JP4910162B2 - Foldable tetrahedral surface model - Google Patents

Description

The present invention relates to a foldable geometric teaching material, a foldable building structure material, a tent, and a container.

Conventionally, in the learning of polyhedron geometry, when building a regular tetrahedron surface model using development diagrams such as cardboard as a geometric teaching material to learn the surface area ratio and volume ratio between polyhedrons with regular tetrahedron as the standard As shown in FIG. 2 and FIG. 3, in the known developed view composed of four equilateral triangles having the equilateral triangle 10 as a unit, the adjacent equilateral triangles are connected as much as possible so that the edges of the regular tetrahedron are connected to each other. It is possible to reduce the margin and the distance between them, and to create a development view that makes the connection easy and accurate. When a regular tetrahedron having a side of 100 mm is created using these development views, the total coupling distance required for mutual coupling of the ridge portions of the regular tetrahedral surface model is 600 mm. With such a development view, the total connection distance of 1200 millimeters when individual equilateral triangles are joined to each other can be easily halved, so that the development view is sufficiently useful.

In addition, in the development shown in FIG. 4 as another known example, the equilateral triangle region 10 as a unit is further expanded to be completely two by introducing an outline 10h other than the regular tetrahedron ridge into the connection. The separated congruent rectangular regions 50 are rearranged. Next, after folding the tetrahedral ridge line 10c in the same direction and coupling the ridge lines 10b, the rectangular outline 10h and the ridge line 10a can be coupled to each other. In this case, the total coupling distance is about 746.4 mm.

In addition, as another improvement measure to further reduce the margin and the connecting distance, as shown in Fig. 5 from "Origami Geometry" (Koji Fushimi, Mitsue Fushimi, Nihon Crihonsha 1984), two congruent rectangles are shown. The region 50 can be synthesized into a development view in which the regions 50 are continuous. In the case of a development of a regular tetrahedron having a side of 100 mm by combining the two rectangular regions 51, the ridge line 10 c of the regular tetrahedron and the outline line 10 h other than the ridge line are mixed in the connection. The total length of the outer circumference, that is, the total coupling distance is about 546,4 mm. In this way, the developed view composed of a series of congruent rectangles makes the total coupling distance even shorter than in the case of FIGS.

However, even in these methods, since the ridge line itself or a part of the ridge line is used as a connecting portion for the development of the regular tetrahedron in common with FIGS. It was easy to occur. Such shifts and distortions directly lead to surface deformation and angular misalignment of the equilateral triangle forming the regular tetrahedron. In particular, when thick paper is used in the developed view, the ridgeline portion and the apex portion are significantly deformed mainly due to the thickness deviation. Therefore, accurate and effective assembly of the development drawing requires particularly delicate attention to the linear connection on the boundary line between the areas with sharp angles near the ridge line part and the apex.

As another known example, four rectangular units having diagonal folds are formed at the ridges of the regular tetrahedron, as seen in the regular tetrahedron in FIG. 1 of the patent drawing of Japanese Patent Publication No. 62-105182. A folding function formed by flexible connection can reduce a margin for fixing. In the case of forming a foldable regular tetrahedron having a side of 100 mm, the coupling distance between the two units when the rectangular unit shown in FIG. 3 is composed of two rectangular units is 200 mm. The distance at which the openings join each other is approximately 692 mm. Due to this increase in the joining distance, the opening is accurately closed at the midpoint of the ridge line of the regular tetrahedron, and at the same time, the unit is relatively thick to maintain continuity between the ridge line and the surface of the regular tetrahedron. Needed material. That is, the folding function in this case increased the total coupling distance required for the minimum coupling between the units and the joining of the openings to 892 mm, and also increased the material weight.
Published patent publication Sho 62-105182 Koji Fushimi and Mitsue Fushimi, “Origami Geometry”, Nippon Critics 1984

Up to now, the number of polygons more than the congruent equilateral triangles constituting the development of the regular tetrahedral surface model could be reduced to a minimum of two, but it does not include the six ridge lines constituting the regular tetrahedron at all. It has been difficult to construct a development view of a regular tetrahedral surface model only by two congruent regular polygons made of outlines. In addition, it is more difficult to construct a development view of a regular tetrahedral surface model that shortens the total coupling distance of the development view of the regular tetrahedron.

