Periodic and non-periodic planar filling shape system based on regular hexagonal

Abstract

To provide a non-cycle planar filling shape system capable of performing a variety of planar filling effects by combination of the same kind of tile substrates which has a simple structure and costs less.SOLUTION: The current invention discloses a periodic and non-periodic planar filling shape system based on a regular hexagonal and includes seven different tile substrate types. The tile substrates of the respective kinds are based on a regular hexagonal structure. Outlines of the tile substrates of the respective kinds comprise six boundary outlines. In a communication figure encircled by the six boundary outlines, areas of a part projecting from the regular hexagonal and a part recessed in the same match and are equal. The six boundary outlines encircle as specific shapes of animals, plants or human images. The specific shapes encircled by the six boundary outlines engage one another to form a planar filling shape. The tile substrates are provided with corresponding figures in correspondence with boundary outlines in the specific shape. The current invention has a simple structure, varies in planar filling effect, and has wide application prospect.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to the technical field of tessellation structures, and in particular to periodic and aperiodic tessellation systems based on regular hexagons. [Background technology]

[0002] Tiling refers to covering an entire plane with multiple patterns without overlapping or gaps. In addition to being widely used in tile tiling, tiling of plane figures is also used in mathematics education. For example, the process of exploring the conditions for polygon tiling can further develop students' reasoning and aesthetic abilities; experiencing the applications of plane figures in real life, such as jigsaw puzzles, can better enlighten children's thinking.

[0003] Tilings are divided into periodic, aperiodic and inverse periodic tilings. If a basic pattern can be constructed using a tiling substrate, and then the basic pattern can be translated along two different directions to cover the entire plane without gaps or overlaps, then such a tiling substrate is called a periodic tiling, and the corresponding tiling is called a periodic tiling. An aperiodic prototile is a tiling substrate that cannot form periodic tilings. Aperiodic prototiles are tiles that can form periodic tilings as well as nonperiodic tilings.

[0004] People have studied deeply the intellectual and decorative functions of tilings, for example, Nobel Prize winner Roger Penrose invented the Penrose tiles (Patent Number: 4133152) which have gained far-reaching fame, and John AL Osborn invented Escher-style art tiles (Patent Numbers: 5481841, 5520388, 5619830). However, the tilings currently discovered have the following defects. First, most periodic, aperiodic or inverse periodic tiling puzzles have more than one type of slice block, and the excessive number of slice block types results in poor artistic beauty of the unity and harmony of the puzzle, and high manufacturing costs. Second, the geometric structure of periodic tilings is too simple and not challenging enough, making them unattractive. Third, compared with periodic tessellations, the construction of aperiodic or inverse periodic tessellations is relatively difficult, but the number of solutions for such tessellations is small and lacking in variety, and players quickly lose interest after understanding the rules of the tessellations.

[0005] Therefore, it is a problem to be solved by those skilled in the art to provide a non-periodic tessellation system that has a simple structure, low manufacturing costs, and can achieve a variety of tessellation effects by combining tile substrates of the same type. Summary of the Invention [Means for solving the problem]

[0006] In view of this, the present invention provides a periodic and non-periodic plane filling system based on regular hexagons, which can provide plane filling forms with various contours and beautiful and rich styles, and has extensive research and development potential, the tile base provided by the present invention is suitable for Escher-style art, has good affinity and intellectual development and enlightenment functions for children, and the tiles not only enable periodic and non-periodic plane filling, but also have a rich plane filling structure method, which can be widely applied in the fields of educational toys and games and art decoration, and has good aesthetics and economic potential.

