TW201805895A - Two-dimensional patterning method composed of squares and triangles - Google Patents

Abstract

The invention relates to a two-dimensional patterning method. The invention utilizes structural designs based on basic units such as squares and triangles with arranging approaches. Accordingly, the invention can obtain a two-dimensional patterning method to fulfill a two-dimensional regular pattern based on regularly arranging approaches. The two-dimensional patterning method further includes a symmetric center configuration step, a first symmetric pattern forming step and a symmetric patterns extending step. The symmetric center configuration step includes configuring a hexagon with a symmetric axis as the symmetric center. The first symmetric pattern forming step includes configuring squares on all sides of the hexagon. The symmetric patterns extending step includes forming N layers of symmetric patterns on the first symmetric pattern.

Description

Two-dimensional patterning method

The invention relates to a patterning method, in particular to a regular patterning method of a two-dimensional plane.

Space and plane aesthetics are an important part of modern people's pursuit of beauty and creativity. Among them, the mosaic art has not only appeared in the decoration of buildings in modern decorative art design, but also began to extend to the field of graphic design and clothing.

Therefore, how to form a specific structure with basic repeating units and extend and fill the two-dimensional space in a periodic manner so that it can show specific rules and aesthetics is the research direction of the inventors.

Therefore, an object of the present invention is to provide a two-dimensional patterning method.

Therefore, the two-dimensional patterning method of the present invention includes a symmetrical center setting step, a first symmetrical pattern forming step, and a symmetrical pattern extending step.

The symmetry center setting step is to set a hexagonal symmetry center with a symmetry axis, define a first symmetry unit whose basic constituent unit is a square, and a second symmetry unit whose basic constituent unit is a regular triangle. The symmetrical unit has at least one square and at least one regular triangle, the side length of the square and the regular triangle are the same, and the side length of the symmetrical center of the hexagon is an integer multiple of the side length of the square and the regular triangle.

The first symmetrical pattern forming step is to set a square from each side of the symmetrical center of the hexagon, and set a regular triangle at each corner of the symmetrical center of the hexagon to form a plurality of first symmetrical units and A first symmetrical pattern formed by staggering a plurality of second symmetrical units, wherein each of the first symmetrical units of the first symmetrical pattern has at least one square, and the square and the hexagonal symmetrical center are in an edge-to-edge manner. Connected, each of the second symmetrical units of the first symmetrical pattern has a regular triangle, and the regular triangle and the hexagonal symmetrical center are connected in a diagonally diagonal manner.

The symmetrical pattern extending step is to sequentially form N layers of symmetrical patterns, N ≧ 1, from the first symmetrical pattern. Each layer of the symmetrical pattern is composed of a plurality of first symmetrical units and a plurality of second symmetrical units. One layer of the symmetrical pattern is formed on the first and second symmetrical units of the previous layer to form a second and one symmetrical unit opposite to its pattern, so as to obtain the symmetrical pattern of each layer, wherein the first A symmetrical unit has at least one square, and the at least one square is connected to the second symmetrical unit in the front layer in an edge-to-edge manner, and any adjacent two first symmetrical units are sandwiched by a plurality of regular triangles and formed The second symmetrical unit of an inverted trapezoid.

In addition, another object of the present invention is to provide a two-dimensional patterning method.

Therefore, the two-dimensional patterning method of the present invention includes a symmetrical pattern providing step and a pattern extending step.

The symmetrical pattern providing step is to provide a plurality of symmetrical patterns. Each symmetrical pattern includes a regular hexagonal symmetrical center, a plurality of squares formed on each side of the regular hexagonal symmetrical center, and a plurality of formed on the six sides. An equilateral triangle at each corner of the center of symmetry of the shape, wherein the sides of the regular hexagon, the square and the regular triangle are the same length, each square and the center of symmetry of the regular hexagon are connected side by side, and Each regular triangle is connected diagonally with the regular hexagon symmetrical center.

The pattern extending step is to arrange the symmetrical patterns in a first direction so that adjacent symmetrical patterns partially overlap each other, and each symmetrical pattern has at least one overlapping portion, and the overlapping portion has a square, and A regular triangle with two opposite sides of the square.