The present invention relates to a foldable regular tetrahedral surface comprising two identical units that form a regular tetrahedron composed of four congruent equilateral triangles by an outline line other than six ridgelines and equally divide the surface area of the regular tetrahedron. The main feature is to construct a model.

As long as the design of the development of regular polyhedron is taken from the definition of the regular polyhedron of ancient Greece, the regular polyhedron had to be composed only of congruent regular polygons. I noticed the possibility of defining a new development that forms a regular polyhedron. The inventor of the present invention has made the above-mentioned publicly known design in which the development of a tetrahedron, which is a minimum polyhedron, is composed only of two symmetrical polygonal units of the same type with fewer than four congruent triangles. It was discovered that there are countless other than rectangular units of the same type. Furthermore, the six ridges of these tetrahedrons are not composed of the joints of the outlines in the developed view, and are also distorted by the discovery of a polygon group that can be shorter than 600 millimeters using the overall length of the outline (total joint distance) as an index. Therefore, an economical method with a folding function that does not require coupling and fixing at the ridge line portions as in the conventional development drawing and a highly accurate development drawing and its assembly was derived.

Development view for forming a foldable regular tetrahedral surface model and forming a regular tetrahedron by two identical units composed of outlines that do not include the six ridges constituting the regular tetrahedron at all The purpose of further shortening the connecting distance of the connecting portion required by is realized by two continuous hexagonal isomorphic units.

Examples will be described with reference to the drawings. The regular tetrahedral surface model 80 shown in FIG. 1 is a diagram in which two regular hexagonal homomorphic units are coupled and integrated at the covalent coupling portion 22 so that the midpoints X and Y of the outline 21 are matched. 6 is developed from the developed view 81 shown in FIG. In the regular hexagonal region 30 of the same unit, a perpendicular bisector connecting the midpoints P and Q of a set of parallel outlines 20 is defined as a first fold line 11 having a length equal to the ridgeline of the regular tetrahedron, Line segment PR equal to half the length of the ridgeline of the tetrahedron connecting the outlines 21 adjacent to the midpoints P and Q and the midpoints R, S, Z, and X of the covalent coupling portion 22 to each other. , PX, QS, and QZ are the second mountain fold lines 12, and a total of five fold lines 11, 12 are symmetrically arranged. As shown in FIG. 7, the regular hexagonal region 30 is folded in half at the first mountain fold line 11, and the symbols B and C, E and F attached to the apexes are made to coincide with each other, and flexible from the parallel outline 20 The joint 20h is formed. Similarly, as shown in FIG. 8, also from the regular hexagonal region 31 of the same unit, the flexible joint 20h is formed from the parallel outline 20, and the two openings R1 and R2 are formed. Next, the flexible joint 20h is flattened, and the first and second fold lines are expanded at substantially the same angle, and the openable and closable opening RA having a zigzag outer periphery made up of the openings R1 and R2. S-B, C-Z-D, K, L-U-G-T-H, I-V-J, F-R (not shown) are formed. Finally, R and V, S and T, U and Z coincide with each other at the midpoint of the ridgeline, and A and H · I, G and B · C, and L and D · K at the outer periphery of the opening. , J and EF are made to coincide with each other, the regular tetrahedral surface model 80 shown in FIG. 1 is completed. The folding function of the foldable tetrahedral surface model 80 is that the ridgeline PN is flexibly integrated with the flexibility of the first fold line 11, the second fold line 12, and the flexible joint 20h. Generated by linkage with added sex. In this embodiment, not only the ridgeline gathering at the apex part is not formed from the mutual joining of the outlines of the developed view, but also each acute apex part formed by gathering three regular tetrahedron ridgelines is folded or Since all of them are formed by folding in a valley, it is possible to accurately reproduce a straight ridge line part and an acute vertex part necessary for the configuration of a regular tetrahedral surface model.

