Where do I find beauty in mathematics

The beauty of math

Transcript

1 The Beauty of Mathematics Thomas Weth The idea that beauty and aesthetics can be at least partially captured and described mathematically may take some getting used to, but has meanwhile become widespread and is widely accepted; One thinks here of measurable proportions, generally perceived as beautiful faces or bodies, of buildings that are built according to the proportions of the golden section, and the like. The description of beauty through mathematics will not be discussed in the following. Rather, the focus should be on the beauty of mathematics or, even more clearly, on the beauty in mathematics. Because as strange as it may seem to a non-mathematician, mathematicians describe many of their theories, propositions and proofs with adjectives that one would otherwise expect in artistic or musical fields, such as B. when speaking of the elegance of a proof, the beauty of a sentence, etc. One of the most famous number theorists of the 20th century, G. H. Hardy, even made beauty a criterion of whether something is good mathematics, because in his opinion the works of the mathematician ... must be beautiful like those of the painter or poet; the ideas must harmonize like the colors or words. Beauty is the first test: there is no place in the world for ugly math. As always, when of beauty, art, aesthetics and others. the talk is, but opinions differ as to whether and why something is beautiful, artistic, aesthetic; a Wagner opera is for some people the highest artistic enjoyment, for others, such as the famous American narrator Mark Twain, it almost causes physical pain. When Lohengrin visited Bayreuth, Twain had already gone through so much before the break that all my spirits were gone and I had only one wish, namely: to be left in peace ... Whether [the German audience] this one At the time, I didn't know whether they naturally appreciated noise or whether they had learned to like it through getting used to it. Naturally, opinions about beauty in mathematics are also divided, because ... as is the case with so many things also applies to a mathematical theory: Beauty can be perceived, but not explained (Arthur Cayley). And to be able to perceive the beauty of mathematics normally seems to be more difficult than the perception of beauty in painting or music, because although mathematics, viewed correctly, ... not only truth, but also 86 1

2 supreme beauty, this is ... a cold and austere beauty like a sculpture, without attraction to any of our weaker sides, without the splendid charms of painting or music, but of sublime purity and rigorous perfection like them can only show the highest art (Bertrand Russell). The attempt to demonstrate the inherent beauty of mathematics to a wider audience with simple means is doomed to failure from the start if the reader closes his sense of clever thoughts, surprising turns of phrase, ... as indicated above. Locking yourself in would be like trying to enjoy a classical concert while covering your ears. Without claiming to be exhaustive, the following examples are intended to make tangible what is mathematically beautiful, whereby the concept of mathematical beauty is to be differentiated into elegance, being moved / amazed, surprising, whereby the boundaries between the categories mentioned or the categories mentioned are fluid Areas partially overlap. Elegance in mathematics Elegance is characterized by outstanding design, sometimes simplicity, restrained artistic expression (minimalism, less is more, e.g. few, selected colors, few clear visual elements, proportions pleasant to the perception) Wikipedia (elegance,) Manage Imagine a competition in which the winner is who can draw a triangle with the largest possible interior angle sum. A first (naive) attempt may be to draw a small and a large triangle and check whether the sum of the interior angles in the small triangle is really smaller than that of the large triangle. 86 2

3 Amazingly, triangles seem to oppose everyday experience: No matter how big a triangle is, the sum of its interior angles always seems to be the same! And as further experiments show: the size of the sum of the interior angles does not seem to depend on either the size of the triangles or the shape of the triangles. In everyday life, this would roughly correspond to the phenomenon that every car costs the same regardless of the power of the engine and its equipment. In the case of triangles, although the few examples suggest an astonishing assumption, it is still a long way from providing evidence that the sum of the interior angles is constant for every triangle. And while looking at a few examples in everyday life is enough for us to come to an approximately accurate judgment (if Bayern Munich wins the first 8 league games, Bayern will certainly be German champions again), mathematics wants to secure all cases. Mathematics wants unequivocal clarity for every conceivable case: No matter how big, no matter how small, no matter how crooked, no matter how: Mathematics wants to prove that the assumption is correct: In every triangle the sum of the interior angles is 180. The problem is thus posed: For an infinite number of possibilities, unequivocal security should be obtained; once and for all, immovable, indubitable, final. It is astonishing that mathematics succeeds in finding a solution to countless such problems. And it is even more astonishing that many of the ideas with which the respective problem solutions succeed are simple, insightful, concise or in short: elegant: In the case of the interior angle problem, three arguments are sufficient. 1. At a straight line crossing, opposite angles are the same size. 2. If you move a straight line h parallel to itself, the angle of intersection with another straight line does not change. With this insightful knowledge, we can now prove the theorem: In every triangle the sum of the interior angles is 180 with a single, skilfully drawn line. You can immediately see that 86 3

