Geometric paths and thickets
While writing this article, I remembered a very old song by Jan Pietrzak, which he sang before his satirical activity in the cabaret Pod Egidą, recognized in the Polish People's Republic as a safety valve; one could honestly laugh at the paradoxes of the system. In this song, the author recommended socialist political participation, ridiculing those who want to be apolitical and turning off the radio in the newspaper. “It’s better to go back to school reading,” the then XNUMX-year-old Petshak sang ironically.
I'm going back to school reading. I am re-reading (not for the first time) the book of Shchepan Yelensky (1881-1949) “Lylavati”. For few readers, the word itself says something. This is the name of the daughter of the famous Hindu mathematician known as Bhaskara (1114-1185), named Akaria, or the sage who titled his book on algebra with that name. Lilavati later became a renowned mathematician and philosopher herself. According to other sources, it was she who wrote the book herself.
Szczepan Yelensky gave the same title to his book on mathematics (first edition, 1926). It may even be difficult to call this book a mathematical work - it was more of a set of puzzles, and largely rewritten from French sources (copyrights in the modern sense did not exist). In any case, for many years it was the only popular Polish book on mathematics - later Jelensky's second book, Pythagoras's Sweets, was added to it. So young people interested in mathematics (which is exactly what I once was) had nothing to choose from ...
on the other hand, "Lilavati" had to be known almost by heart... Ah, there were times... Their biggest advantage was that I was... a teenager then. Today, from the point of view of a well-educated mathematician, I look at Lilavati in a completely different way - maybe like a climber on the bends of the path to Shpiglasova Pshelench. Neither one nor the other loses its charm ... In his characteristic style, Shchepan Yelensky, who professes the so-called national ideas in his personal life, he writes in the preface:
Without touching on the description of national characteristics, I will say that even after ninety years, Yelensky's words about mathematics have not lost their relevance. Mathematics teaches you to think. It is a fact. Can we teach you to think differently, more simply and more beautifully? May be. It's just... we still can't. I explain to my students who don't want to do math that this is also a test of their intelligence. If you can't learn really simple math theory, then... maybe your mental abilities are worse than we both would like...?
Signs in the sand
And here is the first story in "Lylavati" - a story described by the French philosopher Joseph de Maistre (1753-1821).
A sailor from a wrecked ship was thrown by waves onto an empty shore, which he considered uninhabited. Suddenly, in the coastal sand, he saw a trace of a geometric figure drawn in front of someone. It was then that he realized that the island is not deserted!
Quoting de Mestri, Yelensky writes: geometric figureit would have been a mute expression for the unfortunate, shipwrecked, coincidence, but he showed him at a glance proportion and number, and this heralded an enlightened man. So much for history.
Note that a sailor will cause the same reaction, for example, by drawing the letter K, ... and any other traces of a person's presence. Here the geometry is idealized.
However, the astronomer Camille Flammarion (1847-1925) proposed that civilizations greet each other from a distance using geometry. He saw in this the only correct and possible attempt at communication. Let's show such Martians the Pythagorean triangles... they will answer us with Thales, we will answer them with Vieta patterns, their circle will fit into a triangle, so a friendship began...
Writers such as Jules Verne and Stanislav Lem returned to this idea. And in 1972, tiles with geometric (and not only) patterns were placed on board the Pioneer probe, which still traverses the expanses of space, now almost 140 astronomical units from us (1 I is the average distance of the Earth from the Earth). Sun, i.e., about 149 million km). The tile was designed, in part, by astronomer Frank Drake, creator of the controversial rule on the number of extraterrestrial civilizations.
Geometry is amazing. We all know the general point of view on the origin of this science. We (we humans) have just begun to measure the land (and later the land) for the most utilitarian purposes. Determining distances, drawing straight lines, marking right angles and calculating volumes gradually became a necessity. Hence the whole thing geometry (“Measurement of the earth”), hence all mathematics ...