In the regular tetrahedral surface model of 100 mm on one side assembled from the developed view 81 having two continuous hexagonal isomorphic units shown in FIG. The total length of the outlines, that is, the total coupling distance is about 577.35 mm. With this development, it is possible to easily reduce the total connection distance of 600 mm when connecting continuous equilateral triangles as shown in FIGS. 2 and 3 to each other, so that the assembly is sufficiently economical.

In the tetrahedron surface model 80 shown in FIG. 1, the flexible joint between B.CQ, E.F.P, H.I.M, and K.L-N shown in the developed view of FIG. When the flexibility due to the first mountain fold line between 20h and the second mountain fold line or the ridge line PQ is easily erased, the outer peripheries of the openings R1 and R2 are coupled to each other, and the ridge line of the regular tetrahedral surface model 80 is connected. A band or the like may be applied to the equator portion of the regular tetrahedron formed by connecting the symbols V, R, S, T, U, ZX, and YV of the midpoint. If the band is removed, it can be folded back into a flat shape according to the flexible connection and the folding line, and the flexibility can be reproduced.

FIG. 9 and FIG. 10 show another embodiment composed of equilateral hexagonal units 32 and 33. In a regular hexagonal isomorphic unit 32, a perpendicular bisector connecting the midpoints P and Q of a pair of parallel outlines 20 is defined as a first fold line 11 having a length equal to the ridgeline of a regular tetrahedron, Line segments PR, PX, QS, QZ equal to half the length of the ridgelines of the tetrahedron connecting the points P, Q to the midpoints R, S, Z, X of the outline 21 adjacent to each other. A total of five fold lines 11 and 12 are arranged symmetrically with the second mountain fold line 12 as. Next, a regular hexagonal isomorphic unit 32 is folded in half along the first mountain fold line 11 so that the symbols B, C, E, and F attached to the apexes coincide with each other, and a flexible joint is formed from the parallel outline 20. 20 h is formed, and the same type unit provided with an openable and closable opening R <b> 1 consisting only of the outline 21 is formed. Similarly, the same type unit provided with the opening R2 that can be opened and closed from the regular hexagonal same type unit 33 is formed. As shown in the exploded perspective view of FIG. 9, the opening R1 obtained by flattening the flexible joint portion 20h of two identical units and expanding the first mountain fold line 11 and the second mountain fold line 12 at substantially the same angle A combination of a total of ten symbols A, B, C, D, E, and F attached to the apex of the equirectangular unit 32 with the symbols R, S, Z, X attached to the midpoint of 21 Zigzag octagonal outer periphery ASB, CZDXX, F-R-A consisting only of the outline 21 is formed, and the opening R2 is formed of the outline 21 Among the combinations of a total of 10 symbols, symbols G, H, I, J, K, and L, attached to the apex of the equilateral unit 33 with the symbols T, U, Y, V attached to the midpoint A zigzag octagonal outer periphery G-TH-IV-J-Y-K-L-U-G consisting only of the outline 21 is formed. Next, R and V, S and T, U and Z of two isomorphic units that are the midpoints of the ridgeline of the tetrahedron are made to coincide with each other, and A, H, I, and G on the outer periphery of the opening When B · C, L and D · K, and J and E · F are made to coincide with each other, a regular tetrahedral surface model 80 shown in FIG. 1 is finally completed. The separated isomorphous unit shown in FIG. 9 is flexible in all the fold line portions corresponding to the ridgelines of the respective tetrahedral surface model, and is free from the angle by the flexible joint 20h in the outline 20. Since it has a degree, it can be opened and closed individually, and can be easily folded into a flat shape individually. As shown in the exploded perspective view of FIG. 9, the same-type unit folded in a planar shape is raised again, and can be synthesized from the units separated from each other on the outer periphery of the openings R1 and R2, in the ridgeline of the regular tetrahedral surface model 80. If a band such as a rubber band is put on the equator portion 60 of a regular tetrahedron formed by connecting the point symbols V, RS, TU, ZY, and XV, a self-supporting surface model The shape can be maintained. In this case, the isomorphous units 32 and 33 are arranged so that the surface area of the tetrahedral surface model 80 is completely divided into two equal parts with the openings R1 and R2 as the boundary lines.

