Origami pyramid with the ability to connect its five faces in a repeatable manner

Abstract

The invention of the origami pyramid with the ability to connect in five faces and repeatability in geometric space in the form of cubes and n-dimensional prisms in basic sciences, mathematics, and spatial geometry topics is an effective step towards knowing the future of geometric space, implementing for the first time in industrial design. This origami pyramid uses cubes connecting volumes in the form of cubes and prisms with the help of continuous and combined symmetry in the geometric space. In such a way that a cube divides into 27 small cubes, the first cubic symmetry is formed in the center of the edge of the pyramid, the center of spatial symmetry of the cube.

Description

[0001]Title of the Invention: "Origami pyramid with the ability to connect its five faces in a repeatable manner" Technical Field Calculations, design, and construction of spatial geometries Background of the Art Most packaging containers are essential to human life and are made of materials that take a long time to decompose and recycle, particularly polyethylene. As a result, they pose a significant threat to the environment. A functional design should prioritize collection, recycling, and returning to the industry cycle. This will create a market for reusing or replacing other structures in the industry. Bulk tanks and barrels have various uses in human life, such as transporting and storing materials, which reduces the need for polyethylene tanks for such purposes. This also leads to increased recycling and returning to the industrial cycle, such as in automobile manufacturing and packaging. The origami pyramid demonstrates the vast range of inventions that can be designed using mathematical sciences and its sub-branches, including industrial design. The invention aims to expand and eliminate the limitations in connecting the faces of connecting devices in the prefab structures industry. It also aims to enhance brainstorming and intellectual games, such as Lego, which is a popular game that strengthens children's spatial intelligence. Existing packing volumes and connecting pieces are modeled after geometric shapes such as cubes, rectangular cubes, spheres, cylinders, and ovals. Most connecting structures, such as prefabricated walls and intellectual game pieces like Legos, connect on two faces, allowing the folded bricks to be connected and stacked on top of each other. For instance, in a patent application published under number KR1020140078832, an invention related to the production of a pyramid-shaped learning paper block using origami with an angle setting is disclosed. The method comprises several steps, such as geometrically drawing an equilateral or isosceles triangle by setting angles, continuously drawing four equilateral or isosceles triangles to fold the equilateral or isosceles triangle into a pyramid, minimally drawing an adhesive-applying part on the side of the equilateral or isosceles triangle and cutting out the outermost part, and folding the equilateral or isosceles triangle along an internal line and applying an adhesive to the adhesive applying part. A patent application has been published under the number KR20090006654A, which pertains to an invention in the field of a regular hexahedron paper folding toy. This toy can be used as decorative lighting or interior article and can enhance the creativity and brain development of children. The regular hexahedron paper folding toy is composed of four different paper materials, namely the first paper material (1), the second paper material (2), the third paper material (3), and the fourth paper material (4). These materials are folded along a line with a ratio of 2:1 between the bottom plane and height. Each of the four paper materials comes in a different color. The patent application with the number KR100895494B1 describes an invention related to a type of playground equipment called Magic Cube Folding Paper. The invention pertains to playground equipment created by utilizing a cube, specifically by dividing the cube into several small components of the magic module and connecting them to fold mutually. Then, the neighboring magic modules fold several times to create a box-shaped structure. It is about a useful origami play or learning magic cube playground, which can be used to make and then turn it over and over again to form a star shape. When playing origami, the common practice is to fold along the dotted lines of colored paper to create three-dimensional shapes such as animals, airplanes, dice, and other materials used for teaching or playing games. This technique is primarily used for educational and recreational purposes. To fold origami correctly, one must follow a specific sequence of folds and have a thorough understanding of the paper's orientation. This process used to be time-consuming and challenging to learn. Additionally, traditional origami only focused on folding animals, flowers, and other shapes, lacking any real purpose beyond the act of folding itself. In the patent application published under the number KR20150051354A, an invention related to the field of the present invention relates to a paper for fabricating a cubic box, a paper coin bank, and a paper packing box, which includes first to fourth main outer wall sections connected in order along the longitudinal central area bending against each other; and first to fourth wing sections connected to both lateral sides of the first to fourth main outer wall sections bents against each other. The third and fourth wing sections are connected to the first and second wing sections so that they can bend towards each other. The second and third wing sections are joined along a dotted line. According to the present invention, the paper for fabricating a cubic box, a paper coin bank, and a paper packing box enables users to easily make a cubic box in a telescoping manner without the use of separate adhesives and tools; and allows the assembled cubic box to firmly maintain the assembled state for a long time. Summary of the Invention In the study of the spatial geometry of the cube, there are only two origami pyramids known to have a square symmetrical base that can divide the cube into three and six faces. An origami pyramid that has a trapezoid base (asymmetrical) with