Snowflake curve is named after its shape is similar to snowflake, and its making assumption is also similar to snowflake.
The magic of snowflake curve is that its area is limited, but its circumference is infinite!
The perimeter of the snowflake curve is increasing, and there is no boundary, but the whole curve can be drawn on a small piece of paper.
So its area is limited, in fact, its area is equal to 8/5 times of the original triangle.
2. Crystal-a polyhedron in nature
Polyhedron has appeared in mathematical works since ancient times, but its origin is so ancient.
It can almost be related to the origin of nature itself.
Crystals often grow into polyhedral shapes. For example, sodium chlorate crystals are cubic and tetrahedral.
Shape; Chromite crystals have an octahedral shape. What is puzzling is that in marine microbial radiation,
In the skeletal structure of insects, there are actually dodecahedron and icosahedral lenses.
If a polyhedron is like this, all its faces are equal and the angles of these faces are equal, then
This polyhedron is called a regular polyhedron. All faces of a regular polyhedron are the same and all sides are equal, and
All angles are equal. There are countless kinds of polyhedrons, but there are only five kinds of regular polyhedrons. regular polyhedron
Also known as Platonic style, Plato discovered it independently around 400 BC and was named by later generations.
The existence of regular polyhedron was known long before Pythagoras. The Egyptians even put some of them.
Some are used for spectacular buildings and other objects.
3. Practice circle
If we have a compass in our hand, fix one foot, turn the pencil head around the other foot and draw.
There is a circle. However, it is such a simple circle that gives us a lot of inspiration and is fully conveyed.
Used in human production and life. The image of wheels is round, water pipes are round, and many containers are cylindrical.
Shapes, such as washbasin, cup, bucket, etc. Why use a circle? On the one hand, circles give us vision.
On the other hand, circles have many practical properties.
As we all know, a circle is the locus of a point whose distance from a fixed point is equal to a fixed length. That is, a point on the circumference is a circle.
The distance between two hearts is equal. This is one of the most important and basic properties of a circle. Wheels are made up of circles.
Made of this property. The axle is installed in the center of the wheel, and the distance from the wheel edge to the axle is certain.
. When the car is driving, the distance between the wheelbase and the road surface is always the same. In addition, as long as the road surface is flat,
The car will not bump, giving the rider a smooth and comfortable feeling. If we make the wheel square, we will put it in.
The axle is located in the symmetrical center of the wheel. When the car is driving, the distance between the axle and the road surface will be large or small, even when it is walking.
On a flat road, the car will bump up and down, and passengers will feel uncomfortable.
Another characteristic of a circle is that it surrounds a triangle or a square or a circle with materials of the same length, wherein
The largest area is a circle. Similarly, people come to the conclusion that cubes and cylinders are made of materials with the same area.
It will be bigger. Using this characteristic, people have made various cylindrical products: cylindrical barns,
Cylindrical water tower, cylindrical underground pipeline, etc. A circle is a special curve, which has many properties and similarities.
Application, I hope students study hard and develop more circle uses.
4. Mathematical treasures from the ocean
There is a saying that the ocean is the cradle of life. In the ocean, just like on land, the form of life becomes mathematics.
Rich ideas.
People can see many kinds of shell-like spirals. Nautilus and Room and Nautilization
The stone gives an equiangular spiral.
Sea lion snails and other conical shells provide us with examples of three-dimensional spirals. Symmetry fills the ocean-
The shell shapes of Paleozoic animals such as clams, trilobites, lobsters and fish are symmetrical.
Central symmetry was found in radiolarians and sea urchins.
Geometric shapes are equally colorful-pentagons can be seen on sea urchins in the eastern United States, while seagulls
A regular polygon with different sides can be seen in the shape of the tip; The outline of sea urchin is spherical; Involute rule of circle
Similar to the curve formed by bird and clam shells; The shape of polyhedron can be seen clearly in all kinds of radiolarians; sea
The rocks on the edge become round or oval under the eternal beating of the waves; Coral and free jellyfish
Form a curve that bends randomly or almost bisects.
