Science,English, and math 9th

＞＞346

【Pi and irrational number】

《When changing the pi into decimal number, without recurring it continues forever》

√2 and √３ are also irrational number. When expressing them with decimal number, without recurring, the number continues forever.

There is no regularity on the decimal number on the pi.

As to the illustration on the one-seventeen like 346, there is something regular, for it’s the recurring decimal, but as to the illustration on the pi, there is nothing regular, for it’s not the recurring decimal.

When looking at it, it’s obvious, though I can’t show you it. I’m sorry for it.

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＞３５２

【Equation】

《Equation isn’t hard. It’s a quiz in which a pair of scales is used》

Please solve equation of 2x＋3＝x＋5, if setting the question to a elementary school students they’ll say they can’t understand what it means, but if showing them the illustration and asking, how many balls are in the box, lots of them can answer, the author said.

There is a pair of scales. There are three balls and two boxes in the one scale pan and one box and five balls on the other scale pan. Needless to say, the pair of scales is balanced.

But the equation and the pair of scales mean the same thing.

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＞＞352

【Equation】

《Equation isn’t hard, for it’s a quiz in which a pair of scales is used》

Then the weight on the ball is the same, and we don’t consider the weight on the box.

Even if getting rid of the same things from both of the right and left of the scale pan, it could have been balanced, so removing the one box from both of the scale pan.

Then there are three balls and one box on the left scale pan, and five balls on the right scale pan and they are balanced. Moreover if taking away three balls from both of the scale pan, we find one box and two balls are balanced.

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＞＞352

【Equation】

Thus, we find there are two balls in the box.

《The = in the equation means the balance on the scale》

Solving the equation is the same way of thinking on the pair of scales. Trying to avoid breaking the balance which connected left side and right side with the =, transforming the equation, and we have only to express the content of the box simply.

Speaking of the equation, it seems to be something difficult, but its way of thinking is so simple like that.

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【Mystery in shape】

Triangle, regular pentagon, and regular polyhedron 正多面体. Various shapes are handled in math. When appreciating those shapes carefully, mysterious beautiful law is discovered in math. Ancient mathematicians had been fascinated by mysterious features hidden in various shapes. I’m going to show you interesting nature on math hidden in plane figure or in solid figure, the author said like that.

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＞＞357

【Division on triangle】

《A triangle is atom on shape. If gathering the triangle, we can form any kind of shape》

Animated horse is shown. The horse is depicted with lots of triangles. In animation or game which computer is adopted, technical skill of expressing a thing is used by gathering little triangles, though sometimes a square is also used.

Polygons like square and pentagon are cut by diagonal line and we divide them into triangles. In other words, if gathering lots of triangles, we can form any kind of complicated shape.

As a result, the triangle is the base in shape and the…

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＞＞357

【Division by triangle】

As a result, the triangle is base in shape and is a thing like atom in the shape.

《Eyler discovers Catalan number. What’s that?》

Let’s think of way of dividing complicated polygon into triangles.

Polygon of all angles are less 180 degree is called convex polygon. If cutting a convex square with diagonal line, it is divided into two triangles. As being able to draw two diagonal lines in the convex square, so the way of dividing is two.

In convex pentagon, there are five ways of dividing. In convex hexagon, fourteen ways. In convex heptagon, forty two ways.`

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＞＞357

【Discovery by triangle】

Thus the way of dividing increases.

Euler who was born in Swiss in the 18th century has discovered how many ways of dividing convex polygon into triangles.

There is a polygon which is convex（n＋2）. Then the n is 2, for the polygon except for triangle, it starts from a square, I’m wondering. A way of drawing a diagonal line in the convex （n＋2）and dividing into the triangle without crossing. Its number is expressed with an equation in which the n is used. Euler has discovered it.

The equation is expressed with a fraction. Its denominator is （n＋1）!n! And its…

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＞＞357

【Discovery by triangle】

…and its numerator is （2n!）

What does it mean the ! I’m sure no one exclaims, but I’m not sure what it means. Catalan who is other Belgian mathematician reached the same equation from other question in the 19th century, and the equation was named after him, so we call the number of dividing the polygon the Catalan number at present.

As you know, I have little knowledge on math, so to my sorrow, I can’t understand very much. If you can understand it, could you please tell me on the equation in detail?

