutilh NovaesHi Shubhendu,

Thanks! Funny coincidence that we were both thinking about blind mathematicians at roughly the same time :)

on August 2, 2011 at 10:52 am |Shubhendu TrivediIt is indeed funny Catarina,

Though I suppose I am not thinking as deeply as you are. Perhaps just a little bit above being very fascinated.

Nice to find articles by you. I have been reading some of your stuff. Very interesting! :)

on August 2, 2011 at 5:06 pm |Catarina Dutilh NovaesWell. I’m flattered :) But my work is really very ‘philosophical’, I’m (pleasantly) surprised to hear that a ‘real mathematician’ would be interested…

on August 4, 2011 at 5:27 amShubhendu TrivediI’m definitely interested. It is not the

post-modernistkinds. ;-)For the sort of logic-philosophy-foundations work I find this quote very appropriate:

Sans les mathématiques on ne pénètre point au fond de la philosophie.

Sans la philosophie on ne pénètre point au fond des mathématiques.

Sans les deux on ne pénètre au fond de rien. — Leibniz

Without mathematics we cannot penetrate deeply into philosophy.Without philosophy we cannot penetrate deeply into mathematics.

Without both we cannot penetrate deeply into anything.

on August 3, 2011 at 4:42 am |AlisonFascinating. :)

on August 4, 2011 at 5:28 am |Shubhendu TrivediHello there Mississippi Miller! :)

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