Science,English,and math 4th

《A roller coaster and curves on Euler》

『Clothoid』

On the other hand, there is a corse from the straight line to the Clothoid to the circle and to Clothoid to the straight line. When going into the Clothoid from the straight line, the driver has only to cut the steering wheel with the fixed pace gradually.

When driving on the circle, as long as the driver keeps the steering wheel as it was, it’s all right, so it’s easy to drive and the passenger find it comfortable, so we find the Clothoid pleasant.

It’s Autobahn in Germany where the Clothoid has been adopted for roads on cars for the...

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《A roller coaster and curves on Euler》

『Clothoid』

It is Autobahn in Germany where the Clothoid has been used for roads on cars for the first time. The Autobahn is a highway express.

Almost all of the highway expresses have adopted the Clothoid all over the world at present.

《Two giants and breaking away from a circle》

After Pythagoras in the Ancient Greece, orbits on stars in the cosmos which was the symbol for a complete harmony used to be believed firmly that it must have been a circle.

Not only in the Ptolemaic theory 天動説 that the earth was the center on the cosmos, but in the...

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【Welcome to a museum on curves】

《Two giants and breaking away from a circle》

...but the Copernican theory 地動説 advocated by Copernicus in Poland used to think an orbit on starts was a circle, but being based in the circular orbit and trying to express the movements on stars, they needed very complicated theory in both of Ptolemaic and of Copernican.

In addition the place on stars which was done by a large amount of calculation were often different from actual ones.

On the other hand, Johannesburg Kepler who supported the Copernican theory studied the result from observation carefully and...

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【Welcome to a museum on curves】

《Two giants and getting away from a circle》

...and formulated a hypothesis that an orbit on planet was an ellipse. If having thought, he discovered that he could explain movements on planets by far more simply and more accurately rather than regarding the orbit on planets as a circle.

Kepler framed that when all the planets which rotated around the sun including the earth, all of the orbits were ellipse, and foresaw the movements on the planets ahead of several years and created a list of astronomy. We call it the Rudolphine Tables.

Its precision was...

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【Welcome to a museum on curves】

《Two giants and getting away from a circle》

As its precision is thirtieth times as accurate as in the past, the predominant position on Copernican theory was strengthened.

Moreover being based on the theory by Kepler, Newton arrived at common physical laws including the law of universal gravitation, and succeeded in expressing from a pebble on the earth to the movements on planets with an unified way, as a result, the Ptolemaic theory was forgotten completely.

Two giants of Descartes and Kepler who showed up in the former part of the 17th century have...

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【Welcome to a museum on curves】

《Two giants and getting away from a circle》

Two giants of Descartes and Kepler who turned up at the former part in the 17th century have made us the humankind get away from the circle. It means that looking at nature as it is and expressing mathematically. It’s indispensable for modern science.

【Math in which tiles spread all over】

《Geometric design at Alhambra palace 》

There is Alhambra palace at Granada of ancient city in Spain. It used to be a symbol on power and fortune of Muslims who ruled over there, and was a tragic stage where they were defeated...

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【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

...and a tragic stage where Muslims were defeated by the Christians with the battle of Reconquista. It was a the battle on restoration of their territory by the Christians.

Its premises is no less than one hundred and fifty thousand square. It’s equal to three of Tokyo Dome which is a baseball stadium. Not only a king but some two hundred of the nobles used to live there then.

As to the origin of a word on Alhambra, it’s Islam language which means a red fortress. Several magnificent building are lined up in the...

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【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

Several magnificent buildings stand in a line in a tremendous site, and each of them are masterpiece which gathers the best that Islamic architecture embodied. It’s so splendid that it was said the king completed the palace by magic then.

As worshipping an idol is prohibiting by dogma in Islam, decoration wasn’t done with a motif of a persons or animals. Instead geometric designs have developed. We can see the tradition everywhere in Alhambra palace, especially a wall or a ceiling were spread all over by basic...

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【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

...especially a wall or a ceiling are spread all over by basic figures all round, on the contrary, tiles spread all over by different design one by one. They are so excellent that everyone is forced to watch it with its eyes wide open in astonishment.

Dutch painter, and a woodblock artist, Maurice Escher is one of them. He copied the decoration in the palace carefully in full three days, and his inspiration was stimulated greatly by the beauty continues limitlessly.

Speaking of Escher, he is famous for the art...

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【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

Speaking of Escher, he is famous for the recognized authority on the art of illusion like one of his works, a waterfall as if it looks the water flew from lower to upper part, and as to his style in other work, its canvas is filled with different pattern which changed one after another, and it shows us striking originality.