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＞＞362

【Pythagorean theorem】

《Pythagorean theorem which has been discovered in the time of Pythagoras is the base in geometry
》

Pythagorean theorem which is related to right triangle is one of the most famous and the most beautiful one in the theorems of math. Pythagoras who was a mathematician in Ancient Greece discovered the theorem when looking at tiles spreading on the floor of the palace.

There is the legend like that, so we call the theorem Pythagorean theorem.

In the 362, there is a right triangle and three squares. Each side of the square is each side of the right triangle.

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＞＞362

【Pythagorean theorem】

What is the Pythagorean theorem? When adding both of area on the A to B, its numerical value is equal to the area of the C.

If looking at it from other side, when the triangle meets the condition on the Pythagorean theorem, it means the it’s a right triangle. As to the theorem on Pythagoras it’s able to be demonstrated in the way which is shown below.

By the way, I’ve tried to show you it here, but part of it overlapped with the 362, so I can’t. It’s my mistake. I’m sorry for it.

There are three illustrations, and one of them is the one in which there is C…

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＞＞362

【Pythagorean theorem】

…one of them is the one in which there is the C on the middle of it in the left side of the 362, so there are other two illustrations.

《Kind of religious community by Pythagoras which thought numbers rule controlled the world》

Pythagoras had lots of disciples and led the kind of religious group. The members on the religious group believed in that there were numbers on the base for everything and the numbers controlled the world.

There are lots of episodes on the kind of the religious group led by Pythagoras. One of them is a perfect number. What’s that?

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＞＞362

【Pythagorean theorem】

A natural number which is divisible for other one is called a divisor 約数. When researching all the divisors on the natural number and adding all the divisors except for the natural number itself. Then if the total number of the divisor is equal to the natural number, we call it the perfect number.

For example, as to 28, when adding all its divisors, 1＋2＋4＋7＋14＝28, so the 28 is the perfect number. The kind of religious group led by Pythagoras regarded the perfect number as sacred and gave it a special treatment.

I’m going to express the two of illustrations.

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＞＞352

【Pythagorean theorem】

There is a big regular square and two of the regular square of which size is different are in the big one. The small one is A and is located in the upper left in the big one. The bigger one is B and is in the lower right in the big one. The small one and the big one are diagonal.

As a result, there are two rectangles in the big square. The one is upper right and the other is lower left. One side of the rectangle is equal to the one of the square of A and the other side is equal to the one of the other square of B.

Each of the rectangle is equal and each of …

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＞＞362

【Pythagorean theorem】

Each of the rectangle is equal and is divided in half by diagonal line, so each of two of the rectangle is divided into two triangles each other.

The rectangle which locates in the upper left is divided into two triangles and its lower side moves to diagonal position. At the same time, there is other rectangle which locates in the lower left. It’s also divided into two triangles, and its upper side moves to directly above.

Then there is a square which is surrounded by four triangles. It’s the illustration on the left end of the 362. The part of C is equal to…

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＞＞362

【Pythagorean theorem】

The area on the part of C is equal to the total on A and B. It’s the Pythagorean theorem.

It has been taught in the class of math in junior high school, I’m wondering, but I didn’t try to understand it when being a student in junior high school, for I hate math.

To tell the truth, I’ve repented of it.

I’m not sure whether or not numbers controls everything, but numbers are related to our every day life closely, so if studying harder I’ll find it interesting, though it seems to be late, but I can kill time at least.

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＞＞370

【Five pointed star 五芒星 and golden ratio】

The one which is made up with diagonal lines in pentagon is called a five pointed star. When being asked to write a star shape on the paper with a pen, lots of people will draw the five pointed star, the author said.

The five pointed star has been adopted in design for national flag in lots of countries. The cult led by Pythagoras regarded the five pointed star as sacred figure, and adopted it as symbol mark for the cult.

Some numbers which are important and mysterious for math are hidden in the five pointed star. It’s the golden ration.

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＞＞370

【Five pointed star and golden ratio】

The golden ratio is the one which is 1：1618…. And 1.1618… is also called a golden number.

When the length on one side of regular pentagon is 1, what is the numerical value on the length on the diagonal line which makes up the five pointed star?

The members in Pythagorean cult worked on the problem, and they got the numerical value which is 1：1618…, the golden ration.

《The mark on the golden ratio is φ and is named after Parthenon Palace》

The golden ratio is often said it brings something beautiful in construction. Its typical one is the ….

