A periodic and aperiodic tiling planar system based on regular hexagons

Abstract

The application discloses a periodic and non-periodic close-packing plane system based on regular hexagons, which comprises seven different tile base types, each of which is based on a regular hexagon structure, and the outer contour of each tile base is composed of six boundary contours, in the connected graph surrounded by the six boundary contours, the areas of the convex and concave regular hexagon parts are equal, the six boundary contours surround specific shapes of animals, plants or human beings, and the specific shapes surrounded by the six boundary contours are mutually embedded to form the close-packing plane; the tile base is provided with corresponding patterns corresponding to the specific shape boundary contours, and the application has simple structure, various close-packing effects and wide application prospect.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of tiling structure, and particularly relates to a periodic and non-periodic tiling plane system based on a regular hexagon. BACKGROUND

[0002] Tiling refers to covering a plane with several patterns without overlap and gap. Tiling of plane figures is widely used in tiling of ceramic tiles and is also applied in mathematics teaching. For example, through the process of exploring the tiling conditions of polygons, students' reasoning ability and aesthetic sense can be further developed, and the application of plane figures in real life, such as the application of wisdom splicing board, can better inspire children's thinking.

[0003] Tiling is divided into periodic tiling, non-periodic tiling and aperiodic tiling. If a basic pattern can be constructed by using a ceramic tile (prototile), and then the basic pattern is seamlessly and non-overlappedly covered in two different directions, the ceramic tile is called a periodic prototile, and the corresponding tiling is called periodic tiling. Aperiodic prototile refers to a prototile that cannot construct a periodic tiling. Non-periodic prototile refers to a prototile that can construct both periodic tiling and non-periodic tiling.

[0004] People have deeply researched the wisdom and decoration functions of tiling. For example, Roger Penrose, the Nobel Prize winner, invented the influential Penrose tiling (Patent Number: 4133152), and John A. L. Osborn invented the Escher-style artistic tiling (Patent Number: 5481841, 5520388, 5619830). However, the tiling found at present has the following defects. First, most of the periodic, non-periodic or aperiodic tiling puzzles have more than one type of puzzle piece, and the excessive types of puzzle pieces make the puzzle lack unified and harmonious artistic beauty, and the manufacturing cost is high. Second, the geometric structure of periodic tiling is too simple, lacks challenge difficulty, and is not attractive. Third, compared with periodic tiling, non-periodic or aperiodic tiling is more difficult to construct, and the number of solutions of such tiling is often not large, lacks variation, and players will soon lose interest after understanding the tiling rules.

[0005] Therefore, how to provide a non-periodic tiling plane system with simple structure, low manufacturing cost and various tiling effects of the same type of prototile combination is a problem to be solved by those skilled in the art. SUMMARY

[0006] Therefore, the application provides a periodic and aperiodic tiling plane system based on a regular hexagon, which can provide tiling forms with different profiles and beautiful and rich styles, and has wide research and development potential.

[0007] To achieve the above object, the application adopts the following technical scheme: a periodic and aperiodic tiling plane system based on a regular hexagon, which comprises a plurality of tile bases, the tile bases are all based on a regular hexagon structure, the tile bases correspond to a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point and a sixth corner point of the regular hexagon structure, a boundary profile is arranged between any two adjacent corner points, the area of the convex and concave regular hexagon part in the boundary profile of the tile base is equal, a plurality of boundary profiles on the tile base form a specific shape profile edge, the specific shape profile edges of a plurality of tile bases are mutually embedded to form a periodic tiling system or an aperiodic tiling system, a plurality of boundary profiles on the tile base jointly form a specific shape of an animal, a plant or a human image, corresponding patterns are arranged on the tile base corresponding to the specific shape boundary profile, and the tile base has seven types.

[0008] The application has the following beneficial effects: the specific shape pattern is transformed based on the regular hexagon tile base, the six boundary profiles form the boundary of the specific shape of an animal, a plant or a human image, and the area surrounded by the six boundary profiles is the same as that of the original regular hexagon. Compared with the traditional tiling, the single type tile base in the application can realize complex tiling, greatly simplifying the manufacturing process and production cost. In combination with the mirror image, translation and rotation transformation of the tile base, a variety of tiling can be constructed. In combination with the processing design of an artist, the application can provide artistic patterns with different profiles and beautiful and rich styles, has good affinity and intelligence inspiration function for children, and has wide research and development potential. The tile base not only allows periodic and aperiodic tiling, but also has the characteristics of great solving difficulty and a large number of solutions, which makes up for the major defects of the traditional tiling. The application can be widely applied in the fields of intelligence toy games, artistic design and decoration, and has good aesthetic and economic application potential.