the help of the axis of rotational symmetry can divide the cube into four symmetrical faces (101). The second and fourth-order origami pyramids have an axis of rotational symmetry that can divide the cube into four symmetrical faces. One of the two known pyramids with a square symmetrical base is called an oblique pyramid with a rotation around the third-order rotational symmetry axis, and it can divide the cube into three faces. In the case of a vertical origami pyramid with a square symmetrical base, with rotation occurring at an angle of 90 degrees around the center of rotational symmetry, the cube can be divided into six faces. However, in this scenario, the connecting faces are not facing each other. If a pyramid made of origami with a trapezoidal base (101) exhibits two types of symmetry - fourth and second order - each with an angle of 180 degrees, then it can be transformed into a cube (109) (110). The resulting cube should be formed in a way that the connecting faces face each other as if they are holes and protrusions where they are placed and connected. By repeating this pyramid from its eight vertices in the geometric space, a unified and innovative network structure tangent to the eight pyramid vertices can be obtained. This structure is a novel development in mathematical sciences as well as its sub-branches such as physics and chemistry, and is represented as (124). An origami pyramid with an asymmetric trapezoidal base can be designed in the geometric space of a cube to divide it into four symmetrical faces. This is achieved with the help of two rotational symmetry axes: The fourth-order rotational symmetry axis (109) and the second-order rotational symmetry axis (110). The cube is formed with the help of certain axes. There are four rotational symmetry axes in a cube space which are the 4th order symmetry axis (112), the 2nd order symmetry axis (113), the 4th order symmetry axis (114), and the 115th order symmetry axis with a rotation of 180 degrees. These axes are responsible for creating the cube. By utilizing the second and fourth-order rotational symmetry axes, it is possible to create symmetrical connecting surfaces that take the shape of depressions and projections (118). Brief Description of Drawings Fig-1 101- The pyramid base is a right-angled trapezoid whose large base is equal to the height of the perpendicular leg of the trapezoid, and the small base is ½ of the large base. 102- The perpendicular face of the pyramid is a right-angled triangle whose big leg is tangent and equal to the pyramid's height. 103- The oblique face of the pyramid is an isosceles triangle whose two legs are equal to the obliqued leg of the trapezoid and whose vertex is equally tangent to the edge of the pyramid. 104- The perpendicular face of the pyramid is an isosceles right triangle whose two legs are equal to the pyramid's height. 105- The pyramid's oblique face is a right triangle with the vertex tangent equal to the pyramid's edge and the small leg tangent equal to the trapezoid's large base. 106- The height of the top of the pyramid is equal to the large base of the trapezoid, the perpendicular face of the pyramid is a right triangle. 107- The apex of the pyramid is point A, which is the center and perpendicular to the intersection of the small base and the height of the trapezoid. 108- The center of the pyramid's edge is the focus of the symmetry axes in the geometric space of the cube. 109- Axis of rotational symmetry of order four cubes. 110- Axis of rotational symmetry of the order of two cubes. 111- Axis of rotational symmetry of the order of two cubes. fig-2 112- The axis of rotational symmetry of the 180th pyramid around the fourth-order axis of the cube. 113- The axis of rotational symmetry of 180 degrees of the pyramid around the second- order axis of the cube. 114- The rotational symmetry axis of 180 pyramids around the axis of the order of four cubes. 115- The axis of rotational symmetry of 180 degrees of the pyramid around the second- order axis of the cube. 116- Axis of rotational symmetry 180 of the cube around the tangent axis with the bisecting chord line on the two opposite faces of the cube. 117- Axis of rotational symmetry of the cube 180 around the tangent axis with the bisector parallel line in the four faces of the cube. fig-3 118- Showing placement of two inclined faces of the pyramid in the geometric space of the cube with 12 surface contacts inside the cube. 119- Showing the placement of symmetrical forms connected in 12 surface contacts inside the cube. 120- Showing placement of three faces of a pyramid in six faces of a cube with 12 surface contacts. 121- Showing the placement of connecting forms in 12 surface contacts in six faces of the cube. 122- Showing the positions of the cube in two static positions and showing the positions of the inclined surfaces or faces of the pyramid at two angles of 45 and 27 degrees in the geometric space of the cube. fig-4 123- Showing the repetition and placement of the origami pyramid in the geometric space of a large cube in two static positions on the two sides of the cube. 124- Showing the network of paths created in the geometric space of the tangent cube with the vertices of the pyramid. 125- Showing the symmetrical connection of the faces of the pyramid in a hypothetical structure. Detailed Description of Invention The description of the components of this invention is as follows: The first origami pyramid with an asymmetric trapezoidal base (101) is presented. It has applications in industrial tools like polyethylene packaging containers which are widely used. Polymer containers in the shape of an origami pyramid with an asymmetrical trapezoidal base (101) can be useful for concentrating collection and recycling in the industrial cycle. This, in turn, can help reduce greenhouse gas emissions. The origami pyramid, which is created in the geometric space of a cube, has the following defining characteristics: (1) 1- The pyramid's base is a right-angled trapezoid (101) whose large base is equal to the height of the perpendicular leg of the trapezoid, and the small base is ½ of the large base. 2- The perpendicular face of the pyramid is a right triangle (102) whose big leg is the vertical ridge of the top of the pyramid. 