[0007] In order to achieve the above object, the present invention adopts the following technical solution: a periodic and aperiodic tessellation system based on a regular hexagon, comprising a plurality of tile substrates, each of 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 boundary contour is provided between any two adjacent corner points, in the corresponding boundary contour of the tile substrate, the area of ​​the part protruding from the regular hexagon and the part recessed therein are equal, the boundary contours of the tile substrates form contour edges of a specific shape, the contour edges of the specific shapes of the plurality of tile substrates are fitted together to form a periodic tessellation system or an aperiodic tessellation system, the boundary contours of the tile substrates together form a specific shape of an animal, a plant or a human figure, the tile substrates are provided with corresponding texture patterns corresponding to the boundary contours of the specific shapes, and the tile substrates are of seven kinds.

[0008] The beneficial effects of the present invention are as follows: A regular hexagonal tile base is converted into a specific shape figure, and six boundary contours are used to form the boundary of a specific shape such as an animal, plant or human figure, and the area enclosed by the six boundary contours is the same as the original regular hexagon. Compared with the conventional tessellation, the use of a single type of tile base in the present invention can realize complex tessellation, greatly simplifying the manufacturing process and greatly reducing the manufacturing cost. The mirror image, translation and rotation transformation of the tile base can be combined to form a variety of tessellations. Combined with the artist's processing design, the present invention can provide artistic patterns with various contours, beautiful styles and rich styles, which have good affinity and intellectual education and enlightenment functions for children, and have extensive research and development potential. The tile base not only enables periodic and non-periodic tessellation, but also has the characteristics of high solution difficulty and large number of solutions, which makes up for the serious defects of conventional tessellation. The present invention can be widely applied in the fields of educational toy games, art design decoration and the like, and has excellent aesthetics and economic application potential.

[0009] Preferably, a first boundary contour is provided between the first corner point and the second corner point of the first type of tile base, the first boundary contour having a boundary positioning point located outside the regular hexagon and located on the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is symmetrical with respect to the perpendicular bisector of the connecting line between the first corner point and the second corner point, and the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour; A connecting line between the third corner point and the sixth corner point forms a center line, the first boundary contour is mirror-symmetrical with respect to the center line to form a fourth boundary contour, the fourth boundary contour is rotated 120° clockwise around the fourth corner point to form a third boundary contour, a sixth boundary contour is provided between the first corner point and the sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

[0010] Preferably, a first boundary contour is provided between a first corner point and a second corner point of the second type of tile base, the first boundary contour having a boundary positioning point inside a regular hexagon and symmetrical with respect to the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the first boundary contour is mirror-symmetric with respect to 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 120° clockwise around the fourth corner point to form a third boundary contour, and a sixth boundary contour is provided outside the first corner point and the sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

[0011] Preferably, a first boundary contour is provided between a first corner point and a second corner point of a third type of tile base, the first boundary contour having a boundary positioning point inside a regular hexagon and symmetrical with respect to the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour is rotated 120° counterclockwise around a third corner point to form a third boundary contour, the third boundary contour is rotated 120° counterclockwise around a fourth corner point to form a fourth boundary contour, a sixth boundary contour is provided between the first corner point and a sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

[0012] Preferably, a first boundary contour is provided between a first corner point and a second corner point of a fourth type of tile base, the first boundary contour having a boundary positioning point outside the regular hexagon and symmetrical with respect to the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour is rotated 120° counterclockwise around a third corner point to form a third boundary contour, the third boundary contour is rotated 120° counterclockwise around a fourth corner point to form a fourth boundary contour, a sixth boundary contour is provided between the first corner point and a sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

[0013] Preferably, a first boundary contour is provided between a first corner point and a second corner point of a fifth type of tile base, the first boundary contour having a boundary positioning point inside a regular hexagon and symmetrical with respect to the perpendicular bisector of the connecting line between the first corner point and the second corner point, a second boundary contour having the same state as the first boundary contour is provided between the second corner point and a third corner point, a sixth boundary contour having the same state as the first boundary contour is provided between the first corner point and a sixth corner point, the second boundary contour is rotated 120 degrees counterclockwise around the third corner point to obtain a third boundary contour, a fourth boundary contour having the same state as the third boundary contour is provided between the fourth corner point and the fifth corner point, and a fifth boundary contour having the same state as the fourth boundary contour is provided between the fifth corner point and the sixth corner point.