[0009] Preferably, the first edge corner point and the second edge corner point of the first type of ceramic tile substrate are provided with a first boundary contour, the first boundary contour has boundary positioning points, the boundary positioning points are outside the regular hexagon, the boundary positioning points are on the perpendicular bisector of the line connecting the first edge corner point and the second edge corner point, the first boundary contour is symmetrical about the perpendicular bisector of the line connecting the first edge corner point and the second edge corner point, the first boundary contour is rotated counterclockwise by 120° at the second edge corner point to form a second boundary contour, the line connecting the third edge corner point and the sixth edge corner point forms a center line, the first boundary contour is mirror symmetrical about the center line to form a fourth boundary contour, the fourth boundary contour is rotated clockwise by 120° at the fourth edge corner point to form a third boundary contour, and the first edge corner point and the sixth edge corner point are provided with a sixth boundary contour, and the sixth boundary contour is rotated clockwise by 120° at the sixth edge corner point to form a fifth boundary contour.

[0010] Preferably, the first edge corner point and the second edge corner point of the second type of ceramic tile substrate are provided with a first boundary contour, the first boundary contour has boundary positioning points, the boundary positioning points are inside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the line connecting the first edge corner point and the second edge corner point, the first boundary contour is rotated counterclockwise by 120° at the second edge corner point to form a second boundary contour, the first boundary contour is mirror symmetrical about the line connecting the third edge corner point and the sixth edge corner point to form a fourth boundary contour, the fourth boundary contour is rotated clockwise by 120° at the fourth edge corner point to form a third boundary contour, the outside of the first edge corner point and the sixth edge corner point is provided with a sixth boundary contour, and the sixth boundary contour is rotated clockwise by 120° at the sixth edge corner point to form a fifth boundary contour.

[0011] Preferably, the first edge corner point and the second edge corner point of the third type of ceramic tile substrate are provided with a first boundary contour, the first boundary contour has boundary positioning points, the boundary positioning points are inside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the line connecting the first edge corner point and the second edge corner point, the first boundary contour is rotated counterclockwise by 120° at the second edge corner point to form a second boundary contour, the second boundary contour is rotated counterclockwise by 120° at the third edge corner point to form a third boundary contour, the third boundary contour is rotated counterclockwise by 120° at the fourth edge corner point to form a fourth boundary contour, the first edge corner point and the sixth edge corner point are provided with a sixth boundary contour, and the sixth boundary contour is rotated clockwise by 120° at the sixth edge corner point to form a fifth boundary contour.

[0012] Preferably, the first corner point and the second corner point of the fourth type of ceramic tile substrate are provided with a first boundary contour, the first boundary contour has boundary positioning points, the boundary positioning points are outside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the line between the first corner point and the second corner point, the first boundary contour is rotated counterclockwise by 120° at the second corner point to form a second boundary contour, the second boundary contour is rotated counterclockwise by 120° at the third corner point to form a third boundary contour, the third boundary contour is rotated counterclockwise by 120° at the fourth corner point to form a fourth boundary contour, the first corner point and the sixth corner point are provided with a sixth boundary contour, and the sixth boundary contour is rotated clockwise by 120° at the sixth corner point to form a fifth boundary contour.

[0013] Preferably, the first corner point and the second corner point of the fifth type of ceramic tile substrate are provided with a first boundary contour, the first boundary contour has boundary positioning points, the boundary positioning points are inside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the line between the first corner point and the second corner point, the second corner point and the third corner point are provided with a second boundary contour in the same state as the first boundary contour, and the first corner point and the sixth corner point are provided with a sixth boundary contour in the same state as the first boundary contour; the second boundary contour is rotated counterclockwise by 120° at the third corner point to form a third boundary contour, the fourth corner point and the fifth corner point are provided with a fourth boundary contour in the same state as the third boundary contour, and the fifth corner point and the sixth corner point are provided with a fifth boundary contour in the same state as the fourth boundary contour.

[0014] Preferably, the first corner point and the second corner point of the sixth type of ceramic tile substrate are provided with a first boundary contour, the first boundary contour has boundary positioning points, the boundary positioning points are outside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the line between the first corner point and the second corner point, the first boundary contour is rotated counterclockwise by 120° at the second corner point to form a second boundary contour, the second boundary contour is rotated counterclockwise by 120° at the third corner point to form a third boundary contour, the third boundary contour is rotated counterclockwise by 120° at the fourth corner point to form a fourth boundary contour, the third boundary contour is mirrored about the line between the second corner point and the fifth corner point to form a sixth boundary contour, and the sixth boundary contour is rotated clockwise by 120° at the sixth corner point to form a fifth boundary contour.

[0015] Preferably, the first corner point and the second corner point of the seventh type of the ceramic tile base are provided with a first boundary contour, the first boundary contour has a boundary positioning point, the boundary positioning point is outside the regular hexagon, the first boundary contour is rotated counterclockwise by 120° to form a second boundary contour, the second boundary contour is rotated counterclockwise by 120° to form a third boundary contour, the third boundary contour is rotated counterclockwise by 120° to form a fourth boundary contour, the fourth boundary contour is rotated counterclockwise by 120° to form a fifth boundary contour, and the fifth boundary contour is rotated counterclockwise by 120° to form a sixth boundary contour.