Further, as shown in the structure explanatory diagram of the regular tetrahedral surface model 87 shown in FIG. 11, the ridgeline of the regular tetrahedron is composed of four congruent surfaces between the center of gravity O and the outer periphery of the zigzag octagonal opening. Since the boundary surfaces Z1 to Z4 in contact with the four midpoints are virtually present, the structure of the regular tetrahedral surface model 88 shown in FIG. 12 is explained when the volume of the regular tetrahedral surface model 87 is divided into two equal parts. As shown in the figure, the symmetric modules 36 and 37 having a half volume may be formed by positively adding the boundary surfaces Z1 to Z4 to the regular hexagonal unit using the center of gravity.

In another embodiment shown in FIG. 13, the bent line portions corresponding to all the ridgelines of the regular tetrahedron are formed in a V-shaped or U-shaped shape by a plastic integrally formed plate 82 in which two regular hexagonal identical units 38 are continuously arranged. It is possible to form the opening by forming the groove hinges 14 and 15 to be flexibly coupled and flexibly coupling the outline 20. The regular tetrahedral surface model has angular freedom and durability, and can easily reciprocate between a plane and a solid by opening and closing movements of the opening. In this case, since the thickness of the plastic plate at the ridgeline can be adjusted by the V-shaped or U-shaped groove of the hinges 14 and 15, a regular tetrahedral surface model having a more accurate outer shape can be provided. Furthermore, the assembly with the plastic plate is not performed along the ridge line of the regular tetrahedron, but the cross sections of the thickness of the plastic plate are joined to each other within each triangle along the outline of the plastic plate. The surface model has no gaps at the vertices and ridges, and it can be assembled without causing distortion due to thickness.

In another embodiment shown in FIG. 14, a hinge is used only in the parallel grooves 16 and 17 in the ridge line portion of the regular tetrahedron instead of the hinges of the V-shaped or U-shaped grooves, and folding by flexible coupling is performed. A function may be added. In this case, a rubber band or a band may be applied to the equator portion 61 of the regular tetrahedral surface model so that the opening portion is not opened in order to make the foldable regular tetrahedral surface model self-supporting in an assembled state. Further, if the openings are bonded and fixed to each other with a tape, a fastener, and an adhesive, an accurate and stable regular tetrahedron can be obtained.

FIG. 15 is an exploded view of the hexagonal isomorphic unit according to the present invention in which no regular tetrahedral ridge line is used as the outline as a coupling part necessary for assembling the regular tetrahedral surface model. FIG. 86 shows a development view 86 in which is the shortest. The outer circumference of the hexagonal isomorphic units 39 and 40 connected in series is about 570.07 mm. This is obviously shorter than 577.35 mm in the case of the regular hexagonal plate.

