Related papers on shape mosaic

Research paper on graphic mosaic

Name: Xu Haofan School: beijing no.11 high school Class: Class 3, Grade 1.

Thursday, 7 May 2007

Keywords: complete coverage, plane mosaic, mathematical perspective

Introduction: Mathematics is everywhere, and we often encounter some problems related to mathematics in our life. On the tiled floor or wall, adjacent floor tiles or tiles are evenly stuck together, and there is no gap on the whole floor or wall. Why can these shapes of floor tiles or tiles cover the ground without gaps? Can you change some other shapes? In order to solve these problems, we must explore the truth. From a mathematical point of view, a part of the plane is completely covered by non-overlapping polygons; This kind of problem is usually called polygon plane mosaic.

Content: We should explore the truth in daily life and study the related concepts and properties of polygons in graphic mosaic.

For example, a triangle. A triangle is a plane figure composed of three line segments that are not on the same line. Through experiments and research, we know that the sum of the inner angles of the triangle is 180 degrees, and the sum of the outer angles is 360 degrees. The ground can be covered by six regular triangles.

Look at the regular quadrangle, which can be divided into two triangles. The sum of internal angles is 360 degrees, the degree of an internal angle is 90 degrees, and the sum of external angles is 360 degrees. The ground can cover four regular quadrangles.

What about regular pentagons? It can be divided into three triangles, the sum of internal angles is 540 degrees, the degree of one internal angle is 108 degrees, and the sum of external angles is 360 degrees. It cannot cover the ground.

Hexagon can be divided into four triangles, the sum of internal angles is 720 degrees, the degree of one internal angle is 120 degrees, and the sum of external angles is 360 degrees. The ground can cover three regular quadrangles.

A heptagon can be divided into five triangles. The sum of internal angles is 900 degrees, the degree of internal angles is 900/7 degrees, and the sum of external angles is 360 degrees. It cannot cover the ground.

……

It can be concluded that an N-polygon can be divided into (n-2) triangles, the sum of internal angles is (n-2)* 180 degrees, the degree of an internal angle is (n-2)* 180÷n degrees, and the sum of external angles is 360 degrees. If (n-2)* 180÷n can be divisible by 360, then it can be used to pave the way; If not, it can't be used to pave the road.

Not only can you cover the ground with a regular polygon, but you can also cover the ground with more than two or three kinds of graphics.

For example: regular triangle and square, regular triangle and hexagon, square and octagon, regular pentagon and octagon, regular triangle and square and hexagon.

In real life, we have seen all kinds of patterns composed of regular polygons. In fact, many patterns are often composed of irregular basic graphics. Above, we use real-life examples and floor tiles to prove the wonder of graphic mosaic. Next, I want to talk about a printmaker's interest in graphic mosaic: escher is fascinated by every mosaic graphic, whether it is regular or irregular; Moreover, he is particularly interested in a shape he calls deformation, in which the figures change and influence each other, sometimes breaking through the freedom of the plane. His interest began at 1936, when he traveled to Spain and saw the patterns of tiles used locally in Alhambra. He spent several days painting these tiles, and later claimed that they were the most abundant inspiration resources I have ever met. 1957, he wrote an article about mosaic graphics, in which he commented: In the field of mathematics, regular plane division has been studied theoretically ... Does this mean that it is just a strict mathematical problem? In my opinion, no. Mathematicians have opened the door to a vast field, but they have never entered it themselves. They are naturally more interested in the way to open the door than in the garden behind it. Escher used these basic patterns in his mosaics. He used reflection, smooth reflection, transformation and rotation in geometry to obtain more diverse patterns. He also carefully twisted these basic patterns into animals, birds and other shapes. These changes have to be symmetrical three times, four times or even six times to get the mosaic pattern. This effect is both amazing and beautiful. Here are some pictures of graphic mosaics in escher.

What about these mosaic shapes? Are they beautiful? Let's learn the mosaic of graphics better and wander in mathematics and art!

thesis

The so-called graphic stitching is to use one or more graphics of the same size to pave the plane, requiring that there should be no gaps or overlaps between the graphics. In this regard, escher has made remarkable achievements. For example, the following pictures are his representative works.

Next, I will introduce the mosaic of graphics.

Regular plane division is called mosaic, and mosaic graphics are the arrangement of closed graphics, with no overlap and no gaps. Generally speaking, the basic unit of mosaic pattern is polygon or similar conventional shape, such as square tiles commonly used on the floor. However, escher is fascinated by every kind of mosaic, whether it is regular or irregular; Moreover, he is particularly interested in a shape he calls morphs, in which the graphics change and influence each other, sometimes breaking through the freedom of the plane.