[0014] Preferably, a first boundary contour is provided between a first corner point and a second corner point of a sixth type of tile base, the first boundary contour having a boundary positioning point outside the regular hexagon and symmetrical with respect to the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour is rotated 120° counterclockwise around the third corner point to form a third boundary contour, the third boundary contour is rotated 120° counterclockwise around the fourth corner point to obtain a fourth boundary contour, the third boundary contour is mirror-symmetrical with respect to the connecting line between the second corner point and the fifth corner point to form a sixth boundary contour, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

[0015] Preferably, a first boundary contour having a boundary positioning point outside the regular hexagon is provided between a first corner point and a second corner point of the seventh type tile base, the first boundary contour being rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour being rotated 120° counterclockwise around a third corner point to form a third boundary contour, the third boundary contour being rotated 120° counterclockwise around a fourth corner point to obtain a fourth boundary contour, the fourth boundary contour being rotated 120° counterclockwise around a fifth corner point to obtain a fifth boundary contour, and the fifth boundary contour being rotated 120° counterclockwise around the fifth corner point to form a sixth boundary contour.

[0016] Preferably, the distance from the boundary positioning point to the connecting line between the first corner point and the second corner point is smaller than the dimension of the radius of the inscribed circle of the regular hexagon.

[0017] Preferably, six boundary contours of the tile substrate are enclosed as a tile substrate in the shape of an animal, plant or human figure, and the tile substrate is provided with a texture pattern of an animal, plant or human figure corresponding to the shape of the animal, plant or human figure. [Brief description of the drawings]

[0018] [Figure 1] FIG. 1 is a structural schematic diagram of a first embodiment of the present invention. [Diagram 2] 3 is a design flowchart of the first embodiment of the present invention. [Diagram 3] FIG. 2 is a tessellation of the first embodiment of the present invention. [Figure 4] FIG. 2 is an application diagram of the first embodiment of the present invention. [Diagram 5] FIG. 4 is a structural diagram of a second embodiment of the present invention. [Figure 6] FIG. 11 is an application diagram of the second embodiment of the present invention. [Figure 7] FIG. 11 is a structural diagram of a third embodiment of the present invention. [Figure 8] FIG. 11 is an application diagram of the third embodiment of the present invention. [Figure 9] FIG. 11 is a scene decoration diagram of the third embodiment of the present invention. [Figure 10] FIG. 11 is a structural diagram of a fourth embodiment of the present invention. [Figure 11] FIG. 11 is an application diagram of the fourth embodiment of the present invention. [Figure 12] FIG. 11 is a structural diagram of a fifth embodiment of the present invention. [Figure 13] FIG. 13 is a diagram showing a specific shape formed based on a modified method of the fifth embodiment of the present invention. [Figure 14] FIG. 13 is an application diagram of the fifth embodiment of the present invention. [Figure 15] FIG. 11 is a scene decoration diagram of the fifth embodiment of the present invention. [Figure 16] FIG. 13 is a structural diagram of a sixth embodiment of the present invention. [Figure 17] FIG. 13 is a diagram showing a specific shape formed based on a modified method of the sixth embodiment of the present invention. [Figure 18] FIG. 13 is an application diagram of the sixth embodiment of the present invention. [Figure 19] FIG. 13 is a schematic diagram of a multi-pack case constructed in accordance with a sixth embodiment of the present invention. [Figure 20] FIG. 13 is a structural diagram of a seventh embodiment of the present invention. [Figure 21] FIG. 13 is a diagram showing a specific shape formed based on a modified method of the seventh embodiment of the present invention. [Figure 22] FIG. 13 is an application diagram of the seventh embodiment of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0019] Hereinafter, the technical solutions of the embodiments of the present invention will be described clearly and completely with reference to the drawings of the embodiments of the present invention, and it is obvious that the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative labor, fall within the protection scope of the present invention. EXAMPLES

