five times in the eye
At the end of 2020, several events were held at universities and schools, postponed from ... March. One of them was the "celebration" of pi day. On this occasion, on December 8, I gave a remote lecture at the University of Silesia, and this article is a summary of the lecture. The whole party started at 9.42, and my lecture is scheduled for 10.28. Where does such accuracy come from? It's simple: 3 times pi is about 9,42, and π to the 2nd power is about 9,88, and the hour 9 to the 88th power is 10 to the 28th ...
The custom of honoring this number, expressing the ratio of the circumference of a circle to its diameter and sometimes called the Archimedes constant (as well as in German-speaking cultures), comes from the USA (see also: ). 3.14 March “American style” at 22:22, hence the idea. The Polish equivalent could be July 7 because the fraction 14/XNUMX approximates π well, which…Archimedes already knew. Well, March XNUMX is the best time for side events.
These three and fourteen hundredths are one of the few mathematical messages that have remained with us from school for life. Everyone knows what that means"five times in the eye". It is so ingrained in the language that it is difficult to express it differently and with the same grace. When I asked at the car repair shop how much the repair might cost, the mechanic thought about it and said: “five times about eight hundred zlotys.” I decided to take advantage of the situation. "You mean a rough approximation?". The mechanic must have thought I misheard, so he repeated, “I don’t know exactly how much, but five times an eye would be 800.”
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What is it about? Pre-World War II spelling used "no" together, and I left it there. We are not dealing here with overly pompous poetry, although I like the idea that "the golden ship pumps happiness." Ask students: What does this thought mean? But the value of this text lies elsewhere. The number of letters in the following words are the digits of the pi extension. Let's see:
Π ≈ 3,141592 653589 793238 462643 383279 502884 197169 399375 105820 974944 592307 816406 286208 998628 034825 342117 067982 148086 513282 306647 093844 609550 582231 725359 408128 481117 450284
In 1596, a Dutch scientist of German origin Ludolph van Seulen calculated the value of pi to 35 decimal places. Then these figures were engraved on his grave. She dedicated a poem to the number pi and to our Nobel laureate, Vislava Shimborska. Szymborska was fascinated by the non-periodicity of this number and by the fact that with probability 1 each sequence of numbers, such as our phone number, would appear there. While the first property is inherent in every irrational number (which we should remember from school), the second is an interesting mathematical fact that is difficult to prove. You can even find apps that offer: give me your phone number and I'll tell you where it is in pi.
Where there is roundness, there is sleep. If we have a round lake, then walking around it is 1,57 times longer than swimming. Of course, this does not mean that we will swim one and a half to two times slower than we will pass. I shared the 100m world record with the 100m world record. Interestingly, in men and women, the result is almost the same and is 4,9. We swim 5 times slower than we run. Rowing is completely different - but an interesting challenge. It's got a pretty long storyline.
Fleeing from the pursuing Villain, the handsome and noble Good One sailed to the lake. The villain runs along the shore and waits for her to make him land. Of course, he runs faster than Dobry rows, and if he runs smoothly, Dobry is faster. So the only chance for Evil is to get Good from the shore - an accurate shot from a revolver is not an option, because. Good has valuable information that Evil wants to know.
Good adheres to the following strategy. He swims across the lake, gradually approaching the shore, but always trying to be on the opposite side from the Evil One, who randomly runs to the left, then to the right. This is shown in the figure. Let Evil start position be Z1, and Dobre is the middle of the lake. When Zly moves to Z1, Dobro will sail to D.1when Bad is in Z2, good on D2. It will flow in a zigzag manner, but in compliance with the rule: as far as possible from Z. However, as it moves away from the center of the lake, Good must move in larger and larger circles, and at some point it cannot adhere to the principle “to be on the other side of Evil.” Then he rowed with all his might to the shore, hoping that the Evil One would not bypass the lake. Will Good succeed?
The answer depends on how fast Good can row in relation to the value of Bad's legs. Suppose that the Bad Man runs at a speed s times the speed of the Good Man on the lake. Therefore, the largest circle, on which Good can row in order to resist Evil, has a radius that is one times smaller than the radius of a lake. So, in the drawing we have. At point W, our Kind begins to row towards the shore. This must go
with speed
He needs time.
Wicked is chasing all his best feet. He must complete half of the circle, which will take him seconds or minutes, depending on the selected units. If this is more than a happy ending:
The good one will go. Simple accounts show what it should be. If the Bad Man runs faster than 4,14 times the Good Man, it doesn't end well. And here, too, our number pi intervenes.
What is round is beautiful. Let's look at the photo of three decorative plates - I have them after my parents. What is the area of the curvilinear triangle between them? This is a simple task; the answer is in the same photo. We are not surprised that it appears in the formula - after all, where there is roundness, there is pi.
I used a possibly unfamiliar word:. This is the name of the number pi in the German-speaking culture, and all this thanks to the Dutch (actually a German who lived in the Netherlands - nationality did not matter at that time), Ludolf of Seoulen... In 1596 g. he calculated 35 digits of his expansion to decimal. This record held until 1853, when William Rutherford counted 440 seats. The record holder for manual calculations is (probably forever) William Shankswho, after many years of work, published (in 1873) extension to 702 digits. Only in 1946, the last 180 digits were found to be incorrect, but it remained so. 527 correct. It was interesting to find the bug itself. Soon after the publication of Shanks' result, they suspected that "something was wrong" - there were suspiciously few sevens in development. The yet unproven (December 2020) hypothesis states that all numbers should appear with the same frequency. This prompted D.T. Ferguson to revise Shanks' calculations and find the "learner's" error!