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＞＞370

【Five pointed star and golden ratio】

The typical one on the golden ratio is the Parthenon Palace which is built in Athens in the time of the Ancient Greece, and the ratio between depth and width is almost the golden one.

The golden ratio which is 1618… is expressed with φ, so it’s done with 1：φ. The φ is Greek letter and it is said it originates from Pheidias who was a sculptor and directed the construction on the Parthenon Palace.

But it has been unclear whether or not the sculptor in Ancient Greece recognized the golden ratio and used the golden ratio for the Parthenon Palace.

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＞＞370

【Five pointed star and golden ratio】

In addition, the golden ratio causes nature what is called self-similar. The self- similar is a part of something is expanded or contracted in size which is similarity, and the similarity matches with the whole thing.

For example, some of isosceles triangles 2等辺三角形 are in the five pointed star, and the isosceles triangles become similarity and they appear repeatedly. If connecting each of vertex on the isosceles triangles, a beautiful spiral appears. We call it a golden spiral.

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＞＞374

【Five pointed star and golden ratio】

《The ratio on the side of the isosceles triangle is the golden one no matter how far it goes ahead》

The pointed five star is made up with diagonal lines on the pentagon and there are lots of golden ratio hidden in the pointed five star.

What is the ratio between the diagonal line AB and the side on the regular pentagonBC?

When the length on the side of the regular pentagon, BC is 1, the length on the diagonal line of AB is 1618. In short, BC ：AB is equal to 1：1618, it’s the golden ratio. Moreover the isosceles triangle ABC and the other one…

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＞＞374

【Five pointed star and golden ratio】

Furthermore the isosceles ABC and other one BCD are similar, so CD ：BC is equal 1：1618, this is also the golden ratio. Besides, DE：CD and FG：EF are also the golden ratio. All of them is 1：1618,

Spiral appears in nautilus オウムガイ, and it resembles the golden ratio, but it is said the way of winding is different from the golden ratio a little.

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＞＞３７６ and ＞＞３７７

【A regular polyhedron 正多面体 and golden ratio】

《We can make a regular icosahedron 二十面体 with three piece of name card. Its key is also the golden ratio》

We can discover the golden ration in the ones which are close to us. For example, it’s a name card. In general, as to the name card, when its short side is 1, the other long side is 1.618., lots of them are close to the golden ratio.

Making use of its nature, we can create a regular icosahedron with a combination of three pieces of name card. The regular icosahedron is a three dimension which has twenty triangles as face.

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＞＞376 and 377

【Regular polyhedron and golden ratio】

Preparing for three pieces of name card and making a notch each of them. When combining them and making three of them vertical, it becomes the shape like 377, though it doesn’t seem to be the regular icosahedron at first glance.

But when connecting the vertex on three cards, it becomes the regular icosahedron like 376, the author said like that.

《The golden ratio is hidden in the regular icosahedron》

When using the name card, we can make the regular icosahedron. Why? Because the golden ratio hides in the regular icosahedron.

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＞＞376 and 377

【Regular polyhedron and golden ratio】

Let’s pay attention to one side on the regular polyhedron and other one which is opposite. Those sides are parallel and connecting its vortex, a rectangle appears then. When its short side is 1, its long side which is the diagonal line in the regular icosahedron is 1.1618, the golden ratio.

The rectangle of which ratio on length to width is the golden ratio is called a golden rectangle. If cutting out a square of which side is the short side, a small rectangle is left. The small rectangle also becomes the golden one. The nature on self…

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＞＞376 and 377

The nature on the self similar appears there.

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＞＞381

【Mysterious nature on a regular polygon】

《Regular dodecahedron 正十二面体 and regular icosahedron can change into other shape each other》

A cubic which is surrounded by the same shapes on the surface is called a regular polygon.

When comparing a regular dodecahedron which is surrounded with twelve regular pentagons with a regular icosahedron which is surrounded by twelve triangles, there are difference between the shape on the surface and the number on the surface on the two of them.

The regular dodecahedron is the left one and the regular icosahedron is the right one on the picture.

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＞＞381

【Mysterious nature on regular polyhedron】

《Regular dodecahedron and regular icosahedron can change into other shape each other》

But both of the regular dodecahedron and the regular icosahedron have inseparable connection.