The effect of the present invention is that a two-dimensional patterning method that can fill a two-dimensional space with a regular pattern can be obtained in a regular arrangement through the design of the basic unit structure and the arrangement.

Before the present invention is described in detail, it should be noted that in the following description, similar elements are represented by the same numbers.

Referring to FIG. 1, a first embodiment of the two-dimensional patterning method of the present invention is to form a regular pattern as shown in FIG. 1, which is arranged with a regular hexagonal symmetrical center as a center point.

Referring to FIG. 1, the first embodiment of the two-dimensional patterning method includes a symmetrical center setting step 11, a first symmetrical pattern forming step 12, and a symmetrical pattern extending step 13.

The symmetry center setting step 11 is to set a hexagonal symmetry center having a symmetry axis. A first symmetrical unit defining a basic constituent unit as a square and a second symmetrical unit constituting a regular triangle as a basic constituent unit are defined. The first and second symmetrical units have at least one square and at least one regular triangle, respectively. The square and the regular unit The side lengths of the triangles are the same, and the side length of the symmetrical center of the hexagon is an integer multiple of the side lengths of the square and the regular triangle.

The first symmetrical pattern forming step 12 is to set a square from each side of the regular hexagonal symmetry center HC, and set a regular triangle at each corner of the regular hexagonal symmetry center to form a plurality of first symmetry A first symmetrical pattern composed of cells and a plurality of second symmetrical cells alternately, wherein each of the first symmetrical patterns of the first symmetrical cell has at least one square, and the square and the hexagonal symmetrical center are edge-to-edge The two symmetrical units of the first symmetrical pattern each have a regular triangle, and the regular triangle and the hexagonal symmetrical center are connected in a diagonally diagonal manner.

The symmetrical pattern extending step 13 is to sequentially form N layers of symmetrical patterns, N ≧ 1 from the first symmetrical pattern. The symmetrical pattern of each layer is composed of a plurality of first symmetrical units and a plurality of second symmetrical units. And the symmetrical pattern of each layer is formed on the first and second symmetrical units of the previous layer to form a second and a symmetrical unit opposite to its pattern, so as to obtain the symmetrical pattern of each layer. The first symmetrical unit of each layer of the symmetrical pattern has at least one square, and the at least one square is connected to the second symmetrical unit of the previous layer in an edge-to-edge manner, and any two adjacent first symmetrical units are sandwiched between them. A second symmetrical unit composed of a regular triangle and an inverted trapezoid is assumed.

The first embodiment is described below with two specific examples.

For specific example 1, please refer to FIGS. 1 and 2. The symmetrical center setting step 11 of the specific example 1 is to set a regular hexagonal symmetrical center HC with a regular hexagonal structure. Therefore, the sides of the regular hexagonal symmetrical center HC are equal to The sides of the square and the regular triangle have the same length.

The first symmetrical pattern forming step 12 is to set a square from each side of the regular hexagonal symmetry center HC, and set a regular triangle at each corner of the regular hexagonal symmetry center HC to form a plurality of first A symmetric unit 2 and a plurality of second symmetric units 3 are staggered first symmetrical patterns 100, wherein each of the first symmetrical patterns 100 has a square, and the square and the hexagon The symmetry centers HC are connected side by side. Each of the second symmetrical units 3 of the first symmetrical pattern 100 has a regular triangle, and the regular triangles are connected diagonally with the hexagonal symmetry center HC. .

Step 13 of the symmetrical pattern is extended, and a symmetrical pattern 200 forming the first layer (N = 1) is taken as an example. The square may be set to the corresponding front layer in an edge-to-edge manner (the previous layer is the first layer when N = 1). A symmetrical pattern 100) is formed on each of the second symmetrical units 3, and each of the second symmetrical units 3 corresponding to the first symmetrical pattern 100 forms the first symmetrical unit 2 composed of a square, and then, Between any adjacent first symmetrical units 2, a regular triangle is set on the first symmetrical unit 2 corresponding to the first symmetrical pattern 100 in an edge-to-edge manner, and then the other regular triangles are used to fill the regular triangles. Adjacent gaps between the adjacent first symmetrical units 2 are joined to each other. In this way, a second symmetrical unit 3 consisting of three regular triangles forming the inverted trapezoid can be formed between two adjacent first symmetrical units 2 to obtain the symmetry of the first layer (N = 1). Pattern 200. At this time, because the number of squares in the previous layer is 1, the length of the long bottom side of the second symmetrical unit 3 of N = 1 is the sum of the lengths of the two sides of the (1 + 1) squares, and the adjacent The joint angle between the first and second symmetrical units 2 and 3 is 150 degrees.

