What do you think of the American mathematician s discovery of a pentagon that can be seamlessly lai

Perhaps the whole mathematics and the whole mathematical formula are the miracles that happened in China in the Internet era, and the whole mathematical formula is different from any mathematical formula in the past in that the whole mathematical formula is also the law of the whole cosmology, which can inspire people's understanding of life in the universe

It seems that there are new styles of sidewalk tiles for mathematicians.

You can consider what kind of convex polyhedra can be arranged into an entire gap-free space.

The new pentagonal shape discovered by Mr. and Mrs. Cassie Mann. All the pentagons in the diagram are congruent, and the author fills in three colors to indicate that they fill the entire plane in a group of three.

On August 19, according to foreign media reports, the research team of the University of Washington recently discovered a new irregular pentagon, which can be completely covered with a plane after being combined with each other, without overlapping or any gaps, and is the 15th pentagonal shape in the world that can achieve this effect. It has been 30 years since the last discovery of a pentagon with a similar effect, which is equivalent to the discovery of a new atomic particle in the field of mathematics.

According to the report, the research team is composed of Cassie Mann, an associate professor in the Department of Mathematics at the University of Washington, his wife Jennifer, and student Von der Lao.

According to the report, the two eventually used a computer program designed by von der Lao to find the perfect inlay of the pentagon. According to Cassie Mann, the discovery has helped people to thoroughly understand how different shapes are densely laid out on flat surfaces. <>

A regular hexagon can be densely paved because each of its inner corners is 120 degrees, accommodating exactly 3 inner corners at each splice point; The regular pentagon cannot be densely paved, because each of its inner angles is 108 degrees, and 360 is not an integer multiple of 108, and the inner angles at each splicing point cannot guarantee that there are no gaps or overlaps; With the exception of regular triangles, regular quadrilaterals, and regular hexagons, no other regular polygons can be densely laid with planes. Leaving no gaps is dense paving.

We all know that when paving, the floor should be covered with a blank space between the floor tiles. If the floor tile is square, and each corner of it is a right angle, then the 4 squares are put together, and the 4 corners at the common vertex are exactly 360 degrees around. Each angle of a regular hexagon is 120 degrees, and when the three regular hexagons are put together, the sum of the three angles at the common vertex is exactly 360 degrees.

In addition to squares and rectangles, regular triangles can also be used to spread the ground closely. Because each of the inside angles of a regular triangle is 60 degrees, when the 6 regular triangles are put together, the sum of the degrees of the 6 angles at the common vertex is exactly 360 degrees.

It is precisely because after the square and hexagon are combined, the sum of several angles on the common vertex is exactly 360 degrees, which ensures that the ground can be densely paved and is more beautiful. Because only the integer multiples of the inner angles of regular triangles, squares, and regular hexagons are 360°, only these three can be densely paved in regular polygons.

The circle should not be densely paved either, because the circle has no edges, it is surrounded by curves, and it has no degrees of angles. <>

The development of computers has led to the development of mathematics, and many studies that originally only stayed in theory can obtain results through high-speed calculations of computers.

Mathematicians are superb, mathematics is deeper than physics, more difficult to find, purely by imagination.

What do you mean by how to look at it? I'm actually very strange that a scientific problem for Mao should be made into a social problem: what do you think...

I don't have anything to look at myself, but when I bought a house, the carpet in each house was made to be this kind of non-repetitive. That's pretty. Also, don't be so colorful.

If the premise you are asking is all regular polygons.

If the conditions for a regular square are not set, both can be densely laid. As shown in the figure below, it is a densely paved figure of a non-regular pentagon.

And regular pentagons cannot be densely paved.

First of all, you need to know when to lay it secretly.

Dense paving, that is, the inlay of surface figures, splices several or dozens of plane graphics with the same shape and size, and paves them together without leaving gaps and overlapping with each other, which is the dense laying of plane figures, also known as the inlay of plane figures.

The regular pentagon cannot be densely paved, because each of its inner angles is 108 degrees, and 360 degrees is not an integer multiple of 108, and the inner angles at each splicing point cannot guarantee that there are no gaps or overlaps; With the exception of regular triangles, regular quadrilaterals, and regular hexagons, no other regular polygons can be densely laid with planes.