[0016] Preferably, the distance between the boundary positioning point and the first corner point and the second corner point is less than the radius of the inscribed circle of the regular hexagon.

[0017] Preferably, the six boundary contours of the ceramic tile base form an animal, plant or human image-shaped ceramic tile base, and the animal, plant or human image-shaped ceramic tile base is provided with an animal, plant or human image texture pattern corresponding to the animal, plant or human image shape. BRIEF DESCRIPTION OF DRAWINGS

[0018] Figure 1 A structural schematic diagram of a first embodiment of the present application;

[0019] Figure 2 A design flowchart of the first embodiment of the present application;

[0020] Figure 3 A close-packed diagram of the first embodiment of the present application;

[0021] Figure 4 An application diagram of the first embodiment of the present application;

[0022] Figure 5 A structural diagram of a second embodiment of the present application;

[0023] Figure 6 An application diagram of the second embodiment of the present application;

[0024] Figure 7 A structural diagram of a third embodiment of the present application;

[0025] Figure 8 An application diagram of the third embodiment of the present application;

[0026] Figure 9 A scene decoration diagram of the third embodiment of the present application;

[0027] Figure 10 A structural diagram of a fourth embodiment of the present application;

[0028] Figure 11 An application diagram of the fourth embodiment of the present application;

[0029] Figure 12 Structure diagram of the fifth embodiment of the present application;

[0030] Figure 13 Specific shape diagram of the present application formed based on the variation of the fifth embodiment;

[0031] Figure 14 Application diagram of the fifth embodiment of the present application;

[0032] Figure 15 Scene decoration diagram of the fifth embodiment of the present application;

[0033] Figure 16 Structure diagram of the sixth embodiment of the present application;

[0034] Figure 17 Specific shape diagram of the present application formed based on the variation of the sixth embodiment;

[0035] Figure 18 Application diagram of the sixth embodiment of the present application;

[0036] Figure 19 Rich tiling case diagram of the sixth embodiment of the present application;

[0037] Figure 20 Structure diagram of the seventh embodiment of the present application;

[0038] Figure 21 Specific shape diagram of the present application formed based on the variation of the seventh embodiment;

[0039] Figure 22 Application diagram of the seventh embodiment of the present application. DETAILED DESCRIPTION

[0040] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all the other embodiments obtained by those of ordinary skill in the art without creative work fall within the scope of the present application.

[0041] Embodiment 1

[0042] Reference is made to the drawings of the present application Figures 1-5According to the embodiment of the present application, a hexagon-based periodic and aperiodic tiling plane system is constructed, a special tile base is constructed, each tile base is based on a hexagon structure, the outer contour is composed of six boundary contours, the boundary contour can be outside or inside the hexagon as needed, but the area of the convex and concave hexagon part corresponds to equal, thus the area of the tile base is equal to the original hexagon. The six boundary contours form a specific shape of animals, plants or human images, the specific shapes formed by the six boundary contours are embedded to form a tiling plane; the tile base is provided with a corresponding pattern corresponding to the specific shape boundary contour.

[0043] Specifically, the first type of tile base corresponding to the hexagon structure has a first corner point A1, a second corner point A2, a third corner point A3, a fourth corner point A4, a fifth corner point A5 and a sixth corner point A6, a first boundary contour is provided between the first corner point A1 and the second corner point A2, the first boundary contour has a boundary positioning point B1, the boundary positioning point B1 is outside the hexagon A1A2A3A4A5A6, the first boundary contour is symmetric about the perpendicular bisector BB1 of the line between the first corner point A1 and the second corner point A2; the first boundary contour is rotated counterclockwise by 120° at the second corner point A2 to form a second boundary contour between the second corner point A2 and the third corner point A3; the first boundary contour is mirror symmetric about the line between the third corner point A3 and the sixth corner point A6 to form a fourth boundary contour between the third corner point A4 and the fifth corner point A5; the fourth boundary contour is rotated clockwise by 120° at the fourth corner point A4 to form a third boundary contour between the third corner point A3 and the fourth corner point A4; a sixth boundary contour is provided between the first corner point A1 and the sixth corner point A6, the sixth boundary contour is rotated clockwise by 120° at the sixth corner point A6 to form a fifth boundary contour between the fifth corner point A5 and the sixth corner point A6. The process of constructing six boundary contours is shown in Figure 1 .

[0044] More specifically, the distance between the boundary positioning point B1 and the line between the first corner point A1 and the second corner point A2 is less than the radius of the incircle of the hexagon, which ensures that the six contour lines can only intersect at the end points and smoothly form a graphic contour, and the tile base constructed according to the method Figure 2 can construct a tiling plane without overlapping and gaps Figure 3 . If the tile base artistic drawing is changed into a human head shape and a proper head portrait pattern is designed, it can be used to construct a head portrait artistic tiling, as shown in Figure 4 .