[0020] Example 1 As shown in Fig. 1-5 of the present invention, according to the embodiment of the present invention, the periodic and non-periodic tessellation system based on regular hexagons forms a special tile base, each of the tile bases is based on a regular hexagonal structure, and its outer contour is composed of six boundary contours, and the boundary contours may be located outside or inside the regular hexagon as needed, but the areas of the parts protruding from the regular hexagon and the parts recessed into it are correspondingly equal, so that the area of ​​the tile base is equal to the original regular hexagon. The six boundary contours surround a specific shape of an animal, plant or human figure, and the specific shapes surrounded by the six boundary contours fit together to form a tessellation, and the tile base is provided with a corresponding texture pattern corresponding to the boundary contours of the specific shapes.

[0021] Specifically, the first type of tile base 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 a regular hexagonal structure, and a first boundary contour is provided between the first corner point A1 and the second corner point A2, the first boundary contour being symmetrical with respect to a perpendicular bisector BB1 of the connecting line between the first corner point A1 and the second corner point A2, and the first boundary contour is rotated 120° counterclockwise around the second corner point A2 to align the second corner point A2 and the third corner point A3. A second boundary contour is formed between the first corner point A1 and the sixth corner point A6, a fourth boundary contour is formed between the third corner point A4 and the fifth corner point A5 by mirror symmetry with respect to the connecting line between the third corner point A3 and the sixth corner point A6, the fourth boundary contour is rotated 120° clockwise around 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, and the sixth boundary contour is rotated 120° clockwise around 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 by this method is as shown in FIG.

[0022] More specifically, the distance from the boundary positioning point B1 to the connecting line between the first corner point A1 and the second corner point A2 is smaller than the radius of the inscribed circle of the regular hexagon, ensuring that the six contour lines can only intersect at their end points and are smoothly enclosed as one graphic contour, and the tile base constructed by this method (Fig. 2) can form a non-overlapping and gap-free tessellation (Fig. 3). By modifying the tile base from an artistic painting to the shape of a human head and designing an appropriate head image texture pattern, it can be used to construct an artistic tessellation of the head image, as shown in Fig. 4. EXAMPLES

[0023] Example 2 As shown in Figures 5 and 6, the difference from Example 1 is that the boundary positioning point B1 is provided inside a regular hexagon, thereby changing the layout in which a part of the boundary contour protrudes from and is recessed into the regular hexagon. Specifically, the second type of tile base 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 hexagon structure, and between the first corner point A1 and the second corner point A2, a boundary positioning point B1 is provided that is located inside the regular hexagon A1A2A3A4A5A6, and a first boundary contour is provided that is symmetrical with respect to a perpendicular bisector BB1 of the connecting line between the first corner point A1 and the second corner point A2, and the first boundary contour is rotated 120° counterclockwise around the second corner point A2 to be aligned with the second corner point A2 and the third corner point A3. A second boundary contour is formed between the first corner point A1 and the sixth corner point A6, and a fourth boundary contour is formed between the third corner point A4 and the fifth corner point A5 by performing a mirror image symmetry with respect to the connecting line between the third corner point A3 and the sixth corner point A6, and the fourth boundary contour is rotated 120° clockwise around the fourth corner point A4 to form a third boundary contour between the third corner point A3 and the fourth corner point A4, and a sixth boundary contour is provided between the first corner point A1 and the sixth corner point A6, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point A6 to form a fifth boundary contour between the fifth corner point A5 and the sixth corner point A6. The case of the tile base constructed by this method is as shown in FIG. 5. According to this method, the contour diagram of the tile base having a fox shape is designed, and an appropriate fox texture pattern is designed to construct an artistic plane filling of the fox pattern, as shown in FIG. 6. EXAMPLES