Later, calculators and computers helped people. The current (December 2020) record holder is Timothy Mullican (50 trillion decimal places). The calculations took ... 303 days. Let's play: how much space this number would take, printed in a standard book. Until recently, the printed "side" of the text was 1800 characters (30 lines by 60 lines). Let's reduce the number of characters and page margins, cram 5000 characters per page, and print 50 page books. So XNUMX trillion characters would take ten million books. Not bad, right?
The question is, what is the point of such a struggle? From a purely economic point of view, why should the taxpayer pay for such "entertainment" of mathematicians? The answer is not difficult. First, from Seoulen invented blanks for calculations, then useful for logarithmic calculations. If he had been told: please, build blanks, he would have answered: why? Similarly command:. As you know, this discovery was not entirely accidental, but nevertheless a by-product of research of a different type.
Secondly, let's read what he writes Timothy Mullican. Here is a reproduction of the beginning of his work. Professor Mullican is in cybersecurity, and pi is such a small hobby that he just tested his new cybersecurity system on.
And that 3,14159 in engineering is more than enough, that's another matter. Let's do a simple calculation. Jupiter is 4,774 Tm away from the Sun (terameter = 1012 meters). To calculate the circumference of such a circle with such a radius to an absurd precision of 1 millimeter, it would be enough to take π = 3,1415926535897932.
The following photo shows a quarter circle of Lego bricks. I used 1774 pads and it was about 3,08 pi. Not the best, but what to expect? A circle cannot be made up of squares.
Exactly. The number pi is known to be circle square - a mathematical problem that has been waiting for its solution for more than 2000 years - since Greek times. Can you use a compass and straightedge to construct a square whose area is equal to the area of the given circle?
The term "square of a circle" has entered the spoken language as a symbol of something impossible. I press the key to ask, is this some sort of attempt to fill the trench of hostility that separates the citizens of our beautiful country? But I already avoid this topic, because I probably only feel in mathematics.
And again the same thing - the solution to the problem of squaring the circle did not appear in such a way that the author of the solution, Charles Lindemann, in 1882 he was set up and finally succeeded. To some extent yes, but it was the result of an attack from a broad front. Mathematicians have learned that there are different kinds of numbers. Not only integers, rational (that is, fractions) and irrational. Immeasurability can also be better or worse. We may remember from school that the irrational number is √2, a number expressing the ratio of the length of a square's diagonal to the length of its side. Like any irrational number, it has an indefinite extension. Let me remind you that periodic expansion is a property of rational numbers, i.e. private integers:
Here the sequence of numbers 142857 repeats indefinitely. For √2 this will not happen - this is part of the irrationality. But you can:
(fraction goes on forever). We see a pattern here, but of a different type. Pi is not even that common. It cannot be obtained by solving an algebraic equation - that is, one in which there is neither a square root, nor a logarithm, nor trigonometric functions. This already shows that it is not constructible - drawing circles leads to quadratic functions, and lines - straight lines - to equations of the first degree.
Perhaps I deviated from the main plot. Only the development of all mathematics made it possible to return to the origins - to the ancient beautiful mathematics of the thinkers who created for us the European culture of thought, which is so doubtful today by some.
Of the many representative patterns, I chose two. The first of them we associate with the surname Gottfried Wilhelm Leibniz (1646-1716).
But he was known (model, not Leibniz) to the medieval Hindu scholar Madhava of the Sangamagram (1350-1425). The transfer of information at that time was not great - Internet connections were often buggy, and there were no batteries for mobile phones (because electronics had not yet been invented!). The formula is beautiful, but useless for calculations. From a hundred ingredients, "only" 3,15159 is obtained.
he's a little better Viète's formula (the one from quadratic equations) and its formula is easy to program because the next term in the product is the square root of the previous plus two.
We know that the circle is round. We can say that this is a 100 percent round. The mathematician will ask: can something be not 1 percent round? Apparently, this is an oxymoron, a phrase containing a hidden contradiction, such as, for example, hot ice. But let's try to measure how round the shapes can be. It turns out that a good measure is given by the following formula, in which S is the area and L is the circumference of the figure. Let's find out that the circle is really round, that the sigma is 6. The area of the circle is the circumference. We insert ... and see what is right. How round is the square? The calculations are just as simple, I won't even give them. Take a regular hexagon inscribed in a circle with a radius. The perimeter is obviously XNUMX.
Pole
How about a regular hexagon? Its circumference is 6 and its area
So we have
which is approximately equal to 0,952. The hexagon is more than 95% "round".
An interesting result is obtained when calculating the roundness of a sports stadium. According to IAAF rules, straights and curves must be 40 meters long, although deviations are allowed. I remember that Bislet Stadium in Oslo was narrow and long. I write “was” because I even ran on it (for an amateur!), but more than XNUMX years ago. Let's get a look:
If the arc has a radius of 100 meters, the radius of that arc is meters. The area of the lawn is square meters, and the area outside it (where there are springboards) totals square meters. Let's plug this into the formula:
So does the roundness of a sports stadium have anything to do with an equilateral triangle? Because the height of an equilateral triangle is the same number of times the side. It's a random coincidence of numbers, but it's nice. I like it. And the readers?
Well, it's good that it's round, although some might object because the virus that affects us all is round. At least that's how they draw it.