Similarly, when continuing to N = 2, each of the second layer corresponding to the previous layer (the previous layer is the first layer (N = 1) of the symmetrical pattern 200) is edge-to-edge. Two squares are disposed on the symmetrical unit 3, and a first symmetrical unit 2 composed of two squares is formed on each of the second symmetrical units 3 of the symmetrical pattern 200 on the previous layer. Next, two regular triangles are set on each of the first symmetrical units 2 corresponding to the symmetrical pattern 200 of the previous layer in an edge-to-edge manner between two adjacent first symmetrical units 2 formed, Then, other regular triangles are used to fill the gap between the two regular triangles and the first symmetrical unit 2 adjacent to the two regular triangles. In this way, the second symmetrical unit 3 composed of five regular triangles to form the inverted trapezoid can be formed between any two adjacent first symmetrical units 2, and the symmetry of the second layer (N = 2) can be obtained. Pattern 200. Because the number of squares of the first symmetrical unit 2 of the front layer (N = 1) is two, the length of the long bottom side of the second symmetrical unit 3 of the symmetrical pattern 200 of N = 2 is 3 (2 + 1) squares. And the length of one side. Extending in this way, you can get the repeated extension structure shown in Figure 2. It should be noted that in Figure 2, N = 5 is the final arrangement description. When N = 5, each of the first A symmetrical unit 2 has three squares, and each of the second symmetrical units 3 has seven regular triangles.

Specific example 2

With reference to Figures 1 and 3, this specific example 2 is substantially the same as this specific example 1, except that the hexagonal symmetric center HC of this specific example 2 is selected from two groups with different side lengths, and one group of sides The length is m times the side length of the square, and m> 1. In this specific example, the hexagonal symmetry center HC has two sets of side lengths, one of which has the same side length as that of the square and the regular triangle, and the other side has the side length of the square and The side of the regular triangle is twice as long.

In detail, since a group of sides of the hexagonal center of symmetry HC of the specific example 2 is twice as long as a square side, the first symmetrical pattern forming step 12 is on the side of the hexagon of the center of symmetry HC. The long side and the long side are provided with two squares, and the side having the same length as the square is provided with a square, so a first symmetry composed of a plurality of first and second symmetrical units 2, 3 can be obtained. The pattern 100 is just that the first symmetrical units 2 include two kinds of squares.

In the symmetrical pattern extension step 13, the arrangement and arrangement of the symmetrical pattern 200 of each layer is the same as that of the first embodiment, and the difference is that when the number of squares corresponding to the first symmetrical unit 2 of the previous layer is different, it is correspondingly formed. Different numbers of regular triangles, such as this specific example 2, when N = 1, the number of regular triangles of the second symmetrical units 3 of the symmetrical pattern 200 of the first layer will correspond to the number of squares of the previous layer. The number of these first symmetrical units 2 is different. According to this, a regular change arrangement pattern as shown in FIG. 3 can be obtained after the extended arrangement. Figure 3 uses N = 5 as the final arrangement for illustration. When N = 5, it can be obtained that each of the first symmetrical units 2 has 3 squares, and the second symmetrical units 3 correspond to the front layer. The number of squares of the first symmetrical units 2 has 7 and 9 regular triangles, respectively.

With reference to FIG. 4, a second embodiment of the two-dimensional patterning method of the present invention includes a symmetrical pattern providing step 41 and a pattern extending step 42.

The symmetrical pattern providing step 41 is to provide a plurality of symmetrical patterns. Each of the symmetrical patterns includes a regular hexagonal symmetrical center, a plurality of squares formed on each side of the regular hexagonal symmetrical center, and a plurality of regular triangular shapes formed at each corner of the hexagonal symmetrical center. Wherein, the regular hexagon, the square and the regular triangle have the same side length, each square is connected to the regular hexagon symmetrical center in an edge-to-edge manner, and each regular triangle and the regular hexagon symmetrical center are connected. Connected diagonally.