[0024] Example 3 As shown in Figures 7 and 8. Specifically, the third type of tile base 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 a regular hexagonal structure, and between the first corner point A1 and the second corner point A2, a boundary positioning point B1 is located inside the regular hexagon A1A2A3A4A5A6, and a first boundary contour is provided that is symmetrical with respect to a perpendicular bisector BB1 of the connecting line between the first corner point A1 and the second corner point A2, and the first boundary contour is rotated 120° counterclockwise around the second corner point A2 to form a boundary between the second corner point A2 and the third corner point A3. A second boundary contour is formed between the first corner point A1 and the sixth corner point A6, and the second boundary contour is rotated 120° counterclockwise around the third corner point A3 to form a third boundary contour between the third corner point A3 and the fourth corner point A4, and the third boundary contour is rotated 120° counterclockwise around the fourth corner point A4 to form a fourth boundary contour between the fourth corner point A4 and the fifth corner point A5, and a sixth boundary contour is provided between the first corner point A1 and the sixth corner point A6, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point A6 to form a fifth boundary contour between the fifth corner point A5 and the sixth corner point A6. The case of the tile base constructed by this method is as shown in FIG. 7. According to this method, the contour of the asuka-shaped artistic tile base can be designed and an artistic plane filling of the asuka pattern can be constructed, as shown in FIG. 8. The constructed Asuka tessellation can be used to decorate a house, as shown in Figure 9. EXAMPLES

[0025] Example 4 10 and 11, the difference from the third embodiment is that the boundary positioning point B1 is provided outside the regular hexagon, thereby changing the layout in which a part of the boundary contour protrudes from and is recessed into the regular hexagon. Specifically, the fourth type of tile base 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 hexagon structure, and between the first corner point A1 and the second corner point A2, the boundary positioning point B1 is located outside the regular hexagon A1A2A3A4A5A6, and a first boundary contour is provided that is symmetrical with respect to the perpendicular bisector BB1 of the connecting line between the first corner point A1 and the second corner point A2, and the first boundary contour is rotated 120° counterclockwise around the second corner point A2 to become a vertex between the second corner point A2 and the third corner point A3. A second boundary contour is formed between the first corner point A1 and the fourth corner point A4, and the second boundary contour is rotated 120° counterclockwise around the third corner point A3 to form a third boundary contour between the third corner point A3 and the fourth corner point A4, and the third boundary contour is rotated 120° counterclockwise around the fourth corner point A4 to form a fourth boundary contour between the fourth corner point A4 and the fifth corner point A5, and a sixth boundary contour is provided between the first corner point A1 and the sixth corner point A6, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point A6 to form a fifth boundary contour between the fifth corner point A5 and the sixth corner point A6. The case of the tile base constructed by this method is as shown in FIG. 10. According to this method, the contour of the rose-shaped artistic tile base and its texture pattern can be designed, and a rose pattern artistic plane tessellation can be constructed, as shown in FIG. 11. EXAMPLES

[0026] Example 5 As shown in Figures 12 and 13, the fifth type of tile base 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 a regular hexagonal structure, and between the first corner point A1 and the second corner point A2, a boundary positioning point B1 is located inside the regular hexagon A1A2A3A4A5A6, and a first boundary contour is provided that is symmetrical with respect to a perpendicular bisector BB1 of the connecting line between the first corner point A1 and the second corner point A2, and a second boundary contour that is the same as the state of the first boundary contour is provided between the second corner point A2 and the third corner point A3. A boundary contour is provided, a sixth boundary contour having the same state as the first boundary contour is provided between the first corner point A1 and the sixth corner point A6, the second boundary contour is rotated 120° counterclockwise around the third corner point A3 to form a third boundary contour between the third corner point A3 and the fourth corner point A4, the sixth boundary contour is rotated 120° clockwise around the sixth corner point A6 to form a fifth boundary contour between the fifth corner point A5 and the sixth corner point A6, and a fourth boundary contour having the same state as the third boundary contour is provided between the fourth corner point A4 and the fifth corner point A5. The case of a tile base constructed by this method is as shown in FIG. 12. According to this method, the contours of multiple kinds of tile bases can be designed, as shown in the animal and plant contour case shown in FIG. 13. A texture pattern can be designed on the contour of a fish-shaped tile base, and an artistic plane tessellation of a fish pattern can be constructed, as shown in FIG. 14. Fish tessellations can be used to decorate the home, as shown in Figure 15. EXAMPLES