Whether this is fair to mathematicians or not, one thing is true-they point out that among all regular polygons, only triangles, squares and regular hexagons can be used for mosaic. However, many other irregular polygons can also form mosaics after tiling. For example, many mosaics use irregular five-pointed star shapes. Escher used these basic patterns in his mosaics. He used reflection, smooth reflection, transformation and rotation in geometry to obtain more diverse patterns. He also carefully twisted these basic patterns into animals, birds and other shapes. These changes have to be symmetrical three times, four times or even six times to get the mosaic pattern. This effect is both amazing and beautiful.

Graphic Mosaic-Mosaic of Plane Regular Polygons

If you use regular polygons with different sides for stitching, you need to meet two points: one is the length of the sides, and the other is that the sum of the internal angles of a vertex is 360.

What kind of regular polygons are combined together? So if you only cover the plane with a regular polygon, can you use any regular polygon? That was not the case. For example, regular pentagons can only be spelled into the following shapes.

So which regular polygons can be used to pave the plane? We can assume that the number of sides of this regular polygon is a regular polygon at the same vertex. Because these regular polygons enclose a fillet at the same vertex, and the inner angle of each regular polygon is, we get:

(It is a positive integer, not less than 3)

∴

∴ ……(*)

∵ is a positive integer, and∴is an integer not less than 3.

∴ Manufacturing (*) type is established under the following conditions:

∴

∴ Only one regular polygon is used for the floor, and there are only three situations: equilateral triangle, square and regular hexagon, as shown in the following figure:

Mosaic rules of regular polygons

Arranged with three regular polygons

Minimum number of edges: 3

Arrangement: (3, 7, 42) (3, 8, 24) (3, 9, 18) (3, 10, 15) (3, 12,/kloc-0.

Minimum number of edges: 4

Arrangement: (4, 5, 20) (4, 6, 12) (4, 8, 8)

Minimum number of edges: 5

Arrangement: (5,5, 10)

Minimum number of edges: 6

Arrangement: (6, 6, 6)

Arranged with four regular polygons

Minimum number of edges: 3

The combination of 3, 3, 4, 12 produces two completely different arrangements.

The combination of 3, 3, 6 and 6 leads to two completely different combinations.

The combination of 3, 4, 4 and 6 leads to two completely different combinations.

Arrangement: (3,3,4, 12), (3,4,3,12)-(3,3,6,6), (3,6,3,6)-(3,4,4,6)

Minimum number of edges: 4

Arrangement: (4, 4, 4, 4)

Arranged with five regular polygons

Minimum number of edges: 3

The combination of 3, 3, 3, 3 and 6 can only produce one arrangement.

The combination of 3, 3, 3, 4 and 4 produces two different combinations.

Arrangement: (3, 3, 3, 3, 6)-(3, 3, 3, 4, 4), (3, 3, 4, 3, 4)

Arranged in six regular polygons

Minimum number of edges: 3

Arrangement: (3, 3, 3, 3, 3, 3)

Note: The above figure shows that a point is filled with an angle of 360 and arranged as a regular polygon, with 2 1 arrangement, but in fact they only have 17 different combinations. There are four combinations of two different arrangements.

Classification using regular polygon mosaics;

Classification of mosaics:

Mosaic of (1) regular polygons

(A) regular mosaic

(2) Semi-regular mosaic

(iii) Irregular mosaic

(2) Splicing of irregular polygons

Definition: We call a regular mosaic with only one regular polygon.

From the previous discussion, we know that there are only three kinds of regular mosaics: regular triangles, squares and regular hexagons.

As shown in the figure below:

More than one regular polygon is used for mosaic, and there is the same regular polygon arrangement at each vertex. We call it semi-regular mosaic.

As shown in the figure below:

Some mosaics include regular mosaics, which we call semi-regular mosaics. These mosaics are a mixture of regular mosaics or semi-regular mosaics.

For example, in the figure below, 3,6,3,6 are arranged at point 1, while 3,3,6,6 are arranged at point 2. In this kind of mosaic, the arrangement of regular polygons at each vertex is not exactly the same, but there are two arrangements, so it is neither a regular mosaic nor a semi-regular mosaic, which we call irregular mosaic.

At point 1, it is the arrangement of 3, 6, 3, 6, and at point 2, it is the arrangement of 3, 3, 6, 6.