The pattern extending step 42 is to arrange the symmetrical patterns along a first direction, so that adjacent symmetrical patterns partially overlap each other, so that the symmetrical patterns have a plurality of overlapping portions, and the overlapping portions of the portions overlap each other. Each of the overlapping portions has a square, and a regular triangle on two opposite sides of the square.

The second embodiment of the two-dimensional patterning method of the present invention is described below with the following two specific examples (specific examples 3 and 4). It should be noted that the two specific examples have a plurality of identical symmetrical patterns 300 respectively, and the definitions of the symmetrical patterns 300 are as described in the foregoing second embodiment, and therefore will not be described again.

Specific example 3

With reference to FIGS. 4, 5 and 6, the specific example 3 is to arrange the symmetrical patterns 300 along a first direction X, and partially overlap adjacent symmetrical patterns 300 with each other, so that the symmetrical patterns 300 have multiple overlaps. Part 301, and make some of these overlapping parts 301 coincide with each other, wherein each of the overlapping parts 301 has a square and a regular triangle located on two opposite sides of the square, and a part of the overlapping part 301 passes through a positive The triangles overlap with each other to form an overlapping portion 302, and a two-dimensional pattern arranged regularly as shown in FIG. 6 is obtained. It should be noted that the first direction X shown in FIG. 6 is taken as an example of the horizontal direction. However, in actual arrangement, the first direction X may also form an angle with the horizontal direction, and there is no particular limitation.

Specific example 4

Referring to FIGS. 4, 7 and 8, the difference between the specific example 4 and the specific example 3 is that the symmetrical patterns 300 of the specific example 4 are one of two along the first direction X and perpendicular to the first direction X. The direction Y is arranged along the extension. Let the symmetrical patterns 300 arranged along the second direction Y and adjacent to each other have one overlapping portion 301, and the symmetrical patterns 300 arranged along the first direction X and adjacent to each other tangent to each other, and along the first, Arranged in two directions X and Y, a middle portion 303 composed of two regular triangles is sandwiched between two adjacent symmetrical patterns 300, and a two-dimensional pattern arranged regularly as shown in FIG. 8 is obtained.

In summary, the present invention can form a two-dimensional patterning method that can fill a two-dimensional space with a regularly changing pattern through the design of the basic unit structure and the arrangement method to form a regularly changing arrangement. To achieve the purpose of the invention.

However, the above are only examples of the present invention. When the scope of implementation of the present invention cannot be limited by this, any simple equivalent changes and modifications made according to the scope of the patent application and the contents of the patent specification of the present invention are still Within the scope of the invention patent.

2‧‧‧ the first symmetrical unit
11‧‧‧Symmetry center setting steps
3‧‧‧Second Symmetric Unit
12‧‧‧First symmetrical pattern forming step
100‧‧‧ the first symmetrical pattern
13‧‧‧Symmetry pattern extension step
200‧‧‧ symmetrical pattern
HC‧‧‧ Hexagon Symmetric Center
300‧‧‧ symmetrical pattern
41‧‧‧ Symmetrical pattern providing steps
301‧‧‧ Overlap
42‧‧‧Pattern extension steps
302‧‧‧Coincident Department
X‧‧‧ first direction
303‧‧‧Middle
Y‧‧‧ second direction

Other features and effects of the present invention will be clearly presented in the embodiment with reference to the drawings, in which: FIG. 1 is a text flow chart of a first embodiment of a two-dimensional patterning method of the present invention; FIG. 2 is a second embodiment of the present invention Schematic diagram of the arrangement process of the specific example 1 of the two-dimensional patterning method; FIG. 3 is a diagram of the arrangement process of the specific example 2 of the two-dimensional patterning method of the present invention; FIG. 4 is a text flow of the second embodiment of the two-dimensional patterning method of the present invention Fig. 5 is a schematic diagram of a repeating unit of specific example 3 of the two-dimensional patterning method of the present invention; Fig. 6 is a two-dimensional pattern obtained by arranging specific example 3 of the two-dimensional patterning method of the present invention; A schematic diagram of a repeating unit of the specific example 4 of the two-dimensional patterning method; and FIG. 8 is a two-dimensional pattern obtained by arranging the specific example 4 of the two-dimensional patterning method of the present invention.