It is no exaggeration to say that his visit Alhambra palace caused to create a group of works on the latter one like the series on Metamorphose.

In fact the first work on Metamorphose was...

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【Math in which tiles spread all over】

《Geometric design in Alhambra palace》

In fact his first work on Metamorphose was done in the next year after he visited the Alhambra palace. Escher adopted a motif on animal which wasn’t done in Islam and created a work by designs of reputation, so whoever watches the work is attracted by the original world.

《What is a polygon which is filled with a plain?》

To tell the truth, when wanting to fill a plain which spread all over limitless all round, it doesn’t always mean that every shape is all right. For example, if trying to fill the plain with a...

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【Math in which tiles spread out all over】

《What is a regular polygon where a plain is filled with?》

To tell the truth, if trying to fill a plain with a regular pentagon, it’s impossible, for one of angles on the regular pentagon is 108 degree, so gathering three of the regular pentagons in a single apex, 108 × 3 = 324, it’s smaller than 360 degree, so there is a space.

If gathering four, it’s over 360 degree, so one of the regular pentagon is on top of the other.

If thinking over with the same way, the regular polygon which filled with the plain all around is limited the next one.

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【Math in which tiles spread all over】

《What is a regular polygon in which the plain is filled with?》

A scale of a single angle × an integral number = 360 degrees. The regular polygon means the one of which all angles are equal each other, and that kind of regular polygons are regular triangle, regular square, and regular hexagon. It’s just the three kinds.

In general, filling the plain with several kinds of plain figures all around without being the one on top of the other is called tessellation or tiling.

Except for regular triangle, how about other triangle?

In fact every kind of...

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【Math in which tiles spread all over】

《What kind of regular polygon in which a plain is filled with?》

In fact the plain is filled with every kind of triangle as long as it’s the triangle, for preparing for the same shape of two triangles, and turning the one upside down and put together, it changes into a parallelogram.

If the parallelogram is spread out up and down, right and left, the plain is filled with the parallelogram all around.

In fact, as to a square, even if it’s an every kind of a square, the plain is able to be filled with. When turning the one square upside down, and...

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【Math in which tiles spared all over】

《What is the regular polygon in which a plain is filled with?》

In fact when turning a square upside down and put together it with the same shape of the square, it changes into a parallel hexagon of which each side faces each other is parallel, and the plain is filled with the parallel hexagon.

《Inspection on a pentagon》

It turns out that triangle and square which we can choose as we like are the figures, the ones which a plain is be able to filled with until now, and it was already clear in the time of Ancient Greece.

If the triangle, and square...

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【Math in which tiles spread all over】

《Inspection on pentagons》

If the triangle, and square are all right, everyone tends to try to pentagons, but then the situation suddenly becomes complicated.

As the author described, the plain isn’t filled with the regular pentagon, the plain isn’t filled with every kind of the pentagon, but if it’s a warped pentagon which has a character, the plain is filled with the pentagon. For example it has pair of sides which are parallel.

15 kinds of concave pentagons in which a plain is filled with have been discovered until November in 2019. Please look up...

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【Math in which tile spread all over】

《Inspection on pentagons》

Please look up it into pentagonal tiling in Wikipedia if you are interested in it.

A student in German announced a graduation thesis, five kinds of concave pentagons which are able to fill with a plain in university in 1918, and after that the tessellation has been argued mathematically.

He demonstrated that the plain is filled with three kinds of concave hexagon alone in relation to polygon. Then the polygon means the one which has more angle than pentagon. It doesn’t exist in any concave heptagons.

On the other hand...

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【Math in which tiles spread all over】

《An inspection on pentagon》

On the other hand, as to the concave pentagons, he said it was unclear whether or not there were any other ones, except for five kinds which he discovered.

Then a question on tesselation 平面充填 by a single kind of a concave polygon is limited to the pentagon alone.

《A difficult problem which a housewife solved》

After fifty years from the graduation thesis by the German student in 1968, three kinds of concave pentagons were discovered, so the concave pentagon which fills with the plain becomes eight kinds.

The question in....

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【Math in which tiles spread all over】

《A difficult problem which a housewife solved》

A question of filling the plain with the concave pentagon was such a difficult problem that it took no less than a half century to discover three more kinds, but five kinds have been discovered one after another from 1975 to 1977.

Besides, it wasn’t done by any mathematicians, and the one was done by a computer scientist, and the other four were done by a housewife. Her favorite thing was a patchwork. She took interested in a question that the field is filled with tiles by polygons introduced in a magazine.