However, for some time this clear picture of the history of science clouded us. For if mathematics were needed solely for operational purposes, we would not be engaged in proving simple theorems. “You see that this should be true at all,” one would say after checking that in several right triangles the sum of the squares of the hypotenuses is equal to the square of the hypotenuse. Why such formalism?
Plum pie has to be delicious, the computer program has to work, the machine has to work. If I counted the capacity of the barrel thirty times and everything is in order, then why else?
In the meantime, it occurred to the ancient Greeks that some formal evidence needed to be found.
So, mathematics begins with Thales (625-547 BC). It is assumed that it was Miletus who began to wonder why. It is not enough for smart people that they have seen something, that they are convinced of something. They saw the need for proof, a logical sequence of arguments from assumption to thesis.
They also wanted more. It was probably Thales who first tried to explain physical phenomena in a naturalistic way, without divine intervention. European philosophy began with the philosophy of nature - with what is already behind physics (hence the name: metaphysics). But the foundations of European ontology and natural philosophy were laid by the Pythagoreans (Pythagoras, c. 580-c. 500 BC).
He founded his own school in Crotone in the south of the Apennine Peninsula - today we would call it a sect. Science (in the current sense of the word), mysticism, religion and fantasy are all closely intertwined. Thomas Mann very beautifully presented the lessons of mathematics in a German gymnasium in the novel Doctor Faustus. Translated by Maria Kuretskaya and Witold Virpsha, this fragment reads:
In Charles van Doren's interesting book, The History of Knowledge from the Dawn of History to the Present Day, I found a very interesting point of view. In one of the chapters, the author describes the significance of the Pythagorean school. The very title of the chapter struck me. It reads: "The Invention of Mathematics: The Pythagoreans".
We often discuss whether mathematical theories are being discovered (eg unknown lands) or invented (eg machines that did not exist before). Some creative mathematicians see themselves as researchers, others as inventors or designers, less often counters.
But the author of this book writes about the invention of mathematics in general.
From exaggeration to delusion
After this long introductory part, I will move on to the very beginning. geometryto describe how an over-reliance on geometry can mislead a scientist. Johannes Kepler is known in physics and astronomy as the discoverer of the three laws of motion of celestial bodies. First, each planet in the solar system moves around the sun in an elliptical orbit, at one of the foci of which is the sun. Secondly, at regular intervals the leading ray of the planet, drawn from the Sun, draws equal fields. Thirdly, the ratio of the square of the period of revolution of a planet around the Sun to the cube of the semi-major axis of its orbit (i.e., the average distance from the Sun) is constant for all planets in the solar system.
Perhaps this was the third law - it required a lot of data and calculations to establish it, which prompted Kepler to continue searching for patterns in the movement and position of the planets. The history of his new "discovery" is very instructive. Since antiquity, we have admired not only regular polyhedra, but also arguments showing that there are only five of them in space. A three-dimensional polyhedron is called regular if its faces are identical regular polygons and each vertex has the same number of edges. Illustratively, each corner of a regular polyhedron should "look the same". The most famous polyhedron is the cube. Everyone has seen an ordinary ankle.
The regular tetrahedron is less well known, and in school it is called the regular triangular pyramid. It looks like a pyramid. The remaining three regular polyhedra are less well known. An octahedron is formed when we connect the centers of the edges of a cube. The dodecahedron and icosahedron already look like balls. Made from soft leather, they would be comfortable to dig. The reasoning that there are no regular polyhedra other than the five Platonic solids is very good. First, we realize that if the body is regular, then the same number (let q) of identical regular polygons must converge at each vertex, let these be p-angles. Now we need to remember what is the angle in a regular polygon. If someone does not remember from school, we remind you how to find the right pattern. We took a trip around the corner. At each vertex we turn through the same angle a. When we go around the polygon and return to the starting point, we have made p such turns, and in total we have turned 360 degrees.
