Geometry, the root of artificial intelligence and the subject most challenging for young people, is breaking down barriers.
Angles, congruence, similarity, area, and solids encountered through time travel

Why do teenagers find geometry so difficult? Geometry is a discipline that involves thinking about "space" and "relationships" within shapes.
So, geometry is the root discipline of art, technology, and artificial intelligence.
To properly understand geometry, you must ask and experience the question, 'Why did this concept come into being?'
If you only memorize definitions and formulas, you will build a high wall that will make it difficult to become close to others.


"The World of Geometry for Teens" is an encounter with geometry in the form of time travel.
To help understand geometry, the author, who received a Ph.D. in mathematics from Moscow State University in Russia, explains geometry in a story format that transcends time and space.


This book covers all aspects of geometry from the middle and high school level, from the basics to advanced levels.
Each chapter centers around three young protagonists, who engage in discussions and proofs with real mathematicians from ancient times to the present.
In it, you can feel and understand geometry with your body.
However, areas where problems are solved using formulas or using function concepts are excluded.
So, even middle school students can read it, and it helps them solidify their basic geometry and thinking skills.

Humanity has been steadily studying and developing geometry.
Geometry has allowed us to build great pyramids, map the entire Earth, and measure the distance to the Sun while sitting on Earth.
Without knowing the angles, lengths, and shapes of the figures, you couldn't do anything like that! Thanks to this, we now use GPS to find unknown paths, transmit sound and images wirelessly, and send spacecraft outside the solar system.
(syncopation)
Geometry is also a very old discipline.
Even at the most conservative estimate, it is at least 3,000 years old.
It has been at least 2,300 years since Euclid's Elements, which is called the Bible of mathematics, came out, and there is quite a bit of evidence that humans thought about shapes long before that.
In fact, the age of geometry is infinite.
Let's think about the simplest shape, a straight line.
When did straight lines originate? And when did humanity begin to conceive of the "concept" of a "straight line"? Thinking this way, we can immediately see that geometry is infinitely old.
--- From the text “Preface”

“I assumed the two bars below were in a straight line… Okay.
Can you tell me if I understand correctly? A right angle is the measure of a straight angle.
So it didn't lean to either side.
That is, when a straight line is placed on another straight line and the two angles are adjacent to each other, if the sizes of the left and right angles are the same, both angles are right angles.
how is it?"
All three nodded at the same time.
Nicole said in a low voice.
“A right angle is an angle that is not inclined to either side.
So the building is built at right angles and we stand at right angles too.
… how can we verify that these two bars are placed in a straight line?”
--- From the text "Chapter 1 | Geometry of Angles"

Aesop: So a triangle is defined by three sides, three vertices, and three angles.

Euclid: You've summarized it well, just to the point! Now, how do we know if two triangles are congruent? Remember when you said a triangle is defined by three sides, three vertices, and three angles? Seems like the answer lies there?
(syncopation)
Aesop: All three sides must be the same length and all three angles must be the same size.
It doesn't matter where the points, lines, and angles are, as long as they are the same size, they will completely overlap.
A point has no size, so just check its three sides and three angles!
Euclid: Excellent.
(Omitted) If the lengths of the three sides and the sizes of the three angles are all the same, they cannot help but overlap.
Because they determine the triangle.
Now let's ask the question a little differently.
Wouldn't it be possible to know only some of the six elements (three sides, three angles) without having to know all of them?
--- From the text "Chapter 2 | Geometry of Congruence"

There is a person standing on the left and a mirror is placed at point A.
The distance between the person and the mirror is 3, and the distance between the mirror and the tree is 9.
If two triangles are similar (and, in fact, they cannot be similar due to the nature of mirrors), you can measure the height of a tree by knowing the height of a person's eye from the ground. If the similarity ratio is 3, then the tree's height will be three times the height of the person's eye.
Can we imagine it again?
--- From Chapter 3, Geometry of Similarity

To summarize Archimedes' words:
(1) The area of ​​a triangle can be found by the length of the base and the height of the triangle.
(2) All polygons are divided into triangles.

The area of ​​a polygon can also be found from these two facts.
That's because you can divide the polygon into triangles, find the area of ​​each triangle, and then add them.
For example, a square piece of land can be divided into two triangles like this.

--- From the text "Chapter 4 | Geometry of Area"

After much research, Brahma discovered the following facts:

First, if we know the sine values ​​for two angles, there is a way to find the sine value corresponding to the angle that is the sum of the two angles.
Only addition and multiplication are used.

Second, if we know the sine values ​​for two angles, there is a way to find the sine value corresponding to the angle that is the difference of the two angles.
Only multiplication and subtraction are used.
Third, if we know the sine value corresponding to an angle, there is a way to find the sine value corresponding to half the angle.
Use square roots.

For example, if you know the sine values ​​of 45 degrees and 30 degrees, you can find the sine value corresponding to 75 degrees, which is 45 + 30.
Draw a right triangle with one angle of 75 degrees and measure its length without even having to!

--- From the text "Chapter 7 | Geometry of Trigonometric Ratios"