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私、個人的に、
ちょっと、英語のできる人に、
かなり昔から、負い目があります。

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＞＞170

To the person who sent me a message in 170.

170にメッセージをくれた方に。

Hi, you’re welcome.

こんにちは、ようこそ。

I don’t know why you feel a little inferior to a person who can speak English, but everyone tends to feel like that for something, I’m sure. It doesn’t always mean that it limits to English alone.

あなたが何故英語を話せる人に負い目を感じるのかは知らないけど、英語に限らず、誰でもそう感じる事はあると思います。

Even if being able to speak English a little, it’s not so useful as I expected. It’s just that I have a fun when expressing myself in English.

多少英語を話せたとしても、思ったほどには役に立つわけでは無いです。ただ、英語が好きなんで、その時は楽しいと思うだけですよ。

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＞＞169

【What I’ve thought】

Without being able to show its illustration, I’m afraid I can’t send others what I thought, so I’m going to give up the chapter, math in which tiles spread out, so I’ll start the next chapter.

【A correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

An English word of calculation is made up with calc and ation, and the calc means a pebble, and the ation means doing something.

Calculus means 微分積分 in Japanese, and has also meaning of a stone in the kidney.

Thus calculation and calculus are related to the stone, for...

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【A correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

...for when we the human race started to use numbers, the stone used to be a tool for counting numbers.

As our ancestors in the ancient time, they couldn’t count more than three, they thought of three, thirty, and a hundred as many, though there are various theories on it.

But if leading every day life, we frequently come across a scene when we have to count more than three.

For example, there was a farmhouse which kept some cattle, and a routine for the head of the farmhouse was...

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【A correspondence one by one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

...and the routine of the head of the farmhouse was corresponding each of cattle to a stone before putting the cattle out to the pasture. When the cattle returned, the head make sure whether or not all the cattle returned, using the stones.

When we the human race couldn’t use any of large number, we corresponded the one which we wanted to count to the stone by one to one, so it is said using the stone learned to mean the calculation.

What is the correspondence by one to one?

When there are...

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【A correspondence by one to one and a string of Hideyoshi】

《Making a stone and a thing correspond》

When there are two kinds of set, A and B, every element on A corresponds with only element on B, at the same time, every element on B corresponds with only element on A. It’s the correspondence by one to one.

For example, a system on my number has started in Japan from 2015. All individuals who have its certificate of residence in Japan are given different number in twelve digit number.

As long as the one has the certificate of residence in Japan, everyone corresponds with one of my number...

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【A correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

...and if choosing one of my number, everyone can pinpoint the individual, so the set of the individuals who have certificate of residence in Japan and the set of my number correspond by one to one.

On the other hand a set of students who belong to A class in high school, the total number on students are 40, and the set of birthday, one year is made up with 365 days, so there are 365 elements. Those of two set don’t correspond by one to one.

A birthday for a student is only one, but while...

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【Correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

As to a single student’s birthday, it’s only one, but while the number of the students are 40, the number of the birthday is 365, as a result, as to the rest of 325 of the birthday, none of the students correspond with them.

In addition there is a possibility that two or more students correspond with one birthday. It means that there is a possibility that two or more students have the same birthday. The author may add that the probability of having the same birthday in a class made up forty...

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【Correspondence by one to one and a string of Hideyoshi】

《Making a pebble and a thing correspond》

The probability that having the same birthday in a class made up forty students is 89%, the author said. Its high unexpectedly.

《Hideyoshi has a marvelous mathematical sense》

It is well known that Hideyoshi Toyotomi had a splendid mathematical sense, and the author said he is going to show us his episode when he used to be a subject of Nobunaga Oda. Hideyoshi used the correspondence by one to one skillfully and he was put more confidence by Nobunaga then.

Nobunaga ordered his common foot...

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【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

Nobunaga ordered his common foot soldiers 足軽 to count the number of the trees on the mountain behind his territory for an investigation one day.

His subjects obeyed the order from their master and started to count but they fell into a confusion at once, for they separated into groups and counted the number of the trees, but they were not sure that who counted which trees.

When looking at the confusion, Hideyoshi said the subjects like the next.

There are a thousand of strings here.

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【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

You don’t have to count the trees, but tie a string a tree together one by one, all the trees.

Hideyoshi said like that, and the common foot soldiers obeyed and went to the mountain behind the territory again.

After an hour, all the common foot soldiers returned, and Hideyoshi made them collect the rest of the strings and count them. If the rest of the strings were 220, it turned out that the number of the trees were 780.

The more trees are, the harder to count, but Hideyoshi made...