[0027] Example 6 As shown in Figures 16 to 19. Specifically, the tile base 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 a regular hexagonal structure, and between the first corner point A1 and the second corner point A2, a boundary positioning point B1 is located outside the regular hexagon A1A2A3A4A5A6, and a first boundary contour is provided that is symmetrical with respect to a perpendicular bisector BB1 of the connecting line between the first corner point A1 and the second corner point A2. The first boundary contour is rotated 120° counterclockwise around the second corner point A2 to form a second boundary contour between the second corner point A2 and the third corner point A3, and the second boundary contour is provided. The contour is rotated 120° counterclockwise around the third corner point A3 to form a third boundary contour between the third corner point A3 and the fourth corner point A4, the third boundary contour is rotated 120° counterclockwise around the fourth corner point A4 to form a fourth boundary contour between the fourth corner point A4 and the fifth corner point A5, and is mirror-symmetrical with respect to the connecting line between the second corner A2 and the fifth corner point A5 to form a sixth boundary contour between the first corner point A1 and the sixth corner point A6, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point A6 to form a fifth boundary contour between the fifth corner point A5 and the sixth corner point A6.

[0028] The case of the tile base constructed by this method is as shown in Figure 16. According to this method, the contour of the tile base in the shape of an animal, a plant and a human figure can be designed, as shown in the four cases in Figure 17. In particular, by designing a texture pattern on the contour of the lion-shaped tile base, a lion pattern artistic tessellation can be constructed, as shown in Figure 18. The contour of the tile base designed by this method can construct a rich variety of tessellation types, as shown in the 16 local tessellation cases in Figure 19. EXAMPLES

[0029] Example 7 As shown in Figures 20 to 22. Specifically, the tile base 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 a regular hexagonal structure, and between the first corner point A1 and the second corner point A2, a boundary positioning point B1 is located outside the regular hexagon A1A2A3A4A5A6, and a first boundary contour is provided that is symmetrical with respect to a perpendicular bisector BB1 of the connecting line between the first corner point A1 and the second corner point A2, and the first boundary contour is rotated 120° counterclockwise around the second corner point A2 to form a second boundary contour between the second corner point A2 and the third corner point A3, and the second boundary contour is rotated 120° counterclockwise around the third corner point A3 to form a third boundary contour between the third corner point A3 and the fourth corner point A4, the third boundary contour is rotated 120° counterclockwise around the fourth corner point A4 to form a fourth boundary contour between the fourth corner point A4 and the fifth corner point A5, the fourth boundary contour is rotated 120° counterclockwise around the fifth corner point A5 to form a fifth boundary contour between the fifth corner point A5 and the sixth corner point A6, and the fifth boundary contour is rotated 120° counterclockwise around the sixth corner point A6 to form a sixth boundary contour between the sixth corner point A6 and the first corner point A1. The case of the tile base constructed by the above method is as shown in FIG. 18. According to this method, the contour of the tile base of different shapes can be designed, as shown in the case of the contour of the flower and elephant tile base shown in FIG. 21. In particular, by providing an appropriate texture pattern on the outline of the elephant tile base, an artistic plane tessellation of the elephant pattern can be constructed, as shown in FIG. 22.