4 3. the equally marked angles are the same size and complement each other to form a straight (i.e. 180 -) angle. Being moved by mathematics The elegance described so far is only part of the concept of beauty. Minimalism is not a necessary MUST to make beauty possible. One thinks here of sumptuously furnished baroque churches or of works that are played by large orchestras. In these cases, the beauty opens up more in the form of a feeling of emotion, which makes you take a breath, breathe deeper, make the heart beat faster. In some cases, similar real physical reactions also produce mathematical results. Not with the result of the problem =?, But with the appropriate sensitivity, for example in the following example: It has long been assumed that the circle number π (= 3,) is normal, which means that every finite number combination of the same length (e.g. 7558 or 1234) occurs with the same probability in the decimal places of π. An example of a proven normal number is the Champernowne number: 0, provided that π is normal, this has consequences that arouse and stir astonishment: A first finding is that your personal date of birth appears in the decimal places of π. For sure! Because if you convert your date of birth (roughly) into a number (13382), it has 5 digits and occurs with the same probability in the decimal places of π as any other 5-digit number, i.e. like the first 5 decimal places (those with Safety!). The search engine delivers that occurs or begins at the tenth decimal place in π. 86 4

5 In the exhibition Exclusively ... you will find the following picture: Fig. 1: Richard Paul Lohse, Fifteen systematic color series with vertical and horizontal compression, (cat. No. 50, page 123) A digital photo of this picture in the quality shown requires one Storage space of about 10 kb, so it can be digitized by a sequence of about zeros and ones (e.g.:). So this finite number appears with certainty in the decimal places of π. Somewhere far back, but for sure! And the certainty that π contains the image shown here relates to any photo or image. Every imaginable photo, be it a Michelangelo, a Pollock, your personal passport photo or a picture of your personal first day of school, is contained in the decimal places of π! If you are not yet gripped by this omnipotence of π, the following consideration should convince you: Imagine that a cameraman has filmed every moment of your life since you were born; every second, every breath, day and night without a break, until the moment you read these lines. Your life film is z. B. digitized for a DVD player and is thus in a sequence of zeros and ones, so an (admittedly large but) finite number. And so your personal life is in the decimal places of π And not only that; Your film comes in every conceivable variant: One variant originally shows your previous life and that you are now bored with reading this article. In another variant, your original 86 5 is shown

6 life so far and then that you are so moved by this article that you transfer 1000 euros to the author! Be amazed at mathematics From the number 0, my surveys assume "after at least 80 / o of the respondents that it is smaller than 1. A part of beauty" in mathematics results from the fact that it looks behind the horizon like a telescope granted to man by his limited senses or by his common sense. A nice but wrong argument that 0 should actually be greater than 1 goes back to a Greek way of thinking: because 0, is 0.9 + 0.09 + 0,, i.e., an infinite number of summands are added, the adding goes on forever. What other than infinite - according to the Greek way of thinking - can then result? In other words: 0.9 + 0.09 + 0,, = 0, = infinite, so certainly greater than 1. Despite the apparent power of this argument it turns out that it is wrong, because it goes too carelessly with the term infinite or infinitely often and infinitely much. The exact calculation of 0 still gives a surprising and therefore nice result in this sense: As you can easily recalculate, the following applies: 1 = 0, if you add 1 = 0 again on both sides, you get 2 = 0, and so on we get 3 9 = 0,, 4 = 0, etc. until finally = 0, and finally: 9 9 = 0, and there is no doubt about this result. However, the left side of the last equation gives the value 1, as elementary fraction calculation (abbreviation) shows and 86 6