《If cutting out the vertexes on the dodecahedron?》

Cutting out the vertexes from the dodecahedron, and making the cutting plane change into triangle. It’s the illustration of 1.

When making the way of cutting bigger, the regular triangles connect each other. It’s the illustration of 2. Besides, if making the way of cutting still bigger, regular hexagons appear.

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＞＞381

【Mysterious nature on regular poly】

《Regular dodecahedron and regular icosahedron can change into other shape each other》

The polyhedron on which the regular hexagons appear is the illustration of 3.

If making the way of cutting by far bigger, regular triangles appear and the dodecahedron changes into the icosahedron at last. It’s the illustration on the right end

Contrary to it, if starting from the icosahedron, the icosahedron can change into the dodecahedron. We call the nature dual 双対 in mathematical word. Both of the dodecahedron and icosahedron are the dual regular polyhedrons

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＞＞381

【Regular dodecahedron and regular icosahedron can change into other shape each other 】

《There is still more mysterious nature on the polyhedron》

As to the polyhedron which is a plane figure like regular pentagon, regular hexagon, we can make them as we like as long as we increase the number of the vertexes, even if the number of the vertexes increase to hundred, it’s all right.

But as to the polyhedron which is solid figure, it’s limited five kinds. Regular tetrahedron 正四面体, regular hexahedron, regular octahedron 正八面体, regular dodecahedron, and regular icosahedron.

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＞＞381

《There is still more mysterious nature on the polyhedron》

In the time of Ancient Greece when Plato was alive, it has been already recognized that the regular polyhedrons are limited to five kinds alone, so those five kinds of polyhedron are called the cubic of Plato.

Euler who is a genius mathematician in the 18th century also took interest in the polyhedron. When the number on the vertex is V, the number on the side is E, and the number on the face is, an equation of F, V−E＋F ＝2 holds every kinds of polyhedron. We call it the theorem on polyhedron of Euler.

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【Graph and function】

Draw a graph showing the next function.

This kind of questions are lined up in a test of math. When hearing the function or the graph alone, the ones who are poor at math may have felt like avoiding it, but please don’t be worried about it, for the illustration or picture which are showed this time is the ones which we can appreciate easily.

Even if trigonometric function 三角関数, exponential function, differential or integral which seem to be hard to understand, we’d understand them through something visual intuitively.

The author said like that

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【Conic curve】

《Downlight illuminating the wall shows parabola and hyperbola 双曲線》

There is a light which illuminates the wall in the bedroom. In fact, the light has something to do with math.

The shape on the light spreading from the light source is regarded as cone mathematically, and the part of the shape which illuminates on the wall is the same cross section 断面 which the corn is cut down.

If cutting down the cone vertically against the axis on the cone, its cross section is a circle. It’s illustration A.

If tilting it a little, the cross section is ellipse. It’s illustration B.

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＞＞388

【Conic curve】

《Downlight illuminating on the wall shows parabola and hyperbola》

If tilting it still a little more, the ellipse becomes slender and changes into parabola soon. It’s Illustration C. Parabola is a curve like mountain. When throwing something in the air, it shows the curve in the air. We call the curve the parabola.

If tilting it still more again, a hyperbola appears on the cross section. It’s illustration D.

The hyperbola resembles the parabola very much, but they are different curves. There is function on inverse proportion 反比例. It’s expressed with function, y＝one xth.

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＞＞388

【Conic curve】

The graph on y＝one xth becomes the hyperbola.

The shape on the light illuminating on the wall depends on the direction which the illuminating instrument directs on the wall. It becomes the circle, ellipse, parabola, or hyperbola. Either of them appears on the wall.

《Conic curve which mathematicians in Ancient Greece studied》

We call the circle, ellipse, parabola, and hyperbola conic curves collectively. Orbits on celestial bodies influenced by the gravitation of the sun like a planet and comet show the conic curve.

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【Origami and parabola】

《It’s just that lots of straight lines were drawn up, but a curve appears there. How mysterious it is!》

It’s just that lots of straight lines alone are drawn up on the illustration of the left page, but it seems that there is a parabola which is a curve there. Why do we see the parabola in the illustration made up with straight lines alone?

《Let’s fold origami and make a parabola》

Using origami and make sure whether or not the parabola appears.

First of all, put a mark a single point somewhere on the origami with a pencil, and the point is A.