A collapsible tetrahedral surface model 90 composed of four congruent isosceles triangles shown in FIG. 16 is formed in the covalent bond line 26 so that two rectangular isomorphic units match each intermediate point w and z of the outline 25. It is formed from the developed view 91 shown in FIG. In the rectangular region 41 of the same type unit, a perpendicular bisector connecting the midpoints P and Q of a pair of parallel outlines 24 is defined as a first mountain fold line 11 having a length equal to the ridgeline of the tetrahedron, Each of the tetrahedrons connecting two different intermediate points r, s, z, and x, which are provided on the outer shape line 25 adjacent to P and Q with a one-fourth interval of the outer shape line left and right. Line segments Ps, Pz, Qr, and Qx equal to half the length of the ridge line are defined as the second mountain fold line 12, and a total of five fold lines 11 and 12 are arranged symmetrically in the rectangular area 41 of the same type unit. The Next, the rectangular region 41 of the same type unit is folded in half at the first mountain fold line 11, and the symbols a and b, c and d attached to the apexes of the rectangular region 41 of the same type unit are respectively matched, A flexible joint 24 h is formed from 24. Similarly, as shown in FIG. 18, the flexible joint 24h is formed from the parallel outline 24 from the rectangular region 42 of the same type unit, and two openings R3 and R4 are formed. All the flexible joints 24h are flattened and the first and second fold lines are expanded at substantially the same angle, and the openable and closable openings t-g, hv-c, and d are formed by the openings R3 and R4. -Sr-a.bxe.fu (not shown). Finally, the midpoints u and x, t and r, v and s of the ridgeline of the tetrahedron are made to coincide with each other, and the outer periphery e · f−t−t · g · hvc of the opening When d and e · f−x−ba−r−s−c · d coincide with each other, a tetrahedral surface model 90 shown in FIG. 16 is completed. The folding function of the foldable tetrahedral surface model 90 is generated by interlocking in which the flexibility of the ridge line PN is added to the flexibility of the first mountain fold line 11, the second mountain fold line 12, and the flexible joint 24h. Is done. In this embodiment, not only the ridge lines gathered at the apexes are formed from the mutual connection of the outlines of the developed view, but also the sharp apexes formed by gathering three ridge lines of tetrahedrons are all folded or all Since it is formed by folding in a valley fold, it is possible to accurately reproduce the straight ridge line part and the acute vertex part necessary for the configuration of the tetrahedral surface model.

Further, between the center of gravity O shown in FIG. 16 and the equator outer periphery of the opening, a square boundary surface r-s-z- that is composed of the same surface and is in contact with four midpoints of each ridge line of the tetrahedron. Since x or tu-w-v exists virtually, when the surface area of the tetrahedral surface model 90 is divided into two equal parts, it is arranged as the same type units 41 and 42 having a half surface area. May be. Further, when the volume is divided into two equal parts, a symmetric module having a half volume may be formed by positively adding a square boundary surface.

Another embodiment shown in FIG. 19 is a single pattern motif for plane filling consisting of the same periodic single element on all sides of two regular hexagonal isomorphic units. C. Development diagram 100 of a regular tetrahedral surface model using three colors of lizards (white, gray, black) by Escher and having the same units 43 and 44 continuously provided without any gaps. It is. This periodic motif can seamlessly connect the surfaces of two regular hexagonal plate units, and at the same time, even in the case of a surface model standing on a regular tetrahedron, the periodic motif is divided at any joint. It is continuous endlessly.

In another embodiment shown in FIG. 20, an image regarding omnidirectional planar geographic information formed by topological projection onto a regular tetrahedron inscribed in a spherical global geographic information map is shown in FIG. Of the regular tetrahedron surface model shown in FIG. 4B, and then the surface model of the regular tetrahedron that seamlessly fills one side of the two regular hexagonal isomorphic units 45 and 46. An expanded view 101 is shown. In the “azimuth projection” projected onto a flat screen by the projection method of projecting spherical geographic information from two opposing poles, the overall shape becomes two circular maps. The two opposite poles at are located in the center of two regular hexagons, while the geographic information of each hemisphere is replaced with each regular hexagon. Therefore, it can be finally converted into a foldable tetrahedral surface model. In addition to such an all-spherical global geographic information map, an omnidirectional projection formed topologically on a regular tetrahedron inscribed in a cosmic celestial globe information map or an omnidirectional spherical image. The flat image may be seamlessly filled into one side of all two regular hexagonal units. The concept of panorama that captures the entire panoramic view (PANORAMA) means a viewing angle of 360 degrees projected onto a cylinder, but the present inventor has a viewing angle of 720 degrees that captures true omnidirectionality by a regular tetrahedron (this 720 degrees is associated with the sum of the interior angles of the tetrahedron) and was named TETRAMA for the omnidirectional map or omnidirectional image conceptual model of the regular tetrahedron. A wide-angle lens cannot optically exceed an angle of view of 180 degrees or more in terms of optical principle, and has not been able to expand a viewing angle of 360 degrees conceptually beyond that of a panorama, but a tetrahedron is a polyhedron with minimal spherical information by tetramer. If the topological conversion is performed, the maximum viewing angle of 720 degrees existing in terms of geometrical principle can be captured at a unique viewing angle of a human by two consecutive isomorphic units according to the present invention.