[0045] Embodiment 2

[0046] Referring to the accompanying Figures 5-6Unlike Embodiment 1, by setting the boundary positioning point B1 inside the regular hexagon, the layout of some boundary contours being recessed and protruding from the regular hexagon is changed. Specifically, the second type of ceramic tile substrate has a first corner point A1, a second corner point A2, a third corner point A3, a fourth corner point A4, a fifth corner point A5, and a sixth corner point A6 corresponding to the regular hexagonal structure. A first boundary contour is provided between the first corner point A1 and the second corner point A2. The first boundary contour has a boundary positioning point B1, which is located inside the regular hexagon A1A2A3A4A5A6. The first boundary contour is symmetrical about the perpendicular bisector BB1 of the line connecting the first corner point A1 and the second corner point A2. The first boundary contour is formed by rotating the second corner point A2 counterclockwise by 120° to form the second... A second boundary profile is formed between corner point A2 and the third corner point A3; the first boundary profile is mirror-symmetrical about the line connecting the third corner point A3 and the sixth corner point A6, forming a fourth boundary profile between the third corner point A4 and the fifth corner point A5; the fourth boundary profile is formed by rotating the fourth corner point A4 clockwise by 120° to form the third boundary profile between the third corner point A3 and the fourth corner point A4; a sixth boundary profile is formed between the first corner point A1 and the sixth corner point A6, and the sixth boundary profile is formed by rotating the sixth corner point A6 clockwise by 120° to form the fifth boundary profile between the fifth corner point A5 and the sixth corner point A6. An example of a ceramic tile substrate constructed using this method is shown below. Figure 5 Following this method, a contour map of a ceramic tile substrate in the shape of a fox can be designed, along with an appropriate fox pattern, to create an artistic tiling of the fox motif. (See...) Figure 6 .

[0047] Example 3

[0048] See appendix Figures 7-8Specifically, the third type of ceramic tile substrate corresponds to a regular hexagonal structure having a first edge corner point A1, a second edge corner point A2, a third edge corner point A3, a fourth edge corner point A4, a fifth edge corner point A5 and a sixth edge corner point A6, a first boundary contour is provided between the first edge corner point A1 and the second edge corner point A2, a boundary positioning point B1 of the first boundary contour is located inside the regular hexagon A1A2A3A4A5A6, and the first boundary contour is symmetric about a perpendicular bisector BB1 of a line connecting the first edge corner point A1 and the second edge corner point A2; the first boundary contour is rotated counterclockwise by 120° at the second edge corner point A2 to form a second boundary contour between the second edge corner point A2 and the third edge corner point A3; the second boundary contour is rotated counterclockwise by 120° at the third edge corner point A3 to form a third boundary contour between the third edge corner point A3 and the fourth edge corner point A4; the third boundary contour is rotated counterclockwise by 120° at the fourth edge corner point A4 to form a fourth boundary contour between the fourth edge corner point A4 and the fifth edge corner point A5; a sixth boundary contour is provided between the first edge corner point A1 and the sixth edge corner point A6, and the sixth boundary contour is rotated clockwise by 120° at the sixth edge corner point A6 to form a fifth boundary contour between the fifth edge corner point A5 and the sixth edge corner point A6. The ceramic tile substrate case constructed by the method is shown in Figure 7 According to the method, an artistic ceramic tile substrate contour with a flying bird shape can be designed, and a flying bird pattern artistic close-packing can be constructed, as shown in Figure 8 The constructed flying bird close-packing can be used to decorate a home, as shown in Figure 9 .

[0049] Example 4

[0050] Refer to the accompanying Figures 10-11Unlike the embodiment 3, the layout of the part of the border profile concave and convex to the regular hexagon is changed by setting the border positioning point B1 outside the regular hexagon. Specifically, the fourth type of the ceramic tile substrate corresponding to the regular hexagon structure has a first edge corner point A1, a second edge corner point A2, a third edge corner point A3, a fourth edge corner point A4, a fifth edge corner point A5 and a sixth edge corner point A6, the first edge corner point A1 and the second edge corner point A2 are provided with a first border profile, the border positioning point B1 of the first border profile is outside the regular hexagon A1A2A3A4A5A6, the first border profile is symmetric about the perpendicular bisector BB1 of the line between the first edge corner point A1 and the second edge corner point A2; the first border profile is counterclockwise rotated by 120° at the second edge corner point A2 to form the second border profile between the second edge corner point A2 and the third edge corner point A3; the second border profile is counterclockwise rotated by 120° at the third edge corner point A3 to form the third border profile between the third edge corner point A3 and the fourth edge corner point A4; the third border profile is counterclockwise rotated by 120° at the fourth edge corner point A4 to form the fourth border profile between the fourth edge corner point A4 and the fifth edge corner point A5; the first edge corner point A1 and the sixth edge corner point A6 are provided with a sixth border profile, the sixth border profile is clockwise rotated by 120° at the sixth edge corner point A6 to form the fifth border profile between the fifth edge corner point A5 and the sixth edge corner point A6. The case of the ceramic tile substrate constructed by the method is shown in Figure 10 . According to the method, the artistic ceramic tile substrate profile with rose shape and the pattern thereof can be designed, and the artistic close-packing with rose pattern can be constructed, see Figure 11 .