3- The oblique face of the pyramid is a right triangle (103) whose small leg is equal and tangent to the large base of the trapezoid, and its vertex is tangent and equal to the edge of the pyramid. 4- The perpendicular face of the pyramid is an isosceles triangle (104) whose two legs are equal to the pyramid's height. 5- The oblique face of the pyramid is an isosceles triangle (105) whose two legs are equal to the obliqued leg of the trapezoid, and its apex is tangent and equal to the edge of the pyramid. 6- The pyramid's apex (107) is point A, located at the center and perpendicular to the intersection of the small base and the height of the trapezoid. 7- The height of the apex (106) of the pyramid is equal to the large base of the trapezoid of the perpendicular face of the pyramid of a right triangle. The geometric characteristics of an origami pyramid with an asymmetric base of a trapezoid are as follows: The pyramid obtained focuses on symmetry axes located at the center of the pyramid's edge. The cube's geometric space has three rotational symmetry axes, which are located on the two oblique faces of the pyramid. These oblique faces are of the fourth order (109) and second-order (110) in the cube's geometric space. In addition, the pyramid has two inclined surfaces (103) and (105 are angled differently at 45 and 43.63 degrees, respectively, in the cube's geometric space. Rotating the pyramid three times around these axes results in a cube. The cubic space has four axes of rotational symmetry that operate in a specific order. This order follows the 4-2-4-2 motion cycle in the geometric space. The four symmetry axes are as follows: 1. The fourth-order symmetry axis (112) 2. The second-order symmetry axis (113) 3. The fourth-order symmetry axis (114) 4. The second-order symmetry axis (115) All of the axes of symmetry result in a 180-degree rotation within the cube's geometric space. Using these axes of symmetry, you can create symmetrical surfaces in the form of depressions and protrusions of connecting surfaces (119). With three orders of rotational symmetry of the pyramid around the axes obtained in the two oblique faces of the pyramid, we have caused three surface contacts in the two obliqued faces of a pyramid as follows, by placing four pyramids in a cube, twelve internal surface contacts (118) turning into: 1- Two surface contacts occur on the inclined face of the pyramid with an inclination angle of 45 degrees to the horizon around the two axes X-Y, which are known as the axes of rotational symmetry of the fourth order and the second order in the geometric space of the cube. 2- A surface contact occurs on the oblique pyramid's face with an angle of inclination of 43.63 degrees to the horizon around the Z axis, which is the second-order symmetry axis in the geometric space of the cube. The six faces of the cube are formed by using the other three faces of the origami pyramid (120). The two opposite faces of the cube are created by an isosceles right triangle and its parallel (116). The remaining four faces of the cube are formed by placing the vertical face of the pyramid, the right-angled triangle, and the base of the right-angled trapezoidal pyramid (117). The axis of symmetry of the right-angled trapezoidal pyramid in the face of the cube is the line of symmetry of the bisector of the face of the cube. In the other face of the cube, it is the diameter passing through the face of the cube. With a rotational symmetry of 180 degrees around it, there are two surface contacts on each face, making twelve surface contacts in total (120), forming a cube in the outer faces. Like the inner surfaces of the cube on both sides of the tangent axes with parallel lines, the form of relief and depression create symmetrical forms (121), which are placed inside each other due to the symmetry of rotation of 180 degrees around the axes and limit the movement of the faces on each other. Advantageous Effects of Invention The first origami pyramid has an asymmetrical trapezoidal base located in the geometric space of the cube. This base can divide the cube into four symmetrical parts (101). By utilizing the rotational symmetry axes in the geometric space of the cube, the pyramid can continuously expand and connect symmetrically, forming a network of axes that are tangent to its vertices (124). The second advantage of the aforementioned innovative pyramid is the division of the weight force on the cube due to the presence of inclined surfaces and twelve surface contacts built inside the cube (122). The inclined face of a pyramid positions at two different angles concerning the horizon. The first is at a 45-degree angle, while the second is at a 27-degree angle. The angle of inclination impacts the overall mechanical efficiency of the pyramid's design. The third advantage of the Smart design is that there are no negative angles when the part exits from the two-hole molds, allowing making parts in various dimensions as full and empty volumes. One can find examples of such parts in an intellectual game or a bulky water tank produced using Smart design technology. Industrial Applicability Recyclable packaging, such as polyethylene containers, plays a crucial role in the industry. It serves as a packaging material and enables the concentration of recyclable materials for the return to the industry cycle. Moreover, it is used in the production of both temporary and permanent connecting devices that require specific physical characteristics such as flexibility, permeability, lightness, and heaviness. These devices, including platforms, floating bridges, fire-resistant chambers, and those that can withstand cold, heat, vibration, and explosion waves, are highly beneficial to the industry. The metallurgy and material science industry can benefit greatly from the knowledge of new networking and its use in molecular bonds. This step can help in the development of metallurgy science. Additionally, this knowledge is useful in the education and entertainment industries as it can help to improve our understanding of geometric space and raise the spatial intelligence of the younger generation. Other features: One of the methods of implementation and application of origami pyramids in industrial design is the production of parts with millimeter dimensions of solid volumes in injection molds and up to hollow volumes that are used in the construction of water tanks.