22 shows a development 89 in which hexagonal isomorphic units 47 and 48 for forming a foldable tetrahedron composed of four congruent isosceles triangles are continuous. There are an infinite number of collapsible tetrahedral surface models composed of two identical hexagonal isomorphic units according to the present invention other than the regular tetrahedral surface model.

Since the assembly process of the same type unit is easy and the folding function is simple, the present invention is also effective as a teaching material for solid geometry, a lightweight structure of thin material, a container, and a tent. Furthermore, because the tetrahedron can be constructed by continuously connecting two flat plates of the same unit, the same pattern of the periodic design can be continued on one or both sides of the development of the tetrahedral surface model, and the regular division of the plane It is extremely effective for studying the symmetry of polyhedra. Furthermore, the regular tetrahedral surface model, which is a minimal polyhedron composed of two consecutive developments of two congruent regular hexagons that mathematically and seamlessly project spherical information, is an omnidirectional that could not be achieved by panorama. Can be reproduced. Two hexagonal isomorphic units or rectangular isomorphic units that have a maximum recognizable viewing angle (= 720 degrees) by a collapsible tetramer (TETRAMA) for spherical geographic information that captures all directions. Can be provided at.

It is a perspective view of the regular tetrahedral surface model which consists of two continuous regular hexagonal isomorphic units which showed the Example by this invention. Example 1 The developed view which showed the well-known example. It is a development view of a regular tetrahedron showing another known example. It is a development view of a regular tetrahedron showing another known example. It is a development view of a regular tetrahedron showing another known example. It is an expanded view of the regular tetrahedral surface model of Example 1 by this invention. It is an assembly process figure of the regular tetrahedral surface model of Example 1 by the present invention. It is another assembly-process figure of the regular tetrahedral surface model of Example 1 by this invention. It is a disassembled perspective view of the regular tetrahedral surface model which consists of the separated regular hexagon which showed the other Example by this invention. (Example 2) It is an expanded view of the regular tetrahedral surface model of Example 2 by this invention. It is a perspective view for description of the structure of the regular tetrahedral surface model of Example 2 by the present invention. (Example 3) It is another perspective view for description of the structure of the regular tetrahedral surface model of Example 3 by the present invention. It is an expanded view of the regular tetrahedral surface model by integral molding based on the two continuous same type | mold units which showed the other Example by this invention. Example 4 FIG. 6 is a development view of a regular tetrahedral surface model based on two consecutive identical units showing another embodiment of the present invention. (Example 5) It is the expanded view which showed the other Example by this invention. (Example 6) It is a perspective view of the tetrahedral surface model which consists of two continuous rectangular same type units which showed the other Example by this invention. (Example 7) It is an expanded view of the tetrahedral surface model of Example 7 by this invention. It is an assembly process figure of the tetrahedral surface model of Example 7 by this invention. It is the expanded view of the regular tetrahedral surface model which consists of two hexagons which carried out the plane filling of the single pattern which showed the other Example by this invention. (Example 8) FIG. 6 is a development view of a regular tetrahedral surface model made up of two regular hexagons filled with a tetramer in a plane according to another embodiment of the present invention. Example 9 The image about the geographical information of the omnidirectional plane formed by topologically projecting on the regular tetrahedron inscribed in the spherical global geographical information map for explaining the process of creating the embodiment 9 This is a regular tetrahedral map (tetrama). It is a development view of a tetrahedral surface model made up of two continuous hexagonal isomorphic units showing another embodiment of the present invention. (Example 10)