[0051] Embodiment 5

[0052] Referring to the accompanying drawings Figures 12-13Specifically, the fifth type of ceramic tile substrate corresponds to a regular hexagonal structure with a first edge corner point A1, a second edge corner point A2, a third edge corner point A3, a fourth edge corner point A4, a fifth edge corner point A5 and a sixth edge corner point A6, a first boundary contour is arranged between the first edge corner point A1 and the second edge corner point A2, a boundary positioning point B1 of the first boundary contour is located inside the regular hexagon A1A2A3A4A5A6, and the first boundary contour is symmetric about a vertical bisector BB1 of a line connecting the first edge corner point A1 and the second edge corner point A2; a second boundary contour identical to the first boundary contour is arranged between the second edge corner point A2 and the third edge corner point A3, and a sixth boundary contour identical to the first boundary contour is arranged between the first edge corner point A1 and the sixth edge corner point A6; the second boundary contour is rotated counterclockwise by 120° at the third edge corner point A3 to form a third boundary contour between the third edge corner point A3 and the fourth edge corner point A4; the sixth boundary contour is rotated clockwise by 120° at the sixth edge corner point A6 to form a fifth boundary contour between the fifth edge corner point A5 and the sixth edge corner point A6; and a fourth boundary contour identical to the third boundary contour is arranged between the fourth edge corner point A4 and the fifth edge corner point A5. The ceramic tile substrate case constructed by the method is shown in Figure 12 According to the method, various ceramic tile substrate contours can be designed, as shown in the animal and plant contour cases given below Figure 13 The fish-shaped ceramic tile substrate contour can be designed with patterns, and a fish pattern art dense paving can be constructed, as shown in Figure 14 The fish dense paving can be used to decorate a home, as shown in Figure 15 .

[0053] Example 6

[0054] See the attached Figures 16-19Specifically, the porcelain tile substantially corresponds to a regular hexagon structure having a first edge corner point A1, a second edge corner point A2, a third edge corner point A3, a fourth edge corner point A4, a fifth edge corner point A5 and a sixth edge corner point A6, an outer boundary contour is provided between the first edge corner point A1 and the second edge corner point A2, a boundary positioning point B1 of the first boundary contour is located outside the regular hexagon A1A2A3A4A5A6, the first boundary contour is symmetric about a perpendicular bisector BB1 of a line connecting the first edge corner point A1 and the second edge corner point A2; the first boundary contour is rotated counterclockwise by 120° at the second edge corner point A2 to form a second boundary contour between the second edge corner point A2 and the third edge corner point A3; the second boundary contour is rotated counterclockwise by 120° at the third edge corner point A3 to form a third boundary contour between the third edge corner point A3 and the fourth edge corner point A4; the third boundary contour is rotated counterclockwise by 120° at the fourth edge corner point A4 to form a fourth boundary contour between the fourth edge corner point A4 and the fifth edge corner point A5; the third boundary contour is mirror-symmetric about a line connecting the second edge corner point A2 and the fifth edge corner point A5 to form a sixth boundary contour between the first edge corner point A1 and the sixth edge corner point A6; the sixth boundary contour is rotated clockwise by 120° at the sixth edge corner point A6 to form a fifth boundary contour between the fifth edge corner point A5 and the sixth edge corner point A6.

[0055] The porcelain tile base cases constructed by the method are shown in Figure 16 According to the method, the porcelain tile base contour in the shape of an animal, a plant and a human portrait can be designed, which is shown in Figure 17 The four cases given. In particular, the design of the pattern on the porcelain tile base contour in the shape of a lion can construct a lion pattern art dense paving, which is shown in Figure 18 The porcelain tile base contour designed by the method can construct a rich dense paving type, which is shown in Figure 19 The 16 local dense paving cases shown.