Claims

What is claimed: Claim 1) The invention obtained is the first origami pyramid with an asymmetric base (101) in the mathematical science of space geometry, which divides a cube into four symmetrical dimensions with the help of rotational symmetry axes in the spatial geometry of the cube so that the center of the rotational symmetry axes is in the center of the edge. The pyramid and three axes of rotational symmetry are placed on two oblique faces of the pyramid (109), (110), (111), one axis of rotational symmetry of the fourth order and two axes of rotational symmetry of the second order known in the geometric space of the cube, with rotation The symmetry of the pyramid around the three axes obtained on the two oblique faces of the pyramid creates three surface contacts, which causes four pyramids to place in the geometric space of the cube forming a total of twelve surface contacts inside the cube (118). The obtained cube has six faces formed with the help of three other faces of the pyramid, two vertical faces, and the pyramid's base, which creates twelve contact surfaces in the six faces of the cube (118). Claim 2) In the obtained innovative origami pyramid, by repeating the pyramid in a cubic geometric space, we will have a network of tangent axes with vertices in a geometric space (124) with the following dimensional characteristics. The base of the pyramid is a right-angled trapezoid whose large base is equal to the height of the perpendicular leg of the trapezium, while the small base is 1 / 2 of the large base (101), the height of the top of the pyramid is equal to the large base or the height of the trapezoid (106) and the top of the pyramid is perpendicular and tangent to the point The intersection of the small base and the height of the trapezoid is (107). Claim 3) According to the first claim, placing the inclined faces of the pyramid on top of each other after making a big cube creates twelve internal surface contacts and twelve external surface contacts in a small cube (118). Claim 4) According to the first claim, 12 surface contacts are created by using rotational symmetry axes on the faces of the cube and rotating around these axes (121). These surfaces are used in the outer faces of the cube to form symmetrical faces that are connected in the form of protrusions and depressions on both sides of the axes of rotational symmetry. Claim 5) In the obtained pyramid, the focus of the symmetry axes of the cubic geometric space is located in the center of the edge of the pyramid, while the three rotational symmetry axes of the cube are on the two oblique faces of the pyramid, which are of the fourth order and the second order in the cubic geometric space (109) (110) (111) play a role in the formation of the cube. Two inclined surfaces of the pyramid with different angles (45 and 43.63 degrees) are placed in the geometric space of the cube, which makes a cube by rotating the pyramid three times around the obtained axes. In this way, the four axes of symmetry A rotation in the cubic space works in the order of 2-4-2-4 order motion cycle in the geometric space, 1- fourth-order symmetry axis (112) 2- second order symmetry axis (113) 3- fourth-order symmetry axis (114) 4-The axis of symmetry of the second order (115), which all take place in a 180-degree rotation, in the geometric space of the cube. Utilizing the axes of symmetry can create symmetrical surfaces in the form of depressions and protrusions of connecting surfaces (119) (121). Claim 6) The aforementioned origami pyramid uses the cube to connect the volumes in the form of a cube and a prism with the help of continuous and combined symmetry in the geometric space (123). In such a way that it divides a cube into 27 small cubes, the first cubic symmetry is formed in the center of the pyramid edge, the center of spatial symmetry of the cube, and the pyramid divides the obtained cube into 108 symmetrical parts in the shape of a pyramid with an asymmetric trapezoidal base. Claim 7) The first origami pyramid with a trapezoidal base is in the geometric space of a cube, and with the help of the axes of symmetry in the geometric space of the cube, it can expand and connect symmetrically in a continuous manner (125) and with the help of this ability, it creates a network of tangent axes with its vertices. (124)

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