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【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

...but Hideyoshi made the trees which are hard to count to strings which are easy to count correspond, and succeeded in counting the number of the trees in the mountain behind the territory.

It is said that after that not only Nobunaga but all the other subjects raised their hats to him more and more.

When there a thousand of invited guests in a wedding ceremony, we can see whether or not all of the invited guests were present there at a glance, for seats for the invited guests alone ...

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【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

...for the number of seats for invited guests alone are prepared for in an usual wedding ceremony, so if there is an absentee, we have only to see a list of the seating order. We can see who is the absentee right away.

When wanting to know the number of the one who go to a movie theater, if going into there and counting them one by one actually is hard, and some of them may go to a restroom, so we don’t have to do it. There are stubs 半券 from admission tickets at an entrance, so we have...

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【Correspondence by one to one and a string of Hideyoshi】

《Hideyoshi has a marvelous mathematical sense》

We have only to count the number of the stubs.

Thus the correspondence by one to one has been widely used at present so as to count numbers efficiently.

『Supplementation. A way of seeking probability of people who have the same birthday in other forty』

At first we seek for the probability that none of the forty have the same birthday. We choose a one at random, and as to its birthday, even if which birthday it is, it’s all right.

The probability that the next people is different...

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《Hideyoshi has a marvelous mathematic sense》

『Supplementation. A way of seeking the probability that having the same birthday in forty people』

The probability that the next person is different from the first one on the birthday is three hundred sixty four three hundred sixty fifths.

The probability that the third person is different from the other two on the birthday is three hundred sixty three three hundred sixty fifths.

The probability that the fortieth person is different from the other 39 is three hundred and twenty six three hundred sixty fifths.

In the next, multiplying each of...

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《Hideyoshi has a marvelous mathematical sense》

『Supplementation. A way of seeking the probability of having the same birthday in forty people』

In the next, multiplying each of the probability one by one each other, and its value is 0.1087, and it’s the probability that none of them have the same birthday in forty people.

As a result, the probability that having the same birthday in forty people is 1−0.1087 = 0.8912, so it’s about 89%.

To my sorrow, I can understand why it is. Could someone kind please tell me why?

I think Arabian number is by far better than English this time.

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【Correspondence by one to one and a string of Hideyoshi】

《The device by Descartes》

A situation in which the correspondence by one to one isn’t the one alone when counting numbers.

As introducing coordinates, we’ve learned to express a figure and a graph with a numerical formula, for a pair of points on the graph like （1.2）corresponds with one to one.

When looking at the process which Descartes introduced the coordinates, we can understand it’s more significant than counting numbers.

When thinking over geometry, we find it difficult, but Descartes changed the difficulty into a ...

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【Correspondence by one to one and a string of Hideyoshi】

《Device by Descartes》

...but Descartes changed into the difficulty into a mechanical work of changing a numerical formula and handling it.

When being a students it seems that there are lots of people who could solve an equation easily but are poor at geometry.

The author said when looking at his students in his cramming school, almost all of them can solve a linear equation, a quadratic equation, and a simultaneous equation as long as the author taught them how to solve.

Actually as a way of solving those basic equations has been...

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【A correspondence by one to one and a string of Hideyoshi】

《Device by Descartes》

Actually as a way of solving those basic equations has been established, we can get the answer relatively easily as long as we know its process, but a question on figures don’t go ahead so easily.

Even if we understand how to solve a question, we frequently tend to be worried about other question from the scratch. As a result, if trying to solve the question on figures, we need a sense or an inspiration.

Descartes was worried about it as well, so he thought of coordinates which means figures and numbers.

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【A correspondence by one to one and a string of Hideyoshi】

《Device on Descartes》

Descartes thought of correspondence, it means making use of the figure and the number and make them correspond by one to one in order to change of thinking of a question on geometry into solving an equation.

In short, he tried to change something difficult into something simple.

When there is a question which is difficult to solve, taking advantage of the correspondence by one to one, and changing the difficult one into simple one. It has been done in algorithm which is a process of calculating in computer.

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【Correspondence by one to one and a string of Hideyoshi】

《Device on Descartes》

Algorithm makes a computer decrease its burden and lead to a quick outcome accurately. Furthermore, the correspondence by one to one is important in the world of function.

The author said he is going to go over the function here.

If saying y is a function to x, it means that the value on the y is fixed by the value on the x functionally.

Function is 関数 in Japanese, and the word came from China originally. Then 函数 used to be adopted. As to the essence on the function, 函数 is easier to understand than 関数.