When several corners of the figure are put together to form a 360-degree pattern, it can be densely laid. One of the inner angles of a regular pentagon is 108 degrees, and 360 degrees is not a multiple of 108 degrees, so it cannot be densely paved. So quadrilaterals can be densely paved, while pentagons cannot be densely paved.

A few days ago, I had a wonderful lesson on graphic dense paving, and the students found that rectangles, squares, triangles, trapezoids, parallelograms, etc. can be densely paved separately, while circles and regular pentagons cannot be paved separately.

One student asked: Yes, soccer is fine. When I heard this, I admired the children, so why not let the children argue?

B1: That's because football is three-dimensional.

Student 2: Football can not be called dense paving, the dense paving we learn is paved on a flat surface.

The truth becomes clearer and clearer, and I believe in the wisdom of my students. But why can't regular pentagons be densely paved alone?

It can be densely paved, because an inner angle of the regular pentagon is 108 degrees, and 360 degrees is not a multiple of 108 degrees, so it cannot be densely paved.

Regular hexagons can be densely paved.

Regular pentagons cannot be densely paved.

Regular octagons cannot be densely paved.

What determines whether a figure can be densely paved?

The corners of the figure that can be densely laid intersect at one point.

When the angles of these figures intersect at a point, the sum of the degrees of these angles is exactly 360 degrees.

Summarize the law of polygon paving in one sentence?

Polygon paving rule: When several corners of the figure are put together to form a 360 degree, it can be paved.

Why are only triangles, squares and hexagons in regular polygons densely paved, but regular pentagons and regular octagonal floor tiles cannot be densely paved?

The rule of polygonal floor tiles is densely paved on the ground: when several corners of the figure are put together to form a 360 degree, it can be densely paved. And because each of the inside angles of a regular polygon is equal, only three degrees are divisors of 360.

The inner angle of 60 degrees is a regular triangle, the inner angle of 90 degrees is a square, and the inner angle of 120 degrees is a regular hexagon. Therefore, with the same kind of regular polygon, there are only three kinds of regular triangles, squares, and regular hexagons.

It cannot be densely paved because the inside of the pentagon is 180 degrees and 360 degrees, not multiples of 180 degrees, so it cannot be densely paved.

The first is the interior angles of the various regular polygons:

My understanding is that the sum of the internal angles of two different regular polygons is required to reach 360 degrees. When the number of sides is greater than 12, the sum of the two inner angles is greater than 300 degrees, and the remaining is less than 60 degrees, and the triangle cannot be put down, so it can be ignored.

For regular dodecagons, the two inner angles are 300 degrees, and the remaining 60 degrees meet the requirements and are denoted as 2*[12]+[3].

For regular eleven-sided shapes, the two inner angles are degrees, and the remaining degrees are not combined.

For a regular decagonal, the two interior angles are 288 degrees, leaving 72 degrees, and there is no combination that meets the requirements.

For regular nine-grams, the two inner angles are 280 degrees, leaving 80 degrees, and there is no combination that meets the requirements.

For a regular octagon, the two inner angles are 270 degrees, and the remaining 90 degrees meet the requirements and are denoted as 2*[8]+[4].

For a regular heptagon, the two inner angles are degrees, and the remaining degrees are not combined.

For a regular hexagon, the two inner angles are 240 degrees, leaving 120 degrees, which meets the requirements, and is denoted as 2*[6]+2*[3], and it has two permutations.

For a regular pentagon, the two inner angles are 216 degrees, leaving 144 degrees, which seems to meet the requirements, but it is found that this cannot be true by drawing.

For a square, the three inner angles are 270 degrees, and there is no combination that meets the requirements; The two inner angles are 180 degrees, which meet the requirements, and are recorded as 2*[4]+3*[3], and it has two arrangements.

For triangles, the cases of one, two, and three inner angles have been enumerated, the five inner angles add up to 300 degrees, leaving 60 degrees, or triangles, the four inner angles add up to 240 degrees, leaving 120 degrees, meet the requirements, and record it as 4*[3]+[6].

