Monday, January 26, 2015

Mathematics as a form of art...

I came upon with this thesis as I have been mostly obsessed with mathematics day in and day out. I have worked in various technical fields such as electrical, electronics, computers and mechanical but throughout, I have always been involved in tweaking mathematics. One of the key things that I personally felt was that at the end of the day, whenever I derived something such as a formula result, at the end, I tend to beautify the same by aligning the different part of the equations. Even in the book if one sees, most of the final results are beautified and this also helps in remembrance of the same. This is much common activity in the physics textbooks, where one has to remember a lot of formulas. Further, most of the questions asked in a mathematics paper are so much so beautified to create patterns out of it. Even the solution can be beautiful. One of my mathematics teachers was very much obsessed with questions that are beautifully solved. He used to compile all these solution in a file. A great example here would be puzzle. What exactly puzzle is? – It’s an art on building blocks of mathematics. All these thought process along with my readings on art appreciation class led me to a question – ‘Is Math Art?’ Answer that I finally discovered to this open ended question was – ‘Of course it is!’

What can drew one to think is the idea that ‘Is all a tautology?’ i.e. A implies B, B implies C to C implies A. If it's all logic, there really shouldn't be anything to understand. The early one figures out that it's not logic, the better it is. What keeps one from understanding it and that one couldn't follow the logic is that these ideas are so different and new. One can finally realize that it wasn't all about logic but rather about the creative process. Sometimes it gives a feeling of ‘writing a book’ or ‘providing piece of music’ as one solves a math problem, because it's not about the tools, everyone knows about the tools, it's about how one put them together. There's a little bit artistry involved and that's what drew one to it. One gets drawn into geometry just because it really emphasizes the ‘visual’ aspect as well as the ‘aesthetic’ aspect. One honestly can find a kind of beauty in thinking creatively about problems because somehow something doesn't click until two dollars. There's a little bit of mystery to the creative process. What exactly clicks in your brain that takes you from not being able to solve problem to solving it. One really enjoys their thinking about beautiful problems. Mathematics is known for the use of aesthetic terms in describing their way of work. Nobody can deny the fact that beauty lies in the eyes of beholders. A lot of people, irrespective of whether they are from a mathematical background or not, they seem to appreciate the state of art that mathematics provide. Vi Hart, a self described ‘recreational mathemusician’ from National Museum of Mathematics performed dance with music by representing each figure as a digit. She represented a series of numbers using the powers of 2. She also danced for the value of pi in binary. Finally, she made a painting out of that hand dance by having different color for her different fingers. Scott Kim is an artist and use mathematics as his language. He is the author of many American puzzles and has also designed many computer games. He describes his work as the relationship of mathematics with various shapes and sizes.

Mathematics is as valid a form of art, as any other. Now it's difficult to put into words, exactly what it is that makes numbers and symbols so appealing - but according to scientists, it boils down to simple brain chemistry. In a recent study at University College in London, researchers showed a group of mathematicians 60 different mathematical equations, and asked them to rate those equations on a scale of "ugly” to "beautiful," while inside an fMRI scanner. The results showed that the more "beautiful" an equation was - according to the test subject - the more likely it was to elicit activity in the A1 field of the medial orbitofrontal cortex. This is the part of your brain that's typically associated with emotional responses to visual and musical beauty. Thus, people like us respond to numbers and equations, the same way other people do to music or art. But even people with no musical talent can still appreciate good music - so what constitutes beauty in math? And does the appreciation of it require some understanding of its meaning? Well, not necessarily. To test that idea, researchers performed the same study on a control group - with no special appreciation of math. And while those subjects did show a significantly lower emotional response to the equations - a handful of them were still capable of finding their beauty - even with no understanding of what they actually mean. So what makes an equation "objectively beautiful"? Is it just a formula of curves and shapes, maybe symmetry that makes it pleasing to the eye? It's difficult to quantify the exact reasons, but there is one equation consistently rated to be the most attractive - and that's Euler's identity (1+e=0) – perhaps because it contains the 3 most fundamental numbers in the mathematical universe, e, π, and i. It's a pretty hot equation. What's the ugliest, if one asks, according to the mathematicians in this study, it's the Srinivasa Ramanujan's rapidly converging infinite series of π.