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【Correspondence by one to one and a string of Hideyoshi】

《A device on Descartes》

The word of 函 has a meaning of a box. If saying y is a function to x, it means that x was input to a box 函 and a value came out of the box. It’s y, but the box isn’t an inaccurate one. When pushing the specified input button, it corresponds with the specified output goods, it’s a reliable box as if it were a vending machine in a town.

But as to some ordinary vending machine, some buttons often correspond with a single goods alone which is popular among us, so we can’t specify the goods which button we pushed.

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【Correspondence by one to one and a string of Hideyoshi】

《Device on Descartes》

The situation resembles the next one.

Each of cause corresponds with a outcome, but we can’t specify the cause from the single outcome.

《Imaging from a secret tool by Doraemon》

When our sweetheart was in a bad mood, but we didn’t know why it was, then we were forced to be worried about it.

Or when a score on golf was getting worse, if we know why the score became bad, we can improve the situation.

If it’s fixed that something causes anything definite, it’ll be helpful for us. For example, our sweetheart...

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【Correspondence by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

For example, our sweetheart is in a bad mood in the morning, or when shooting with a driver, if straining himself, he is apt to put slice on a ball.

In addition if we can specify the cause from the outcome, we’ll be encouraged with it.

When we can specify the outcome from the cause, and we can do the cause from the outcome, then the correspondence by one to one is formed between the cause and outcome.

At the same time, when y is a function to x, and at the same time x is the function to y...

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【Correspondence by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

...then x and y corresponds with one to one and it is said the inverse function exists in math.

As to a function, if the inverse function exists, after x is changed into y through the box 函, and returning through the box, y changed into x again.

If imaging the Gulliver tunnel which is one of secret tools by Doraemon, it may be easy to understand. It has two entrance. The one is big and the other small. If going into from big one, we are smaller, but if going into the small one, we return...

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【Correspondence by one to one and a string of Hideyoshi】

《Imaging one of secret tools on Doraemon》

...if returning from small one to big one, we return to the same size.

By the way, I have little knowledge on Doraemon, and I’ve never heard the Gulliver tunnel until now. Others call me strange, but I’m not so eccentric.

The correspondence by one to one on the function has been used in computer for compression when exchange some data between computers.

A way of compress the data is two, the one is a Lossless compression and a Lossy compression.

After compressing a file, we can restore...

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【Corresponding by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

After compressing a file, we can restore on original file from the compressed one. It’s called the lossless compression.

On the other hand, once the file is compressed, we can’t restore the original file. It’s the lossy compression.

By the way, I have little knowledge on computer, so when expressing, I find it boring, so I’m going to give it up.

Math has developed with the correspondence by one to one before numbers were born, so being able to understand the correspondence by one to one...

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【Correspondence by one to one and a string of Hideyoshi】

《Imaging from a secret tool on Doraemon》

...so being able to understand the correspondence by one to one means having the most basic ability on math like being able to understand a relation between orders, to observe, or to abstract.

【An estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

The author said he remembered a conversation with his friend when he was a freshman in university. The one who started to talk was the author.

The author said, how many jeans do you have?

His friend, oh!...

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【An estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

His friend, oh! Suddenly, what’s up?

The author, I bought a pair of jeans yesterday and I’m wondering how many jeans others have.

His friend, I’ve had three at present. When having worn out, I’ll throw it away.

The author, I’m the same one, by the way, how many jeans do you buy in a year?

His friend, Mmmm....a pair of jeans in a year.

The author, Are the others like that?

His friend, There are the ones who buy more, but there are others who rarely buy, but as to our generation, we are the average

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【Estimation on Fermi and something approximate】

《Estimating a scale of market on jeans》

His friend, There is a possibility that the older we are the less we buy, so considering the whole nation, a person buy a pair of jeans in two years.

The author, Oh, yes, the whole population in Japan is about one hundred twenty million, so multiply one hundred twenty million by zero point five is equal sixty million, so the whole Japanese buy sixty million pairs of jeans a year. How much is the average price on a pair of jeans?

His friend, Some of them were priced at ten thousand yen like Levis or...

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【Estimation on Ferm and something approximate】

《Estimating a scale of market on jeans》

His friend, Some of their price is ten thousand yen like Levis and Edwin, but others are more reasonable like for children, and the ones which aren’t big name brands, if an average price on a pair of jeans is five thousand yen, the scale of market on jeans in Japan is multiply sixty million by five thousand is equal three hundred billion, so it’s about three hundred billion.

The author, GDP, the gross domestic product a year in Japan is five trillion, so the scale of market on jeans is equal to about...

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