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In the world-famous water city of Venice, there is a SanMarco square.

At one end of the square is a magnificent church 82 meters wide. In front of the church is an open space. This open area often attracts visitors from all over the world to play a strange game:

Cover your eyes and walk from one end of the square to the church at the other end to see who can reach the front of the church!

Oddly enough, although the distance is only 175 meters, not a single tourist is lucky enough to do so! All like the following picture, walked into an arc, or left or right, skewed to one side!

A similar phenomenon appears more miraculously in the works of Mark Twain, a famous American writer. In the book Journey abroad, Mark Twain (MarkTwain,1835-1910) describes a night trip that lasted for 4.7 miles, but all happened in a dark room! Here are the highlights of this touching story:

I woke up and felt thirsty. A good idea came to my mind-get dressed, go to the garden for a change of air, and wash my face by the fountain.

I got up quietly and began to look for my clothes. I found a sock, but I don't know where the second one is. I got out of bed carefully and crawled around for a while, but found nothing! I began to fumble farther and farther away. I couldn't find my socks, but bumped into the furniture. When I went to bed, there was not so much wood around me. What about now? The whole room is full of wood, especially the most chairs, as if there are chairs everywhere! Is it possible that two more families have moved here during this period of time? I couldn't see any of these chairs in the dark, but I kept banging my head against them. Finally, I made up my mind that I could live without a sock. I stood up and walked to the door, I thought, only to see my hazy face in a mirror unexpectedly.

It was clear that I had lost my way, and I had no impression of where I was. If there is only one mirror in the room, it will help me to discern the direction. But unfortunately there are two sides, and this is as bad as having a thousand sides!

I want to walk along the wall to the door and start my new attempt. Unexpectedly, he knocked down a painting. The picture is not big, but it makes a noise like falling a huge picture. Grace (my neighbor in the other bed in the same room) didn't turn over. But I felt that if I went on, I was bound to wake him up. I began to try another way, and I went to find the round table again-I had just walked next to it several times-to touch my bed from there.

If you find a bed, you can find a glass container that holds water, so you can at least quench your unbearable thirst! The best way is to crawl with your arms and knees. I have tried this method, so I have more trust in it.

Finally, I found the table-my head touched it-and made a loud noise. So I stood up again, stretched out my five fingers, stretched out my hands to balance my body, and walked on like this. I touched a chair, then a wall, another chair, then a sofa, my walking stick, another sofa. This surprised me because I knew very well that there was only one sofa in the room! I bumped into the table again, and it hurt again, and then I hit some chairs.

It was only then that I remembered how I should have gone. Because the table is round, it can't be the starting point of my "travel".

With a fluke, I went to the space between the chair and the sofa, but I was stuck in a completely strange situation. On the way, I touched the candlestick on the fireplace, then the lamp, and finally, the glass retort containing water fell to the ground with a bang!

"Ha ha!" I thought to myself, "I found you in the end, my baby!"

"there's a thief! catch the thief!" Grace shouted wildly.

The whole house was immediately full of people, and the innkeeper, tourists and servants ran in with candles and lanterns.

I looked around, and I was standing by Grace's bed! There was only one sofa against the wall, and only one chair I could touch-I walked around it like a planet all night and touched it like a comet!

According to my step-by-step calculation, I know that I walked a total of 4.7 miles that night!

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The above story of Mr. Mark Twain is undoubtedly greatly exaggerated, but what he describes about a person who has lost his way in the dark can happen.

Readers can also see descriptions of many people spinning around in the desert or snow because they have lost their way in other books. All these almost joking encounters finally attracted the attention of scientists.

In 1896, Norwegian physiologist Goodbell made an in-depth discussion on the problem of closing his eyes. After collecting a large number of examples, he analyzed and said that all this is due to people's own two legs in trouble! The habit that has been formed over the years makes each person's step with one foot sticking out is a negligible distance longer than that of the other.

It is this very small step difference x that causes this person to walk out of a large circle with a radius of y. This is shown in the following figure.

Now let's study the functional relationship between x and y.

Suppose that the distance between one's feet is d. Obviously, when a person is spinning, the two feet actually walk out of two concentric circles with a radius difference of d. Let the average step size of the person be l. Well, on the one hand, this person has more distance on his outer feet than on his inner feet.

On the other hand, this distance is equal to the product of the number of steps of the person walking around and the step difference, that is,

Simplify and simplify

For the average person, dice 0.1 meters, lads 0.7 meters, substitute (in meters)

This is the radius formula of the lost person. Today, the difference between the two steps of a lost man is 0.1 mm. Only this small difference will cause him to circle in a range of about 3 kilometers.

The relationship between the variables XBI y in the above formula is mathematically called the inverse proportional function relation. The inverse proportional function is generally in the form of yardk / x, where k is a constant.

Its image is two curved curves, which are mathematically called equilateral hyperbolas. Inverse proportional function is very useful in industry, national defense, science and technology and other fields.

Go back to the game in St. Mark's Square that we talked about at the beginning of this section.

Let's first calculate what the minimum arc radius should be when people close their eyes and get to the church CD from the M point in the middle of the square.

As shown in the figure, notice the edge BC=175AM=MB=41 (in meters) of the rectangular ABCD. Then the above problems are undoubtedly equivalent to the following propositions in geometry: given BC and MB, find the size of the radius R of MC.

Because

So

That is to say, if a tourist wants to succeed, the radius of the arc he walks must be not less than 394 meters. Now let's calculate again, in order to meet the above requirements, the visitors' two-step difference needs to be limited. According to the formula

Because

So

This shows that the difference between the two feet of visitors must be less than 0.35mm, otherwise there is no hope of success!

However, it is impossible for ordinary people to do such a small step difference between their feet with their eyes closed, which is why no one can walk blindfolded to the front of the church in the game.

Author: Zhang Yuannan Zhang Chang

Editor: Zhang Runxin

This article comes from the official account of Wechat: Origin Reading (ID:tupydread), author: Zhang Yuannan Zhang Chang

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