                                            Figure 1                                                                             Figure 2

Data visualization is a way of representing information and an artistic and interesting way. Scientists might be happy with scatter plots in black-on-white laps, but if one wants to communicate information and data to the public, he or she would want it to be enjoyable as well as colorful. To quote one of the chief icons of such form is Martine Kaminsky. It is quite interesting to look at his artistic work, finding beautiful artistry in the randomness of pi. His painting as seen in Figure 1 is pi in a beautiful colorful way. What make it so special is that, it is done in the simplest way one could have think off. He takes the digits of pi and gives each digit a different color to start with three. Three is an orange color there, then one is a red, four is a yellow, one again is red and then five is green, nine is purple and so on. it makes a really beautiful poster, one can kind of see the randomness of it, but one can’t see it as any particular pattern to the colors; which is which reflects the randomness of the digits of pi is well. To take that further, he started to kind of make the center of the circle using the color of the next digit. Probably he did so, just to make it a bit prettier at produced. He might have found it interesting because he had to join up digits that had the same colors in them. In a research like this, they care to go a bit further and in this art, she connects digits adjacent digits in his poster that have same color, so actually they are the same number, which makes the disconnected networks. One doesn’t need to use any great mathematical truth defined underneath this. This is a thought and Martine is very clear about himself that this is less about the mathematics and more about the beauty of it. But it is intriguing as well. The other way one could do this is if he put the digits of pi in a spiral as seen in the Figure 2. This looks similar to tiling in a Roman bath house. Something that makes one happy, maybe that's because of the colors she used. But to take this a step further is one of Brady's Patriots. It makes us think that we should just appreciate the beauty of that before we describe it and not merely bringing the beauty of it by describing it and not doing the justice to appreciate it. Now what he's done there is connected the digits together as you go through the digits of pi. So he started at three and he has connected three to one and then connected one to four and so on. He has also given each digit a color as well to a psychopath. He's made a package by putting numbers in a circle zero one two three four five up to nine and one have made a path and makes this beautiful piece of art. About data visualizations, which is hugely important, it goes way back! Florence Nightingale once had to represent the statistics that she had in Crimean War, where she was a nurse and the deaths that she was suffering on the wards. It was because they weren’t clear enough and she had to represent it to the state. She was a mathematician herself and she used diagrams for representing the data in a visual way. A simpler example is a pi graph representing data in a visual way. What these guys know and has become quite a thing recently is to make it beautiful as well. This is important because to communicate especially to the general public, one should be able to look at the information, understand it and enjoy looking at it and then only one appreciates it better. While talking about serious mathematics subjects like combinatorics, which is very visual as well and one can and do the maths through pictures, diagrams, networks, graphs and things like that. In the present example, we saw how pi is pretty random. Martine actually compared pi with a few randomly generated numbers. She generated all sorts of ways to randomly generated numbers and did the same sort of artistry and one can see the similar sorts of patterns coming from pi has the appearance of any of the randomly generated number. Another piece of art by Martine is the same idea again i.e. circular paths, but here there is little dots on the outside just to show that where the lines are coming from and where they going to. If it's 3.141, then above the three, is shown in with a little dot that it's going to the number one; and above the number one shows that it is coming from the number three. It's basically showing where it is going to and where it is coming from. So the size of the dots means that it occurs more often. There is a big purple-topped in the piece. Purple means nine and what it is showing is that there's a sequence in pie where there is just 9999, called the Feynman point, which is eighty six consecutive nine's, which is 762nd digit in pi. One can see it right there in the piece which are just six consecutive nines and it makes a big blob because it's coming from 9 repeatedly. Martine has done this with other numbers as well. He has done it for the golden ratio, another famous mathematical number, also e, another famous mathematical number and what he was interested in was to find out where they coincide. When he did this, he lined up these three special mathematical numbers and looked at where they had the same digit. Now, because these numbers are kind of random, he wanted to find out when they have the same digit. So, for all three numbers to have the same digit that happens with the probability of one in a hundred, when one line them up he got something, which he called the accidental similarity number and he discovered that yes they do lined up with the same digit about one in a hundred which is what probability tells us. So the average gap between linings up is about 100. He started to experiment with his accidental similarity number and made some piece of art based on that as well especially circular ones which look pretty.

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