7 one has the result that is surprising for many, which seems to contradict common sense so completely: 1 = 9 9 = 0, ingenious in mathematics In mathematics, not least intelligent considerations are considered beautiful that are not so obvious as that one You can ultimately come to them yourself through diligence, but to which one needs ingenious ideas (see, for example, the proof of the internal angle sum in the triangle), i.e. considerations that require a brilliant mind. One of these proofs (which was voted one of the top ten mathematical proofs in a survey of mathematicians) should be sketched here (not in all technical details). Everyone learns in school what a prime number is, namely a number that is exactly divisible by 2 (different) numbers: by itself and the number 1. B. 3 is a prime number, 5 as well or 101. No prime numbers, on the other hand, are 1 (since it has only one factor, namely 1) or 6 (= 3 2). Hardly anyone will doubt that there may be a great many prime numbers, perhaps even an infinite number. But how can you be sure of that? How do you prove beyond doubt that there are very many, whatever that may be, or that there are even an infinite number of prime numbers? The answer is a more than 2000 year old beautiful train of thought to the sentence: There are infinitely many prime numbers. To prove this, assume that there are only finitely many prime numbers, that is, in order of size: 2, 3, 5, 7, ..., p. Without telling us why and for what purpose, Euclid, the author of this proof, then gives us his brilliant idea: Multiply all these prime numbers together and finally add 1. So you get the number: x = (p) + 1 certainly 2 is not a divisor of x, because otherwise 2 would be a divisor of 1. Likewise, 3 is not a divisor of x, because otherwise 3 would be a divisor of 1. Likewise, 5 is not a divisor of x, because otherwise 5 would be a divisor of 1, and so on until finally p is not a divisor of x, because otherwise p would be a divisor of 1. So x is not divisible by a single prime number. But since x is divisible by 1 and itself, x is a prime number (or divisible by a prime number), but (since it is 86 7

8 is greater than p) is not included in the above set of all prime numbers. The assumption that there are only finitely many prime numbers leads to a contradiction and must therefore be wrong. This shows (by a so-called indirect proof): The assumption that there are finitely many prime numbers is wrong. Accordingly, only the conclusion remains: there is not just a finite, but an infinite number of prime numbers. This is clearly shown in the prime number picture by Suzanne Daetwyler (see Fig. 2). Such tricky but logically correct trains of thought read a logically well-structured detective novel for an interested party as well as for others; for those interested, they simply offer a pleasure and are perceived as beautiful. 86 8

9 PRIME NUMBER IMAGE by Suzanne Daetwyler Definition A natural number greater than 1 is called a prime number if it can only be divided by 1 and itself. Sequence of prime numbers The first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... The number 1 is not counted among the prime numbers, as otherwise some regularities about prime numbers would not apply or they would be different would have to be formulated. One example is the so-called uniqueness of the prime factorization: the prime factorization of a natural number is unambiguous except for the order of the factors. For example, 420 = If the number 1 were allowed in the prime numbers, there would be different decompositions, e.g. 4 = 22 = = ... Prime numbers can be seen as the basic building blocks (atoms) of natural numbers. The largest prime number Euclid was able to prove around 2300 years ago that there are an infinite number of prime numbers. Although new prime numbers are constantly being found with the help of increasingly faster computing, a formula for generating prime numbers does not yet exist. Applications Very large prime numbers have important fields of application in encryption methods and secret codes. Ulams Spirale The Polish mathematician Stanislaw M. Ulam heard a lecture in the autumn of 1963 which he later described as long and very boring. As a distraction, he scribbled the natural numbers counterclockwise, starting with the number 1, in a spiral on a checkered grid. When he then marked the prime numbers, to Stanislaw M. Ulam's astonishment he noticed that there were () diagonal line patterns. Should there be regularities in the distribution of prime numbers? Since then, variations of the so-called Ulam spiral have been examined again and again when it comes to distributions of prime numbers. S. M. Ulam became known through 86 9

10 his participation in optimization processes for nuclear weapons and the development of the Monte Carlo method. Fig. 2: Suzanne Daetwyler's picture of prime numbers was designed by Prof. H.G. Weigand designed and implemented in the layout by Jan Wörler. Final recommendation If even a single one of the above examples has found interest in the reader, has given him the satisfaction of having understood something, recognized that he has looked beyond his previous horizon, then I like the attempt to depict the fascination, the beauty of mathematics, already call it a success. But it may be that none of the above examples reached one or the other reader, be it because they found them banal or simply uninteresting. So that it has not been possible to convey something of the beauty of simple but intelligent, elegant, surprising thoughts and considerations. Finally, let me make the following comparison (with a wink): Imagine you are faced with the task of having to present the beauty of the visual arts to a broad audience with the help of children's drawings alone. You will certainly reach one or the other listener with pictures and works created by children. However, you are burning to be able to offer more, to present pictures of really great painters, to speak about their painting techniques and compositional principles in order to convince your audience.Regardless of whether you belong to the former or the latter; In any case, reading Experience Mathematics (by Davis and Hersh) will be intellectually enriching, beautiful read: for the former, because they have already developed a feeling for mathematical beauty and for the latter, because when reading the beauty of mathematics, they do not Children’s pictures, but get conveyed with the help of correct mathematics. Literature Davis, Ph. J., Hersch, R., experience in mathematics, Birkhäuser Verlag, Basel