In all probability you have already heard about this famous phenomenon called the blind spot in our eyes. Whether you have or not, try this. It is a beautiful and simple demonstration of an amazing fact about your eyes. Given below is a picture.
Now close your left eye with one of your hands, and look at the red diamond on the left. Nothing strange so far. But now, start moving towards the screen, all along keeping your left eye closed and focusing on the red diamond. At some point when you move towards the screen, the blue circle on the right will disappear. The key is to keep your eye focused on the diamond. Did it disappear or not? If not, try again. You will get it the second or third time.
You can also make the diamond disappear, by closing your right eye and focusing your left eye on the blue circle. Now the diamond will disappear as you move closer to the screen.
This disappearance is a beauty. It is almost unbelievable. How many ever times I see it, I am not bored. You can also play around a bit, when you hit the blind spot, by shifting your focus alternately from one object to another. The blue dot is definitely there, but why is it hidden from us?
The Retina is that part of our eye which receives light from the outer world. This has the same function as that of a film in a film based camera. In a digital camera, this function is performed by what is called the CCD or CMOS which is, again, light sensitive, and converts the light received into electrons and charges. Based on the light that the retina receives, a series of chemical reactions happen which results in our brain perceiving the image. The paths through which these messages travel from the eyes to the brain are called the nerve fibres.
Now imagine how a digital camera would be designed.
The light rays(red lines) come and hit the “front” side of the CMOS which is light sensitive and the wires (pathways), represented by green lines, will be on the “back” side to carry the converted messages to the processing device and finally onto the memory card. This seems simple and it is natural to expect this in our eyes too.
But here is where evolution plays a joke on us. Our eyes have evolved in an unexpected way. The pathways that carry the messages from the light sensitive device (our retina) do not originate from the “back” side of the retina, but from the very “front” side where light rays hit initially. So at some point it has to go “back” through the retina to take the messages through the brain. If that is confusing, look at the picture below, which shows how a camera designed in this unexpected way would look. The light rays come and hit the “front” side. The pathways that carry the information come out from the same side, regroup, and then pass through the light blue circle along the vertical line so that it can go “behind” the device to transmit information to the memory card.
Yes, it is terribly messy. In our eyes, brain pathways are on the same side as the light sensitive side, and hence it has to go back through the retina, at some point. And at that point (analogous to the light blue circle in the pic above), light sensitive cells do not exist. Hence light that comes in that direction is lost. That is why the blue circle or the red diamond suddenly goes missing.
In the picture above from Wikipedia, a vertebrate eye, and an octopus’ eye is shown. The part numbered 4 in the left half of the picture shows the blind spot. That is the point where the nerves go back through the retina to go to the brain.
But look at the octopus’ eye. The light receptors are on the front, and the nerve fibres are at the back. This is how an eye should ideally be. Our eyes are imperfect. But we still manage. This is one of those quirks of evolution, like the laryngeal nerve, I mentioned sometime back. The blind spot does not cause us much harm, since we have two eyes, and I also remember it being mentioned that, our eyes wiggle a bit to compensate for this blind spot, though I am not sure where I read/heard it.
The blind spot itself is very interesting, but there is another curious thing about the little experiment we did. When the circle or the diamond disappears, you don’t see a hole in your vision, do you? What you see there is the green background. That is surprising. If no input goes to the eyes at the point where the blind spot exists, where did the green come in from? The answer is that the brain fills it up. It is now well known that what our eyes see is not always what our brain perceives. The brain does a lot of filling up and interpreting of what actually enters our eyes. If you dont trust me, look at these illusions on the website of the neuro-scientist V.S.Ramachandran, who has written a fabulous book, “Phantoms in the brain”, which talks about, among other things, phantom limbs, where people “feel” non-existent limbs. It is a must-read.
While on the topic of illusions, let me finish the post, with this incredible illusion that I was introduced to by the thalaivar, Richard Dawkins
Wasn’t that mind blowing?
If you are mathematically challenged, please don’t panic. This piece is indeed about prime numbers, but my intent here is not to give a proof of some theorem, but only to discuss a couple of interesting points, one each from Physics and Biology, that greatly increased by admiration for prime numbers.
Prime numbers are those numbers that cannot be divided by any number other than 1 and itself. Thus, examples of prime numbers are 7, 11, 13, 17 and so on as they cannot be divided by other numbers, whereas numbers like 12, 14, 15, 16 are all composite (non-prime) numbers.
