Tomrocksmaths.files.wordpress.com



Seeing the Beauty in MathematicsEleanor RulerThe human fascination for beauty can be applied to almost anything. People, places, colours, clothing, music, literature – the list goes on. So it stands to reason that the standard of ‘beauty’ can also be applied to mathematics. Most people would probably laugh if I said that mathematics is beautiful, but then again beauty itself is subjective. What is considered by one person to be beautiful may not be by someone else, or what is considered beautiful in one time, may not be in another time. Just look at the 80s – big hair and mullets were all the range back then, but nowadays most people look back on it with second hand embarrassment, and wonder ‘why did we ever think that was cool?’ When talking about mathematics in itself being beautiful, what comes to mind first, for me, are whole numbers. The number 3 is much more satisfying to look at than 5.6781, probably because of its simplicity, but also because I know it is much easier to multiply 3 by 3, than it is to multiply 5.6781 by 5.6781. It’s why mathematicians opt for fractions over decimals, and why irrational numbers are given their own symbols: it’s easier to calculate with, but also it does have a nice simplicity associated with it. So, could this concept of beauty be extended to mathematical formulas?32639002516505Figure 1 - Einstein's General Relativity EquationFigure 1 - Einstein's General Relativity Equation3314700196405500 The concept of a mathematical formula itself being beautiful is quite a strange one, because it’s not something that people would typically hold to any standard of beauty. However, I’m not the first person to talk about beauty in mathematics, so let’s look at a mathematician called Paul Dirac. Dirac subscribed to the principle of ‘Mathematical Beauty’, believing that all laws of nature could be written mathematically, and if they looked ‘ugly’ they were most certainly wrong. Before, it used to be known as the principle of simplicity – stating that if laws of nature were written simply, then they were most likely correct. However, when Einstein came along with his theory of general relativity, partnered with a new complicated looking equation – which is undeniably correct – the principle of simplicity had to be adjusted slightly. Most mathematicians, Dirac included, believe the general relativity equation to be inherently beautiful. I, a mathematician, would agree with him, although on showing the equation to my sister – a non-mathematician – she most certainly did not regard the equation as beautiful. So we’re back to the same problem, how can we ever regard the principle of mathematical beauty to be correct if beauty is subjective, and what one person may consider to be beautiful, another will not? Then again, humanity does tend to agree on a lot of ‘aesthetically pleasing’ images. Take colour, for example. People have spent their lives researching colour theory, and why certain colours look better together than others. And although an artist may care more about the compatibility of colours than a non-artist, most ordinary people will agree on which colours are more pleasing together and which are not. If I said that purple, blue and pink complemented each other nicely, you would probably agree with me, but if I told you orange, blue and yellow work well together, you probably wouldn’t agree so much. And it’s not just colour; music theory is the same. Certain notes will blend together to create much more beautiful and harmonic sounds, whereas other notes played together will just sound like a loud, uncoordinated noise – like the cat just jumped onto the piano. 33972502387600Figure 2 - TetradecahedronFigure 2 - Tetradecahedron347980057150010922002305050Figure 3 - DodecahedronFigure 3 - Dodecahedron99695064135000Perhaps we can look at this by taking a branch of mathematics that relates more to art and everyday life: geometry. Below is a picture of a dodecahedron, and next to it is a picture of a tetradecahedron. Which one is more satisfying to look at? If you said the dodecahedron, you would be in the majority. This is because a dodecahedron is what is known as a regular polyhedron, or polytope. (A polyhedron is a polytope specifically in 3D space) A regular polyhedron is a shape whose faces are all regular polygons, where all side lengths are equal and all angles are equal. This provides the shapes with the highest degree of symmetry in a 3-dimensional space. Therefore, we must conclude that it is it’s symmetry that makes a dodecahedron more ‘aesthetically pleasing’ to look at than a tetradecahedron. So there we have it, there is a mathematical reason for why polytopes, like the dodecahedron, are more satisfying to look at than non-polytopes. So does this mean that there is a mathematical explanation for why certain mathematical expressions are more beautiful than others? If we go all the way back to Dirac, he suggested that everything might have its own mathematical counterpart. Not just descriptions of nature: everything. He even went as far as to state that someone with a complete knowledge of mathematics may be able to calculate the future. He agreed that, yes, the mathematical calculations would have to abide by a strict set of rules, which would make the formulas unimaginably complicated, and something that we – as a human species – would never be able to comprehend. But complicated doesn’t equal ugly – they may still be subject to the principle of mathematical beauty. So, no matter how complicated the maths gets, it can still be considered ‘beautiful’ – however we are still stuck with our original problem. Beauty is subjective, unlike the complex nature that makes mathematics so undeniably objective. -1397001819910Figure 4 - Quadratic FormulaFigure 4 - Quadratic Formula-13970077025500So maybe there is no mathematical formula for beauty, and I don’t believe there ever can be – not one that we, as a species, could come up with anyway. I will never be able to mathematically explain why I find the quadratic formula so satisfying to look at, when many of my peers would completely disagree with me, especially those who aren’t mathematically inclined. And so, perhaps, much like how a quadratic equation has multiple correct answers, something could be both beautiful and ugly at the same time. It’s all a matter of perspective. Sources:The Relation Between Mathematics and Physics - Paul DiracGraham Farmelo on Paul Dirac and Mathematical BeautyHow Is Math Beautiful? | Quanta Magazine(92) Perfect Shapes in Higher Dimensions - Numberphile - YouTubeRegular polytope - Wikipedia ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download