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＞＞391

【Origami and parabola】

《It’s just that lots of straight lines are drawn up, but a curve appears there. How mysterious it is’!》

In the next, drawing a straight line which doesn’t pass on the straight line and put a mark a single point somewhere on the straight line. It’s the point B. It’s the illustration 1.

Then folding the origami so as to overlap A and B. It’s the illustration 2.

After folding the origami, there is a crease, so opening the origami and drawing a line on the crease. It’s the illustration 3.

After that, moving the point of B on the straight line gradually.

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＞＞391

【Origami and parabola】

So, there is other crease, and drawing other straight line on the crease. Drawing lots of straight lines on the lots of creases, then the parabola appears there.

《We can understand the definition on the parabola from the origami》

What is the parabola? It’s defined that a set 集合 on mathematics that a point which is the focus and other points on some straight lines which don’t pass on the focus, and the focus and the other points on the straight lines are the same distance.

In the example of the origami, the point A is equal to the focus.

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＞＞391

【Origami and parabola】

《We can understand the definition on parabola from origami》

The creases are folded so as to overlap the focus and other points on the straight lines. The distance from the focus to the crease and the one from the point on the straight line to the crease is equal.

Those points gather and they become tangent 接線 on the point of the parabola.

As a result, if moving the point B more and more and making lots of creases, it means drawing the tangent which are plenty on the parabola, so the parabola rises to the surface.

【Trigonometric function 三角関数】

《The one…》

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＞＞394

【Trigonometry function】

《The one which appears on a spiral staircase is a curve on trigonometry function》

There is a spiral staircase attached to a tower, a lighthouse or inside an art museum. Its design is excellent and we can it in lots of other constructions.

When looking at the spiral staircase from its side, a beautiful curve which undulates toward both left and right on the handrail. In fact, the curve is very important in math.

《If a point on the circumference rotates…?》

The one which is deeply related to the curve is trigonometry function. The illustration in 394 is a…

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＞＞394

【Trigonometry function】

《If a point on the circumference rotates…?》

The illustration in the 394 is one of graph on the trigonometry function, y＝sin x. When reading the sin, we say it サイン in Japanese.

There is a circle of which radius is 1. Let’s suppose a point on its circumference rotated anitclockwise. With the rotation on the point, the position on the height of the point changes as if it undulated like the one which is shown in 394.

The one in 394 is the graph of y＝sin x which is the trigonometry function. Changing the direction on the graph in 394 from horizontal to vertical.

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＞＞394

【Trigonometry function】

It becomes the same curve with which appears on the handrail of the spiral staircase.

《Rotation and trigonometry function is a set》

Why does the shape which is the same with the one in the trigonometry appear in the spiral staircase?

If looking at the spiral staircase, its handrail is equivalent to the circumference on the circle.

When going up the stairs, the angle of rotation also goes ahead, so the position on the handrail is the same one with the graph of y＝sin x when looking at it from its side.

If happening to see the spiral case, please remember the…

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＞＞394

When happening to see the spiral staircase, please remember the trigonometry function, the author said like that, but to tell the truth, I have little knowledge on the trigonometry function.

Speaking of trigonometry function, except for the sin, there is cos and tan. To my sorrow, I’m not sure that not only sin but cos and tan.

The trigonometry function is 三角関数 in Japanese, but any triangle doesn’t appear in the explanation, but the phrase of 三角 is included in the Japanese one. Why?

I hope I can understand the trigonometry function very well someday in future.

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【Exponential function】

《The form on grand piano is related to exponential function》

A grand piano which is in a music room in school. As to its form, the side of low tone which is left side of keyboard is long, on the other hand, the side of high tone which is the right side of keyboard is short. Mathematical reason is hidden in the form.

There are some strings inside the grand piano, and making the strings vibrate, we can hear the sound of the piano. While the low tone corresponds with long strings, the high tone does with short strings.

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＞＞399

【Exponential function】

《The form on grand piano is related to exponential function》

As to the ド which is the highest tone in the grand piano is no more than 5 centimeters. When 1 octave is lower, how much is the length on the string of the ド?

《When 1 octave is lower, how long is the string?》

If the weight and tension on the string is the same, the length on the string becomes double whenever being lower 1 octave, so if the highest ド is 5 centimeters, the ド which is 1 octave lower is 10 centimeters.

Whenever 1 octave lower, the length on the string becomes 20, 40, 80 centimeters.

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