10 Triangular area 10a constituting a tetrahedron Outline line 10b serving as a ridge line Outline line 10c serving as a ridge line Mountain fold line 10h serving as a ridge line Junction 11 First mountain fold line 12 Second mountain fold line 14 U-shaped or V-shaped groove hinge 15 U-shaped or V-shaped groove hinge 16 Hinge introduction portion 17 provided at the ridge line portion Hinge introduction portion 20 provided at the ridge line portion Parallel-facing outline 20h Flexible joint 21 One set of parallel-facing outlines 22 Coupling part 23 of the same type unit 23 Outline line 24 facing in parallel 24h outline line 24h facing in parallel Flexible joint 25 Outline line 26 facing in parallel Coin coupling part 30 of the same type unit Regular hexagonal region 31 of the same type unit A regular hexagonal region 32 of the same type unit A regular hexagonal type unit 33 A regular hexagonal type unit 34 A regular hexagonal type unit 35 having a boundary surface A positive surface having a boundary surface set Same unit of square shape 36 Same type module consisting of regular hexagonal shape unit 37 Same shape module consisting of same shape unit of regular hexagon 38 Regular hexagonal region 39 of integrally molded plastic plate 40 Hexagonal region 40 of same type unit Unit hexagonal area 41 Isomorphic unit rectangular area 42 Isomorphic unit rectangular area 43 Isomorphic unit regular hexagonal area 44 Isomorphic unit regular hexagonal area 45 Isomorphic unit regular hexagonal area 46 Isomorphic unit regular hexagonal area 47 The hexagonal region 48 of the isomorphic unit 60 The hexagonal region 60 of the isomorphic unit 60 The tetrahedral equator 61 The tetrahedral equator R1 The opening R2 The opening R3 The opening R4 The opening A The symbol B attached to the apex B Symbol C attached to the vertex Symbol D attached to the vertex E Symbol attached to the vertex F Symbol attached to the vertex G Symbol attached to the vertex H Symbol attached to the vertex I Symbol attached to the vertex J Symbol attached to the vertex L Symbol attached to the vertex Symbol P attached to the vertex Appended to the midpoint of the outline Symbol Q attached to the midpoint of the outline line Symbol N attached to the midpoint of the outline line Symbol R attached to the midpoint of the outline line Symbol S attached to the midpoint of the outline line Symbol Z attached to the midpoint of the symbol Symbol U attached to the midpoint of the contour line Symbol T appended to the midpoint of the contour line Symbol V appended to the midpoint of the contour line Appended to the midpoint of the contour line Symbol X attached to the midpoint of the outline line Symbol r attached to the midpoint of the outline line s Symbol s attached to the midpoint of the outline line Symbol t attached to the midpoint of the outline line Symbol v attached to the intermediate point Symbol u attached to the intermediate point of the outline line Symbol x attached to the intermediate point of the outline line Symbol z attached to the intermediate point of the outline line Inside the outline line Symbol w attached to the point Symbol Z1 attached to the midpoint of the contour line Boundary surface Z2 that bisects the volume of the tetrahedral surface model Boundary surface Z3 that bisects the volume of the tetrahedral surface model Boundary surface Z4 that bisects the volume of the tetrahedral surface model Boundary surface 80 that bisects the volume of the regular tetrahedral surface model 80 Regular tetrahedral surface model 81 by a hexagonal isomorphic unit Regular 4 by a regular hexagonal isomorphic unit Fig. 82 Development of the faceted surface model. Molded integrally using a continuous regular hexagonal unit.
Development view of a regular tetrahedral surface model of a plastic plate 83 Development view using a hinge for the ridgeline of a regular tetrahedral surface model 84 Regular tetrahedral surface model exploded perspective view of a separated regular hexagonal isomorphic unit 85 Separated regular hexagon Fig. 86 Development of a regular tetrahedral surface model by a hexagonal isomorphic unit. Fig. 87 Explanatory diagram of a regular tetrahedral surface model by a hexagonal isomorphic unit. Consists of regular tetrahedral surface model by unit
Module structure explanation diagram 89 Consists of four congruent isosceles triangles with continuous hexagonal isomorphic units
Development view of tetrahedral surface model 90 Tetrahedral surface model 91 by continuous rectangular isomorphic unit 91 Development view of tetrahedral surface model by continuous rectangular isomorphic unit 100 Continuous regular hexagon with plane filling of a single lizard pattern By the same type unit
Development of regular tetrahedral surface model 101 Continuous hexagonal isomorphic unit filled with omnidirectional map (tetrama)
Fig. 102 Development of a regular tetrahedral surface model based on Fig. 102 Regular tetrahedral map of omnidirectional geographic information formed by topological projection onto a regular tetrahedron inscribed in a spherical global geographic information map (Tetrama)