[0030] The transformation of the regular hexagonal boundary contours in the embodiments 1 to 7 must ensure that the area does not change, and periodic or non-periodic tessellation can be realized. According to specific needs, the designer can intentionally design the contour of the tile substrate to match a certain animal, plant or human figure, that is, in the situation where the structural flow of the tile substrates 1 to 7 is the same, the designer can still design different boundary contours to construct tile substrates with different contours, as shown in the cases in Figures 13, 17 and 21. Another prominent advantage of the present invention is that a single type of tile substrate can be used to construct a wide variety of tessellations, as shown in Figure 19.

[0031] In summary, the present invention has five prominent advantages over conventional tessellations: first, conventional tessellations require the combination of multiple tile substrates to form complex and abundant tessellations, whereas the tile substrate disclosed in the present invention can realize complex tessellations with only one type, which greatly simplifies the manufacturing process and greatly reduces production costs; second, conventional tessellations have a relatively simple structure and mostly only allow periodic tessellations, whereas the tile substrate disclosed in the present invention can not only form periodic tessellations, but also non-periodic tessellations; third, conventional tile substrates mostly only allow the formation of a limited number of tessellation structures, whereas the tile substrate disclosed in the present invention allows multiple types of joining methods and can form numerous types of tessellations with different structures, which is much more diverse than conventional tessellations and is particularly suitable for the formation of highly complex systems; and fourth, the joining method allowed by conventional tile substrates is determined, so the joining difficulty is low. However, the tile base disclosed in the present invention allows for multiple connection methods, which can satisfy many local tessellations but not the entire tessellation, and as the scale of the tessellation increases, the difficulty of joining increases sharply. Fifth, while the tile base of conventional tessellations cannot be deformed and its boundaries have a fixed shape, the boundaries of the tile base disclosed in the present invention can be freely transformed within the structural reference frame, which makes it have outstanding potential for artistic design and broad prospects for application in the market.

[0032] The device and the method of use disclosed in the embodiments correspond to the methods disclosed in the embodiments, so the description is relatively simple, and reference should be made to the method section for relevant parts.

[0033] The above description of the disclosed embodiments enables one skilled in the art to realize or use the present invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Thus, the present invention is not intended to be limited to these embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A periodic and aperiodic tessellation system based on regular hexagons, comprising: a plurality of tile substrates, each of 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 boundary contour is provided between any two adjacent corner points; in the corresponding boundary contour of the tile substrates, the areas of the parts protruding from the regular hexagon and the parts recessed into it are equal; the boundary contours of the tile substrates form boundary edges of specific shapes; the boundary edges of the specific shapes of the plurality of tile substrates fit together to form a periodic tessellation system or an aperiodic tessellation system; the boundary contours of the tile substrates together form a specific shape of an animal, a plant or a human figure; and the tile substrates are provided with corresponding texture patterns corresponding to the boundary contours of the specific shapes.

2. Between the first corner point and the second corner point, a first boundary contour is provided, the first boundary contour having a boundary positioning point located outside the regular hexagon and located on a perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour being symmetrical with respect to the perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour being rotated 120° counterclockwise around the second corner point to form a second boundary contour, and the connecting line between the third corner point and the sixth corner point constitutes a center line, 2. The periodic and nonperiodic plane filling system based on regular hexagons as claimed in claim 1, characterized in that the first boundary contour is mirror-symmetric with respect to the center line to form a fourth boundary contour, the fourth boundary contour is rotated 120° clockwise around a fourth corner point to form a third boundary contour, a sixth boundary contour is provided between the first corner point and a sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

3. 2. The periodic and nonperiodic plane filling system based on regular hexagons according to claim 1, wherein a first boundary contour is provided between the first corner point and the second corner point, the first boundary contour having a boundary positioning point inside the regular hexagon and symmetrical with respect to a perpendicular bisector of a connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the first boundary contour is mirror-symmetric with respect to a connecting line between the third corner point and the sixth corner point to form a fourth boundary contour, the fourth boundary contour is rotated 120° clockwise around the fourth corner point to form a third boundary contour, and a sixth boundary contour is provided outside the first corner point and the sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