But α is 180 degrees' complement of the angle we want to compute, and is therefore
We have found the formula for the angle (a mathematician would say: measures of an angle) of a regular polygon. Let's check: in the triangle p = 3, there is no a
Like this. When p = 4 (square), then
degrees is fine too.
What do we get for a pentagon? So what happens when there are q polygons, each p having the same angles
degrees descending at one vertex? If it were on a plane, then an angle would form
degrees and cannot be more than 360 degrees - because then the polygons overlap.
However, since these polygons meet in space, the angle must be less than the full angle.
And here is the inequality from which it all follows:
Divide it by 180, multiply both parts by p, order (p-2) (q-2) < 4. What follows? Let's be aware that p and q must be natural numbers and that p > 2 (why? And what is p?) and also q > 2. There aren't many ways to make the product of two natural numbers less than 4. We'll list them all. in table 1.
I don't post drawings, everyone can see these figures on the Internet... On the Internet... I will not refuse a lyrical digression - perhaps it is interesting for young readers. In 1970 I spoke at a seminar. The topic was difficult. I had little time to prepare, I sat in the evenings. The main article was read-only in place. The place was cozy, with a working atmosphere, well, it closed at seven. Then the bride (now my wife) herself offered to rewrite the entire article for me: about a dozen printed pages. I copied it (no, not with a quill pen, we even had pens), the lecture was a success. Today I tried to find this publication, which is already old. I remember only the name of the author... Searches on the Internet lasted a long time... a full fifteen minutes. I think about it with a smirk and a little unjustified regret.
We go back to Keplera i geometry. Apparently, Plato predicted the existence of the fifth regular form because he lacked something unifying, covering the whole world. Perhaps that is why he instructed a student (Theajtet) to look for her. As it was, so it was, on the basis of which the dodecahedron was discovered. We call this attitude of Plato pantheism. All scientists, down to Newton, succumbed to it to a greater or lesser extent. Since the highly rational eighteenth century, its influence has drastically diminished, although we should not be ashamed of the fact that we all succumb to it in one way or another.
In Kepler's concept of building the solar system, everything was correct, the experimental data coincided with the theory, the theory was logically coherent, very beautiful ... but completely false. In his time, only six planets were known: Mercury, Venus, Earth, Mars, Jupiter and Saturn. Why are there only six planets? Kepler asked. And what regularity determines their distance from the Sun? He assumed that everything was connected, that geometry and cosmogony are closely related to each other. From the writings of the ancient Greeks, he knew that there were only five regular polyhedra. He saw that there were five voids between the six orbits. So maybe each of these free spaces corresponds to some regular polyhedron?
After several years of observation and theoretical work, he created the following theory, with the help of which he calculated quite accurately the dimensions of the orbits, which he presented in the book "Mysterium Cosmographicum", published in 1596: Imagine a giant sphere, the diameter of which is the diameter of the orbit of Mercury in its annual motion around the sun. Then imagine that on this sphere there is a regular octahedron, on it a sphere, on it an icosahedron, on it again a sphere, on it a dodecahedron, on it another sphere, on it a tetrahedron, then again a sphere, a cube and, finally, on this cube the ball is described.
Kepler concluded that the diameters of these successive spheres were the diameters of the orbits of other planets: Mercury, Venus, Earth, Mars, Jupiter, and Saturn. The theory seemed to be very accurate. Unfortunately, this coincided with the experimental data. And what better evidence of the correctness of a mathematical theory than its correspondence with experimental data or observational data, especially "taken from heaven"? I summarize these calculations in Table 2. So what did Kepler do? I tried and tried until it worked out, that is, when the configuration (order of spheres) and the resulting calculations coincided with the observational data. Here are modern Kepler figures and calculations:
One can succumb to the fascination of the theory and believe that the measurements in the sky are inaccurate, and not the calculations made in the silence of the workshop. Unfortunately, today we know that there are at least nine planets and that all coincidences of results are just a coincidence. A pity. It was so beautiful...
