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Archive for the ‘Vision’ Category

This post is of general interest.

I was reading Prof. Alexei Sossinsky ‘s coffee table book on KnotsKnots: 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: A Wild Knot

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

1. Start with a solid torus say T_1.

2. Place inside it four smaller tori linked two by two to make a chain. Let’s call this chain T_2.

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 T_3

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

A = T_1 \cap T_2 \cap T_3 \cap \dotsb

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 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).

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.

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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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I would try to get more systematic about my posts from now on. For every two non-technical posts I would keep two technical posts.

This post would also be the first in a series of posts that in which I intend to write about some Visual Illusions only.

Before getting into subject of this post, it would be helpful to have a quick recap of the background.

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The Blind Spot:

Consider a horizontal cross section of the human eye as shown below.

HorzontalSectionOfRightEye

As seen in the above, the innermost membrane is the Retina, and it lines the walls of the posterior portion of the eye. When the eye is focused, light from the focused object is imaged onto the Retina. It thus acts as a screen. Pattern vision is caused by the distribution of discrete light receptors called rods and cones over the retinal surface.

Each eye has about 6-7 million cones, located primarily in the central portion of the Retina and they are highly sensitive to color. Humans can resolve fine details with cones as each cone is connected to its own nerve end. The vision due to cones is called Photopic or bright-light vision.

The number of rods is about 75-150 million andare distributed throughout the retina. The amount of details that can be resolved by rods is lesser as several of them are connected to the same nerve unlike in the cones. Vision due to rods is simply to give an overall picture of the field of view. Objects that seen in bright day light appear as color-less forms in moonlight as only the rods are stimulated. This type of vision is called Scotopic or dim-light vision.

As seen in the figure there is a portion on the retina which has no receptors (rods or cones), thus will not cause any sensation. This is called the blind spot.

Now because of the blind spot a certain field of vision is not perceived. We however do not notice it as the brain fills it with details from the surroundings or using information from the other eye.

The blind spots in both the eyes are arranged symmetrically so that the loss in field of vision in one eye will compensate for the other. This is shown by the figure below.

illustration-blind-spot[Image Source]

If the brain would not fill the lost field of vision with surrounding details and information from the other eye, then the blind spot would appear something like the black dot on the image below.

Blind Spot view

[Image Source]

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Now that means, if you close one eye then you can indeed detect the presence of the blind spot as the brain would not have sufficient information about the lost field of vision (though it would be good enough for us to not notice it normally). The presence of the blind spot can be demonstrated by the simple figure below.

Demo of Blind Spot

Click on the above image to enlarge

Now enlarge the above image and close your right eye and focus your left eye on the X only. Don’t try to look at the O on the left. You’d just notice it at the periphery. The object of interest should only be X.

Now move towards the screen, at a certain point you will not see O in the periphery. If you go ahead of this point or behind it you’ll see O again, this specific point (a range actually) where you can not see O indicates the presence of the blind spot.

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The Vanishing Head Illusion:

This leads to some interesting illusions, one of the most interesting being the so called vanishing head illusion.

As in the above figure. If the O is replaced by a head, the person would appear headless if the head falls on the blind spot.

Check the video below in full screen for best results.

View in Full Screen

We notice that Richard Wiseman on the left indeed appears headless and that field of view is filled up by the orange background when the blind spot falls.  Then he does something even more interesting. He uses a black bar and moves it up and down in front of his face.  Now instead of seeing the bar as discontinuous, the brain manages to show the bar as a continuous entity!

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