Suppose you are asked to list out all the even numbers, you will immediately reel them off, 2, 4, 6, 8 and so on till you are asked to stop. If you are asked for the list of squares, you will again start listing out squares 1 (1^2), 4(2^2), 9(3^2), 16(4^2), 25(5^2) and so on. It won’t be as easy as listing out even numbers, but you clearly know how to proceed, and given a calculator/computer, you can go on producing perfect squares, theoretically, forever.
Can you do the same thing with prime numbers? The curious thing about prime numbers is that you cannot keep producing them the way you produce even numbers or perfect squares. They don’t follow a pattern. This might not sound very strange, but it is indeed so. Think of it. Can you think of any property of numbers, that you cannot express as a pattern? It is tough. In fact, generating genuine random numbers (numbers without any pattern) is one of the toughest problems in computer science. To be sure, you can definitely look at each number, try dividing it by all numbers less than it and if it is divisible by none, then you can conclude it is a prime number and not if otherwise. This is the simplest way to do it, though you can find many more efficient algorithms on the Internet. But this process cannot go on for long, since with big numbers, it becomes increasingly tough.
Now if you wonder why this inability to easily generate prime numbers is important, how would you react if I said prime numbers could be a means of communication with extra terrestrial intelligence (ETI)? This idea is that of the great science communicator Carl Sagan. Prime numbers are a candidate for alien-human signaling because no naturally occurring phenomenon can be complex enough to generate prime numbers. Imagine yourself receiving some radio waves from somewhere in space and that they follow a pattern of increasing frequency of consecutive even numbers. Could this be a sign from some intelligence in space? It could be. But the fact that generating even numbers is a simple task greatly increases its probability of occurring naturally, without the need of an intelligent being. Thus the chances of some intelligent brain being the origin of those waves are reduced. But imagine if these waves were of frequencies that are in sync with prime numbers. Because there is no obvious pattern that could generate prime numbers, it would be very tough for nature on its own to produce prime numbers one after another. Such a signal has a very high likelihood of being able to be produced only by an intelligent source. Carl Sagan used this idea in his science fiction novel Contact, where ETI contacts Earth by listing out the series of first 261 prime numbers.
Fascinated as I was by Carl Sagan’s idea, I found Cicadas even more wonderful. Cicadas are plant eating insects. If anything can claim to have generated prime numbers naturally without a thought process, Cicadas would be it. There are numerous species of cicadas, most of them having 2 to 8 year life cycles. But there are some species of Cicadas which have life cycles of 13 and 17 years. That is, these cicadas lie underground as larvae for 12 years (or 16 years), and in the 13th year (or the 17th year) come out as adults, mate, lay their eggs, and disappear. They are so well synchronised, that scientists can predict beforehand when their outbreak will happen and give a warning, so that farmers, among others, can take suitable preventive measures.
Considering the topic in hand, you will see that these two species are special not because their life cycles are longer than usual, but because the lengths of their life cycles are prime numbers. Many biologists have suggested that the life cycles being prime numbers helps the Cicadas, since any predators having a shorter life cycle than these will have lesser likelihood to prey on them. To make that clear, imagine a 15 year life cycle cicada. If its predator has a 3 or 5 year life cycle then the outbreaks of predator and prey may coincide and the prey (cicadas) might go extinct soon. But if it is a 17 year cicada the outbreaks of the shorter life cycle predators, say of length 3 years, has a lesser chance of coinciding with the outbreak of the prey (17 is not divisible by 3, or for that matter any number lesser than 17).
There is no conclusive proof in favour of this hypothesis. In fact this hypothesis leaves many questions unanswered, but the fact that there are 3 species with 13 year life cycles, and 3 more with 17 year life cycles, seems to suggest that there is indeed some advantage in prime numbered life cycle lengths. Some Mathematical models of insects have also suggested that there could indeed be an advantage for prey species to have prime number life cycles. By the way, just to be clear, I am not suggesting that Cicadas do number division in their heads, like we do, and find out prime numbers. If true, it would only mean that evolution favoured Cicadas which have prime numbered life cycles over those with non-prime number lifecycles. The life cycle length is part of the Cicadas’ genetic makeup. But the fact that prime numbers can come up naturally seems to contradict Carl Sagan’s conjecture, though it is still tough to imagine some natural process producing a long list of the consecutive prime numbers.
Nobody, least of all I, knows if ETI, if and when it contacts us, will indeed use prime numbers to say “Are you there?” (In all probability they won’t, since there are many other values they can use, like the digits of pi, which is, 3.1415925654…. Moreover, for all we know, they might just decide to land here directly). Neither do we know, yet, if Cicadas do indeed have an advantage in having a life-cycle length of prime numbers. But the very possibility that these questions could be connected, and that too in such fascinating ways by the simple concept of prime numbers that we learnt in school, is exciting. Is it not?