11‧‧‧Symmetry center setting steps

12‧‧‧First symmetrical pattern forming step

13‧‧‧Symmetry pattern extension step

Claims (8)

A two-dimensional patterning method includes: a symmetry center setting step, setting a hexagonal symmetry center having a symmetry axis, defining a first symmetry unit whose basic constituent unit is a square, and a first symmetry unit whose basic constituent unit is a regular triangle. Two symmetrical units, the first and second symmetrical units having at least one square and a regular triangle, the sides of the square and the regular triangle are the same length, and the side length of the hexagonal symmetry center is the side length of the square and the regular triangle A first symmetrical pattern forming step, a square is set from each side of the symmetrical center of the hexagon, and a regular triangle is set at each corner of the symmetrical center of the hexagon to form a plurality of first symmetrical patterns A symmetrical pattern composed of a symmetrical unit and a plurality of second symmetrical units, wherein each of the first symmetrical patterns of the first symmetrical unit has at least one square, and the square and the symmetrical center of the hexagon are separated by edges. The opposite sides are connected in a way that each of the second symmetrical units of the first symmetrical pattern has a regular triangle, and the regular triangle and the The hexagonal symmetry centers are connected diagonally and diagonally; and a symmetrical pattern extending step, sequentially forming N layers of symmetrical patterns from the first symmetrical pattern, N ≧ 1, and each layer of the symmetrical pattern is composed of a plurality of first A symmetrical unit and a plurality of second symmetrical units are staggered, and the symmetrical pattern of each layer is formed on the first and second symmetrical units of the previous layer to form a second and a symmetrical unit opposite to its pattern, so that each layer is obtained. The symmetrical pattern, wherein each layer of the first symmetrical unit of the symmetrical pattern has at least one square, and the at least one square is connected to the second symmetrical unit of the previous layer in an edge-to-edge manner, and any two adjacent first symmetry A second symmetrical unit composed of most regular triangles and formed into an inverted trapezoid is sandwiched between the units. The two-dimensional patterning method according to claim 1, wherein the first and second symmetrical units adjacent to the same layer are joined to each other, and the joining angle is 150 degrees. The two-dimensional patterning method according to claim 1, wherein the symmetrical center of the hexagon is a regular hexagon and the side length is the same as that of the square and the regular triangle. The two-dimensional patterning method according to claim 1, wherein the hexagonal symmetrical center has at least two groups of different side lengths, and the side length of one group is m times the side length of the square, and m> 1. A two-dimensional patterning method includes: a symmetrical pattern providing step, providing a plurality of symmetrical patterns, each symmetrical pattern including a regular hexagonal symmetrical center, and a plurality of squares formed on each side of the regular hexagonal symmetrical center , And a plurality of regular triangles formed at each corner of the symmetrical center of the hexagon, wherein the regular hexagon, the square and the regular triangle have the same length, and each square and the regular center of hexagon are symmetrical Connected in an edge-to-edge manner, and each regular triangle and the regular hexagonal symmetry center are connected in a diagonally diagonal manner; and a pattern extension step, aligning the symmetrical patterns in a first direction so that adjacent The symmetrical patterns of are partially overlapped with each other, and each of the symmetrical patterns has at least one overlapping portion, and the overlapping portion has a square and a regular triangle located on two opposite sides of the square. The two-dimensional patterning method according to claim 5, wherein in the pattern extending step, at least part of the symmetrical patterns have a plurality of overlapping portions, and some of the overlapping portions overlap each other in a regular triangle shape. The two-dimensional patterning method according to claim 5, wherein in the pattern extending step, the symmetrical patterns are also arranged in a second direction that intersects the first direction, so that the symmetrical patterns are arranged in the first direction and each other Adjacent symmetrical patterns have one overlapping portion, and any adjacent symmetrical patterns arranged along the second direction are tangent to each other, and an intermediate portion is sandwiched between two adjacent symmetrical patterns. The two-dimensional patterning method according to claim 5, wherein each of the middle portions is composed of two regular triangles.