The golden rectangle and the golden ratio also appear in marine life-where there are regular pentagons, there are us.
You can find the golden ratio. There are many pentagons in the patterns of sea urchins in the eastern United States. What about the golden rectangle
Directly manifested in shellfish with small rooms such as nautilus.
Swimming at the bottom of the sea can give people a real three-dimensional sense. People can swim almost effortlessly.
In three directions of space.
In the ocean, we can even find mosaic patterns. A lot of fish scales are a kind of perfection.
Beautiful mosaic.
Waves consist of cycloids and sinusoids. The movement of waves is like an eternal movement. sea wave
Waves vary in shape and size, sometimes strong and irresistible, sometimes gentle and calm,
But they are always beautiful and controlled by mathematical principles (cycloid, sine curve, statistics). Finally,
Is there no reason to think that the sand in the sea once inspired the ancients to form infinite thoughts? When we are interested in everyone.
When a mathematical thought is deeply studied, it will be found that they are complex and interrelated. Whenever I am studying by myself,
When they are found in nature, they gain new meanings and connections.
5. The golden section makes beauty.
The key to music harmony lies in its frequency, and the key to stage design lies in its center.
Where the weight is placed, the music is placed and the center of the stage is placed, and the effect is the best.
What about the effect? Artists are often considered by artists. However, mathematicians tell us that as long as
If you put it at the golden section, you will achieve your goal. It's amazing. Many things are just used.
The golden section is easy to solve. It embodies its beauty in architecture, art and even music.
beautiful
As early as 100 years ago, the German psychologist Fauchner carefully made rectangles of various proportions, and
A "rectangular exhibition" was held and many friends were invited to participate. After the visit, everyone was asked to vote.
Choose the most beautiful rectangle. The proportions of the four rectangles finally selected are: 5×8, 8× 13, 13×2 1,
2 1× 34. After calculation, the aspect ratio is 0.625, 0.6 15, 0.6 19 and 0.6 18 respectively. this
These ratios are all around 0.6 18. In fact, about 500 BC, Pythagoras in ancient Greece.
They became interested in this problem. They found that when the aspect ratio of a rectangle is 0.6 18, its shape
The most beautiful shape. So we named 0.6 18 "golden number", which is the origin of golden number. As mentioned earlier.
This number is a wonderful number, waiting for you to explore its mystery.
6. The mathematical genius of animals
A beehive is a strictly hexagonal cylinder with a flat hexagonal opening at one end and a closed one at the other.
The bottom of a hexagonal rhombic pyramid consists of three identical diamonds. The obtuse angle of the rhombus forming the chassis is 109 degrees 28 minutes.
All acute angles are 70 degrees and 32 minutes, which is firm and material-saving. The thickness of the honeycomb wall is 0.073 mm, with an error.
Very small.
Red-crowned cranes always move in groups, forming a "human" shape. The angle of the herringbone is 1 10 degrees.
More accurate calculation also shows that half of the herringbone angle, that is, the angle between each side and the advancing direction of the crane group, is
54 degrees, 44 minutes and 8 seconds! And the angle of diamond crystal is exactly 54 degrees, 44 minutes and 8 seconds! Is it a coincidence or something big?
Natural "tacit understanding"?
Spider's gossip web is a complex and beautiful octagonal geometric pattern, so people use a ruler.
It is also difficult for a compass to draw a symmetrical pattern like a spider web.
In winter, when a cat sleeps, it always hugs its body into a ball, and there is math in it, because the ball makes its body.
The surface area is the smallest, so the heat emitted is the least.
The real "genius" of mathematics is coral. Coral writes a "calendar" on itself, which is there every year.
I have carved 365 stripes on my body wall, which is obviously one stripe a day. Strange to say, Gu Sheng.
Biologists have found that corals 350 million years ago "painted" 400 watercolors every year. Astronomers told me
Tell us that at that time, the earth was only 2 1.9 hours a day, not 365 days a year, but 400 days.