I thought I understood Spectral Clustering well enough till I came across these two paragraphs:

Graph Laplacians are interesting linear operators attached to graphs. They are the discrete analogues of the Laplace-Beltrami operators that appear in the Hodge theory of Riemannian manifolds, whose null spaces provide particularly nice representative forms for de Rham cohomology. In particular, their eigenfunctions produce functions on the vertex set of the graph. They can be used, for example, to produce cluster decompositions of data sets when the graph is the 1-skeleton of a Vietoris-Rips complex. We ﬁnd that these eigenfunctions (again applied to the 1-skeleton of the Vietoris-Rips complex of a point cloud) also can produce useful ﬁlters in the Mapper analysis of data sets

– From Prof. Gunnar Carlsson’s survey Topology and Data. (More on this survey as a manifesto for “Topology and Data” in a later post). That aside, I do like how the image on the wiki entry for Vietoris-Rips complex looks like:

A little less intimidating ( now this almost borders on “ofcourse that’s how it is”. I am interested in the same reaction for the paragraph above some months later):

A related application [of the graph laplacian] is “Spectral Clustering”, which is based on the observation that nodal domains of the first eigenvectors of the graph laplacian can be used as indicators for suitably size-balanced minimum cuts.

– From Laplacian Eigenvectors of Graphs linked in the previous post. While this isn’t really as compressed as the lines above, they made me think since I did not know about Courant’s Nodal domain theorem. Like I did in the previous blog post, I would highly recommend this (about 120 page) book. It soon covers the Nodal Domain theorem and things make sense (even in context of links between PCA and k-means and Non-Negative Matrix Factorization and Spectral Clustering, at least in an abstract sense).

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)

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).

Lev Pontryagin (1908-1988)

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.

Euler (1707-1783) on a Swiss Banknote

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 AMSarticle 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).

Morin’s Surface

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.

The Mind’s Eye by Oliver Sacks (Click on image to view on Amazon)

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.

One of the favorite maxims of my father was the distinction between the two sorts of truths, profound truths recognized by the fact that the opposite is also a profound truth, in contrast to trivialities where opposites are obviously absurd.