4. 2. The periodic and nonperiodic plane tessellating system based on regular hexagons according to claim 1, wherein a first boundary contour is provided between the first corner point and the second corner point, the first boundary contour having a boundary positioning point inside the regular hexagon and symmetrical with respect to a perpendicular bisector of a connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour is rotated 120° counterclockwise around the third corner point to form a third boundary contour, the third boundary contour is rotated 120° counterclockwise around the fourth corner point to form a fourth boundary contour, and a sixth boundary contour is provided between the first corner point and a sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

5. 2. The periodic and nonperiodic plane tessellation system based on regular hexagons according to claim 1, wherein a first boundary contour is provided between the first corner point and the second corner point, the first boundary contour having a boundary positioning point outside the regular hexagon and symmetrical with respect to a perpendicular bisector of the connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour is rotated 120° counterclockwise around the third corner point to form a third boundary contour, the third boundary contour is rotated 120° counterclockwise around the fourth corner point to form a fourth boundary contour, and a sixth boundary contour is provided between the first corner point and a sixth corner point, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

6. 2. The periodic and nonperiodic plane tessellation system based on regular hexagons according to claim 1, wherein a first boundary contour is provided between the first corner point and the second corner point, the first boundary contour having a boundary positioning point inside the regular hexagon and symmetrical with respect to a perpendicular bisector of a connecting line between the first corner point and the second corner point, a second boundary contour having the same state as the first boundary contour is provided between the second corner point and the third corner point, a sixth boundary contour having the same state as the first boundary contour is provided between the first corner point and the sixth corner point, the second boundary contour is rotated 120 degrees counterclockwise around the third corner point to obtain a third boundary contour, a fourth boundary contour having the same state as the third boundary contour is provided between the fourth corner point and the fifth corner point, and a fifth boundary contour having the same state as the fourth boundary contour is provided between the fifth corner point and the sixth corner point.

7. 2. The periodic and aperiodic plane filling system based on regular hexagons according to claim 1, wherein a first boundary contour is provided between the first corner point and the second corner point, the first boundary contour having a boundary positioning point outside the regular hexagon and symmetrical with respect to a perpendicular bisector of a connecting line between the first corner point and the second corner point, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour is rotated 120° counterclockwise around a third corner point to form a third boundary contour, the third boundary contour is rotated 120° counterclockwise around a fourth corner point to obtain a fourth boundary contour, the third boundary contour is mirror-symmetrical with respect to a connecting line between the second corner point and the fifth corner point to form a sixth boundary contour, and the sixth boundary contour is rotated 120° clockwise around the sixth corner point to form a fifth boundary contour.

8. 2. The periodic and nonperiodic tessellation system based on regular hexagons according to claim 1, wherein a first boundary contour is provided between the first corner point and the second corner point, the first boundary contour having a boundary positioning point outside the regular hexagon, the first boundary contour is rotated 120° counterclockwise around the second corner point to form a second boundary contour, the second boundary contour is rotated 120° counterclockwise around a third corner point to form a third boundary contour, the third boundary contour is rotated 120° counterclockwise around a fourth corner point to obtain a fourth boundary contour, the fourth boundary contour is rotated 120° counterclockwise around a fifth corner point to obtain a fifth boundary contour, and the fifth boundary contour is rotated 120° counterclockwise around the fifth corner point to form a sixth boundary contour.

9. A periodic and nonperiodic plane filling system based on regular hexagons according to any one of claims 2 to 8, characterized in that the distance from the boundary positioning point to the connecting line between the first corner point and the second corner point is smaller than the dimension of the radius of the inscribed circle of the regular hexagon.

10. The periodic and aperiodic plane filling system based on regular hexagons described in any one of claims 2 to 8, characterized in that the six boundary contours are together enclosed as a tile base in the shape of an animal, plant or human figure, and the tile base is provided with a texture pattern of an animal, plant or human figure corresponding to the shape of the animal, plant or human figure.

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