In addition, between the different permutations of the same combination, it can be arbitrarily combined and seamlessly connected, so there are many situations and will not be analyzed.

Paving conditions: Each inside corner of the quadrilateral should appear only once at each splice point, and the equal edges should coincide with each other. If it is not convenient when laying in dense, the marking method can be adopted.

The so-called "dense paving" refers to any kind of figure, if it can be laid on the plane without gaps and without overlapping, this paving method is called "dense paving". A dense graphic is a graphic that can be densely laid. Splicing with plane figures of exactly the same shape and size, and laying them together without leaving gaps or overlapping with each other, this is the dense paving of plane figures, also known as the inlay of plane figures.

Because some graphics are individually densely paved, images of the same size are spliced together, and the joints can just form a perimeter angle, and there are no gaps and no overlapping together; However, some images cannot be stitched together to form a perimeter, so they cannot be densely paved separately.

The key to the dense paving of the figure is that the polygons are spliced together around a point, and the sum of the corners at the junction is exactly equal to 360°. When the sum of the corners can form a perimeter angle (i.e., 360°), the figure can be laid separately; If not, it cannot be densely laid alone.

For example, the angles of trapezoids, regular triangles, and regular hexagons can be summed, so they can be densely paved; A circle is made up of a closed curve with gaps between circles, so it cannot be densely paved.

Except for regular triangles, regular quadrilaterals, and regular hexagons, no other regular polygons can be densely paved. Because one or several plane figures of the same shape and size are spliced, and they are paved together without leaving gaps or overlapping with each other, which is the dense paving of plane figures. There must be no gaps, and because the week is 360°, it must reach 360° to be completely densely laid.

Splicing with several or dozens of plane figures with the same shape and size, leaving no gaps and overlapping between each other, this is the dense paving of plane figures, also known as the inlay of plane figures.

At the common vertices of the densely paved figure, the degrees of all angles add up to 360 degrees. If the figures do not coincide and there are no gaps, it is a dense pavement.

Images: trapezoids, parallelograms, squares, rectangles, triangles, and hexagons can all be individually paved.

b Regular hexagon.

A regular hexagon can be densely paved because each of its inner corners is 120°, accommodating exactly 3 inner corners at each splice point.

A regular pentagon cannot be densely paved because each of its inner angles is 108°, while 360° is not an integer multiple of 108°, and the inner angles at each splicing point are not guaranteed to be free of gaps or overlaps.

Each internal angle of a regular heptagon is 7=, and each of the internal angles of a regular octagon is 8=135°, both of which are not divisible by 360°, so they cannot be densely paved.

a. Each inner angle of the regular pentagon is 180°-360° 5=108°, which is not divisible by 360° and cannot be densely paved;

b. Each inner angle of the regular hexagon is 120°, which can be divisible by 360° and can be densely paved;

c. Each inner angle of the regular heptagon is: 180°-360° 7=9007, which cannot be divisible by 360° and cannot be densely paved;

d. Each inner angle of the regular octagon is: 180°-360° 8=135°, which is not divisible by 360° and cannot be densely paved

Therefore, choose B

Wrong. Each inner angle of a regular hexagon is 120°, divisible by 360°, and can be densely paved.

Each inner angle of a regular pentagon is 180°-360° 5=108°, which is not divisible by 360° and cannot be inlaid separately.

The characteristic of a densely paved figure at a splicing point is that when the inner corners of several figures are spliced together, the sum is equal to 360°, and the equal edges coincide with each other, while the regular pentagon does not have such a characteristic.

Wrong. The regular hexagon can be densely paved because of its apex angle of 120°, 360 120=3; The top angle of the regular pentagon is 108°, 360 divided by 108 is not an integer, only 360 degrees divided by the angle of the figure is an integer can be densely laid, so the regular pentagon cannot be densely laid.

Not the regular pentagon, circle can not be densely paved.

Regular hexagons, parallelograms, regular triangles, isosceles trapezoids can be densely paved.

The number of angles in a regular hexagon is 120,360 can be divided, so a regular hexagon can, and the number of angles inside a regular pentagon is 108,360, and a regular pentagon cannot.