[0056] Example 7

[0057] Referring to the accompanying Figures 20-22Specifically, the ceramic tile substantially corresponds to a regular hexagon structure having a first edge corner point A1, a second edge corner point A2, a third edge corner point A3, a fourth edge corner point A4, a fifth edge corner point A5 and a sixth edge corner point A6, an outer boundary contour is provided between the first edge corner point A1 and the second edge corner point A2, a boundary positioning point B1 of the first boundary contour is located outside the regular hexagon A1A2A3A4A5A6, and the first boundary contour is symmetrical about a perpendicular bisector BB1 of a line connecting the first edge corner point A1 and the second edge corner point A2; the first boundary contour is counterclockwise rotated by 120° at the second edge corner point A2 to form a second boundary contour between the second edge corner point A2 and the third edge corner point A3; the second boundary contour is counterclockwise rotated by 120° at the third edge corner point A3 to form a third boundary contour between the third edge corner point A3 and the fourth edge corner point A4; the third boundary contour is counterclockwise rotated by 120° at the fourth edge corner point A4 to form a fourth boundary contour between the fourth edge corner point A4 and the fifth edge corner point A5; the fourth boundary contour is counterclockwise rotated by 120° at the fifth edge corner point A5 to form a fifth boundary contour between the fifth edge corner point A5 and the sixth edge corner point A6; and the fifth boundary contour is counterclockwise rotated by 120° at the sixth edge corner point A6 to form a sixth boundary contour between the sixth edge corner point A6 and the first edge corner point A1. The ceramic tile base body constructed by the above method is shown in Figure 18 According to the method, ceramic tile base bodies with different shapes can be designed, as shown in Figure 21 the flower and elephant ceramic tile base body contour examples given. In particular, a suitable design pattern can be designed on the elephant ceramic tile base body contour to construct an elephant pattern art dense paving, as shown in Figure 22 .

[0058] The regular hexagonal boundary contour transformation of embodiments 1-7 all ensure that the area is unchanged, so that periodic dense paving or non-periodic dense paving can be achieved. According to specific needs, designers can consciously design ceramic tile base body contours that fit certain animals, plants or portraits, that is, under the condition of following the same construction process of ceramic tile base bodies 1-7, different ceramic tile base bodies with different contours can still be constructed by designing different boundary contours, as shown in the examples given in Figure 13 , Figure 17 and Figure 21 . Another outstanding advantage of the present application is that a very rich dense paving can be constructed with one type of ceramic tile base body, as shown in Figure 19 .

[0059] In summary, the present application has the following five advantages over the traditional tiling: first, the traditional tiling requires multiple tile bases to form complex and rich tiling, but the present application only needs one type of tile base to achieve complex tiling, which greatly simplifies the manufacturing process and production cost; second, the tile base structure of the traditional tiling is relatively simple, and most of them only allow periodic tiling, but the tile base disclosed in the present application can not only construct periodic tiling, but also construct non-periodic tiling; third, most traditional tile bases only allow a limited number of tiling structures, but the tile base disclosed in the present application allows multiple splicing methods and can construct thousands of tiling structures with different structures, which is relatively more than the traditional tiling, and is particularly suitable for constructing complex systems; fourth, the splicing method allowed by the traditional tile base is determined, so the splicing difficulty is small. However, the tile base disclosed in the present application allows multiple connection possibilities, which can derive many error splicing possibilities that meet local tiling but do not meet global tiling, and as the size of the tiling increases, the splicing difficulty will increase sharply; fifth, the tile base of the traditional tiling cannot be deformed, and its boundary is a fixed shape, but the boundary of the tile base disclosed in the present application can be freely transformed within the framework of the construction criteria, which makes it have outstanding potential for artistic design and has broad market application prospects.

[0060] For the device and the use method disclosed in the embodiments, since they correspond to the method disclosed in the embodiments, the description is relatively simple, and the relevant part can be referred to the method part.

[0061] The above description of the disclosed embodiments enables a person skilled in the art to implement or use the present application. Various modifications to the embodiments will be apparent to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to the embodiments shown herein, but will conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A periodic and aperiodic tiling planar system based on regular hexagons, characterized in that, The system comprises multiple ceramic tile substrates, each based on a regular hexagonal structure. Each substrate has a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point, and a sixth corner point corresponding to the hexagonal structure. A first boundary profile is provided between the first and second corner points, and this first boundary profile has boundary positioning points. The distance between the boundary positioning points and the first and second corner points is less than the radius of the inscribed circle of the regular hexagon. A boundary profile is provided between any two adjacent corner points. In the boundary profiles corresponding to the ceramic tile substrates, the areas of the protruding and recessed portions of the regular hexagon are equal. The multiple boundary profiles on the ceramic tile substrates form specific shape contour edges. These specific shape contour edges of the multiple ceramic tile substrates interlock to form a periodic tiling system or an aperiodic tiling system. The multiple boundary profiles on the ceramic tile substrates together form... The ceramic tile substrate features a specific shape resembling an animal, plant, or human figure, with corresponding patterns set on the boundary outline of that shape. The boundary positioning point is located outside a regular hexagon, on the perpendicular bisector of the line connecting the first and second corner points. The first boundary outline is symmetrical about the perpendicular bisector of the line connecting the first and second corner points. The first boundary outline is rotated 120° counterclockwise around the second corner point to form a second boundary outline. The line connecting the third and sixth corner points forms a center line. The first boundary outline is mirror-symmetrical about the center line to form a fourth boundary outline. The fourth boundary outline is rotated 120° clockwise around the fourth corner point to form a third boundary outline. A sixth boundary outline is provided between the first and sixth corner points. The sixth boundary outline is rotated 120° clockwise around the sixth corner point to form a fifth boundary outline.