Claims (8)

In constructing a tetrahedral surface model composed of four congruent triangles other than unequal triangles with two isomorphic units, each of the two hexagonal isomorphic units having three parallel-facing outlines, the three A set of parallel bisectors connecting the midpoints of a set of parallel-facing outlines among the sets of parallel-facing outlines as a first mountain fold line, and two sets of parallel-facing outlines adjacent to each midpoint Two sets of four straight lines having the same length connecting each midpoint of the line are used as the second mountain fold line, and the first mountain fold line is folded and overlapped, and the one set of parallel external lines are joined to each other to be flexible. Forming a joint, bending the second fold line, flattening the flexible joint, and opening the first fold line and the second fold line at approximately the same angle to form all of the tetrahedrons in the tetrahedron; Forming four congruent triangular subregions, Each of the openings is formed so as to form an opening composed of two sets of parallel facing outer lines adjacent to the point, and finally only the first mountain fold line and the second mountain fold line form a ridge line of the tetrahedron. A collapsible tetrahedral surface model characterized in that the tetrahedral surface model is configured by completely matching the outer peripheries of each other. The foldable tetrahedral surface model according to claim 1, wherein the hexagonal isomorphic unit is a regular hexagonal isomorphic unit. The hexagonal isomorphic unit having three sets of parallel-facing outlines is a rectangular isomorphic unit consisting of two sets of parallel-facing outlines, and one set of parallel out of the three sets of parallel-facing outlines. A vertical bisector connecting the midpoints of the outlines facing each other is a vertical bisector connecting each midpoint of any one of the two parallel outlines, Two sets of four straight lines of the same length connecting the midpoints and the midpoints of the two sets of parallel-facing outlines adjacent to each other are connected to the left and right on the one set of parallel-facing outlines adjacent to the midpoints. 2. The foldable tetrahedral surface according to claim 1, wherein two sets of four straight lines having the same length are respectively connected to two different midpoints having a quarter interval of the outline. model. In each of the two isomorphic units, the isomorphous unit is formed such that one second fold line of the two sets of the same length is continuous in a straight line to form one ridge line of the tetrahedron. The collapsible tetrahedral surface model according to any one of claims 1 to 3, wherein the two are joined together by an external line to be integrated. 5. A flat filling pattern consisting of a single element or a plurality of periodic elements is continuously provided without gaps on all one side or both sides of the two same-type units. The foldable tetrahedral surface model according to any one of the above. Topologically a spherical global geographic information map, a cosmic celestial globe information map, or a tetrahedron that is inscribed in an omnidirectional spherical image for all one side or both sides of the above two units of the same type The foldable tetrahedral surface model according to any one of claims 1 to 4, wherein an omnidirectional planar image formed by projecting onto a plane is seamlessly filled with a plane. The foldable tetrahedral surface model according to any one of claims 1 to 6, wherein the first mountain fold line and the second mountain fold line are flexiblely connected. The collapsible tetrahedral surface model according to any one of claims 1 to 7, wherein a symbol is provided at the midpoint or vertex of the outline of the two identical units.