Similarly, we still use regular or semi-regular mosaic arrangement to represent this new type of irregular mosaic. We use the symbol "/"to separate each regular or semi-regular mosaic. For example, the mosaic in the above picture is recorded as 3.6.3.6/3.3.6.6..

Mathematicians define those mosaics composed of two or three different regular mosaics as irregular mosaics, and there are at least 14 kinds of irregular mosaics. How to determine this? In fact, as long as you spend a little patience and experiment with the known 2 1 regular or semi-regular arrangement, you can get the above conclusion.

Let's take a concrete look at these irregular mosaic patterns.

A mosaic of two or three different regular regular polygons.

There are two different regular arrangements (9 different mosaics).

3.3.6.6 / 3.6.3.6

3. 12. 12 / 3.4.3. 12

3.3.3.3.3.3 / 3.3.4. 12

3.3.3.4.4 / 3.4.6.4

3.3.3.3.3.3 / 3.3.4.3.4 # 1

3.3.3.3.3.3 / 3.3.4.3.4 #2

Note: Although the above two mosaics use the same regular arrangement, the overall composition is still different.

What graphics are spliced on the football surface? The surface of football consists of 12 regular pentagons and 20 regular hexagons.

Because the internal angle of a regular Pentagon is 108 degrees, and the internal angle of a regular hexagon is 120 degrees, ***348 degrees, it cannot be made into a plane.

No, in order to connect into a sphere.

Floor mosaic

In fact, people in life are more concerned about the problem of laying floor tiles. When we observe the floors of various buildings, we can find that the floors are often inlaid with various regular polygons and become beautiful patterns. When we observe the floors of various buildings, we can find that the floors are often inlaid with various regular polygons and become beautiful patterns.

Usually at home, in shops, in the central square, into hotels, restaurants and many other places will see tiles. They usually have different shapes and colors. In fact, there is a math problem, "Mathematics in Tiles".

On the tiled floor or wall, adjacent floor tiles or tiles are evenly attached together, and there is no gap on the whole floor or wall. Why can these shapes of floor tiles or tiles cover the ground without gaps? Can you change some other shapes? In order to solve these problems, we must explore the truth and study the related concepts and properties of polygons.

……

From this, we come to the conclusion. N- polygon can be divided into (n-2) triangles, the sum of internal angles is (n-2)* 180 degrees, the degree of one internal angle is (n-2)* 180÷2 degrees, and the sum of external angles is 360 degrees. If (n-2)* 180÷2 can be divisible by 360, then it can be used to pave the way; If not, it can't be used to pave the road.

Not only can you cover the ground with a regular polygon, but you can also cover the ground with more than two or three kinds of graphics.

For example: regular triangle and square, regular triangle and hexagon, square and octagon, regular pentagon and octagon, regular triangle and square and hexagon. ...

In real life, we have seen all kinds of patterns composed of regular polygons. In fact, many patterns are often composed of irregular basic graphics.

Ceramic tiles, such ordinary things, have such interesting mathematical mysteries, not to mention other things in life?

In life, mathematics is everywhere.

First, the floor is paved with regular polygons: three types.

(3, 3, 3, 3, 3) Floor pattern

(4, 4, 4, 4) Floor pattern (6, 6, 6) Floor pattern

Second, two kinds of regular polygons are used for floors: six kinds.

(3, 12, 12) floor mode

Three kinds of regular polygons are used to pave the floor: eight kinds.

If two regular polygons with different sides are used for mosaic, the sum of the internal angles of the regular polygons must be 360 at the overlapping vertices. In order to simplify the study, let's first look at the situation of laying the floor with two specific polygons.

Question 1: Now a master worker has two kinds of regular polygon tiles: regular triangle and square. Can you help him design a floor pattern?

Students, please cut out several regular triangles and squares of the same size from cardboard, and then try to spell them out. I believe you can spell it.

Did you spell the picture below?

Question 2

If the master only has hexagonal and triangular tiles to hit the floor, can it be realized? If yes, there are several situations; If not, explain why.

Think about it, can we use equation calculation instead of doing puzzles to study the above problems?

In fact, we can calculate as follows

There are x regular triangles and y regular hexagons at a point.

60x+ 120y=360

x+2y=6

There are two sets of integer solutions

So there should be three schemes.

draw

Question 3: If the master only uses squares and regular hexagons, can he spell the floor? Please think for yourself.

(2) What if there are more than two regular polygons on the floor?

Let's answer the above question.

Suppose that m kinds of regular polygons with edges of,,,, and can be embedded in the whole plane.