2. A periodic and aperiodic tiling planar system based on regular hexagons, characterized in that, The invention relates to a ceramic tile base body, which is based on a regular hexagon structure, and has a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point and a sixth corner point corresponding to the regular hexagon structure, a first boundary contour is arranged between the first corner point and the second corner point, the first boundary contour has a boundary positioning point, the distance between the boundary positioning point and the connecting line between the first corner point and the second corner point is less than the radius of the inscribed circle of the regular hexagon, a boundary contour is arranged between any two adjacent corner points, the area of the convex and concave regular hexagon part in the corresponding boundary contour of the ceramic tile base body is equal, the plurality of boundary contours on the ceramic tile base body form a specific shape contour edge, the specific shape contour edges of the plurality of ceramic tile base bodies are mutually embedded to form a periodic tiling system or a non-periodic tiling system, the plurality of boundary contours on the ceramic tile base body jointly form a specific shape of an animal, a plant or a human image, the corresponding pattern is arranged on the ceramic tile base body corresponding to the specific shape boundary contour, the boundary positioning point is inside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated counterclockwise by 120° at the second corner point to form a second boundary contour, the first boundary contour is mirror symmetrical about the connecting line between the third corner point and the sixth corner point to form a fourth boundary contour, the fourth boundary contour is rotated clockwise by 120° at the fourth corner point to form a third boundary contour, a sixth boundary contour is arranged outside the first corner point and the sixth corner point, the sixth boundary contour is rotated clockwise by 120° at the sixth corner point to form a fifth boundary contour.

3. A periodic and aperiodic tiling planar system based on regular hexagons, characterized in that, The invention relates to a ceramic tile base body, which is based on a regular hexagon structure, and has a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point and a sixth corner point corresponding to the regular hexagon structure, a first boundary contour is arranged between the first corner point and the second corner point, the first boundary contour has a boundary positioning point, the distance between the boundary positioning point and the connecting line between the first corner point and the second corner point is less than the radius of the inscribed circle of the regular hexagon, a boundary contour is arranged between any two adjacent corner points, the area of the convex and concave regular hexagon part in the corresponding boundary contour of the ceramic tile base body is equal, the plurality of boundary contours on the ceramic tile base body form a specific shape contour edge, the specific shape contour edges of the plurality of ceramic tile base bodies are mutually embedded to form a periodic tiling system or a non-periodic tiling system, the plurality of boundary contours on the ceramic tile base body jointly form a specific shape of an animal, a plant or a human image, the corresponding pattern is arranged on the ceramic tile base body corresponding to the specific shape boundary contour, the boundary positioning point is inside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated counterclockwise by 120° at the second corner point to form a second boundary contour, the second boundary contour is rotated counterclockwise by 120° at the third corner point to form a third boundary contour, the third boundary contour is rotated counterclockwise by 120° at the fourth corner point to form a fourth boundary contour, a sixth boundary contour is arranged between the first corner point and the sixth corner point, the sixth boundary contour is rotated clockwise by 120° at the sixth corner point to form a fifth boundary contour.

4. A periodic and aperiodic tiling planar system based on regular hexagons, characterized in that, The invention relates to a ceramic tile base body, which is based on a regular hexagon structure, and has a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point and a sixth corner point corresponding to the regular hexagon structure, a first boundary contour is arranged between the first corner point and the second corner point, the first boundary contour has a boundary positioning point, the distance between the boundary positioning point and the connecting line between the first corner point and the second corner point is less than the radius of the inscribed circle of the regular hexagon, a boundary contour is arranged between any two adjacent corner points, the area of the convex and concave regular hexagon part in the corresponding boundary contour of the ceramic tile base body is equal, the plurality of boundary contours on the ceramic tile base body form a specific shape contour edge, the specific shape contour edges of the plurality of ceramic tile base bodies are mutually embedded to form a periodic tiling system or a non-periodic tiling system, the plurality of boundary contours on the ceramic tile base body jointly form a specific shape of an animal, a plant or a human image, the corresponding pattern is arranged on the ceramic tile base body corresponding to the specific shape boundary contour, the boundary positioning point is outside the regular hexagon, the first boundary contour is symmetrical about the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated counterclockwise by 120° at the second corner point to form a second boundary contour, the second boundary contour is rotated counterclockwise by 120° at the third corner point to form a third boundary contour, the third boundary contour is rotated counterclockwise by 120° at the fourth corner point to form a fourth boundary contour, a sixth boundary contour is arranged between the first corner point and the sixth corner point, the sixth boundary contour is rotated clockwise by 120° at the sixth corner point to form a fifth boundary contour.

5. A periodic and aperiodic tiling planar system based on regular hexagons, characterized in that, The invention relates to a ceramic tile base body, which is based on a regular hexagon structure, and has a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point and a sixth corner point corresponding to the regular hexagon structure, wherein a first boundary contour is arranged between the first corner point and the second corner point, and the first boundary contour has a boundary positioning point, and the distance between the boundary positioning point and the connecting line between the first corner point and the second corner point is less than the radius of the inscribed circle of the regular hexagon, and a boundary contour is arranged between any two adjacent corner points, and the area of the convex and concave regular hexagon portions in the boundary contour corresponding to the ceramic tile base body is equal, and the specific shape contour edges of the plurality of ceramic tile base bodies are mutually embedded to form a periodic tiling system or a non-periodic tiling system, and the plurality of boundary contours on the ceramic tile base body jointly form a specific shape of an animal, a plant or a human image, and the corresponding patterns are arranged on the ceramic tile base body corresponding to the specific shape boundary contour, and the boundary positioning point is inside the regular hexagon, and the first boundary contour is symmetrical about the perpendicular bisector of the connecting line between the first corner point and the second corner point, and the second boundary contour between the second corner point and the third corner point is in the same state as the first boundary contour, and the sixth boundary contour between the first corner point and the sixth corner point is in the same state as the first boundary contour, and the third boundary contour is obtained by counterclockwise rotation of the second boundary contour by 120° at the third corner point, and the fourth boundary contour between the fourth corner point and the fifth corner point is in the same state as the third boundary contour, and the fifth boundary contour between the fifth corner point and the sixth corner point is in the same state as the fourth boundary contour.

6. A periodic and aperiodic tiling planar system based on regular hexagons, characterized in that, The system includes multiple ceramic tile substrates, each based on a regular hexagonal structure. Each ceramic tile substrate has a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point, and a sixth corner point corresponding to the regular hexagonal structure. A first boundary profile is provided between the first and second corner points. The first boundary profile has a boundary positioning point. The distance between the boundary positioning point and the line connecting the first and second corner points is less than the radius of the inscribed circle of the regular hexagon. A boundary profile is provided between any two adjacent corner points. In the boundary profiles corresponding to the ceramic tile substrates, the areas of the protruding and recessed regular hexagonal portions are equal. The multiple boundary profiles on the ceramic tile substrates form specific shape profile edges. The specific shape profile edges of the multiple ceramic tile substrates interlock to form a periodic tiling system or an aperiodic tiling system. Multiple boundary contours on the ceramic tile substrate together form a specific shape of an animal, plant, or human figure. Corresponding patterns are set on the ceramic tile substrate to the boundary contours of the specific shapes. The boundary positioning points are located outside the regular hexagon. The first boundary contour is symmetrical about the perpendicular bisector of the line connecting the first corner point and the second corner point. The first boundary contour is rotated 120° counterclockwise with the second corner point to form the second boundary contour. The second boundary contour is rotated 120° counterclockwise with the third corner point to form the third boundary contour. The third boundary contour is rotated 120° counterclockwise with the fourth corner point to obtain the fourth boundary contour. The third boundary contour is mirror-symmetrical about the line connecting the second corner point and the fifth corner point to form the sixth boundary contour. The sixth boundary contour is rotated 120° clockwise with the sixth corner point to form the fifth boundary contour.

7. A periodic and aperiodic tiling planar system based on regular hexagons, characterized in that, The invention relates to a ceramic tile substrate, comprising a plurality of ceramic tile substrates, each of which is based on a regular hexagon structure, the ceramic tile substrate corresponding to the regular hexagon structure having a first corner point, a second corner point, a third corner point, a fourth corner point, a fifth corner point and a sixth corner point, a first boundary contour being provided between the first corner point and the second corner point, the first boundary contour having a boundary positioning point, the distance between the boundary positioning point and the first corner point and the second corner point being less than the radius of the inscribed circle of the regular hexagon, a boundary contour being provided between any two adjacent corner points, the ceramic tile substrate corresponding to the boundary contour having equal areas of convex and concave regular hexagon portions, the plurality of boundary contours on the ceramic tile substrate forming a specific shape contour edge, the specific shape contour edges of the plurality of ceramic tile substrates being mutually embedded to form a periodic tiling system or a non-periodic tiling system, the plurality of boundary contours on the ceramic tile substrate collectively forming a specific shape of an animal, a plant or a human figure, the ceramic tile substrate corresponding to the specific shape boundary contour being provided with a corresponding design, the boundary positioning point being outside the regular hexagon, the first boundary contour being rotated counterclockwise by 120° at the second corner point to form a second boundary contour, the second boundary contour being rotated counterclockwise by 120° at the third corner point to form a third boundary contour, the third boundary contour being rotated counterclockwise by 120° at the fourth corner point to form a fourth boundary contour, the fourth boundary contour being rotated counterclockwise by 120° at the fifth corner point to form a fifth boundary contour, and the fifth boundary contour being rotated counterclockwise by 120° at the fifth corner point to form a sixth boundary contour.

8. A periodic and aperiodic hexagonal-based tiling planar system according to any one of claims 1-7, characterized in that, The six boundary contours collectively form a ceramic tile substrate in the shape of an animal, a plant or a human figure, the ceramic tile substrate corresponding to the shape of the animal, the plant or the human figure being provided with a texture pattern of the animal, the plant or the human figure.

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