JPH07291261A - Method of forming polyhedron and use of material therefor - Google Patents

Abstract

PURPOSE:To obtain a method of forming a solid convex polyhedron in combination with a piece of sheet of a specific shape, the combination material and a method for use thereof. CONSTITUTION:Each side of a piece of sheet having the shape of each of the faces for forming a convex polyhedron is provided with a tongue-like projecting part at its left or right half and each projecting part of one sheet piece adjacent to each side of the other sheet piece is laid on the surface side of the latter sheet piece an fixed in combination therewith in a shrinking manner. This method permits a rigid solid to be formed without using any adhesive and is applicable to the formation of a container or a cubic puzzle.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention Generally speaking, the present invention relates to a method of assembling sheet-shaped pieces of a specific shape to form a polyhedron and its material, a container using the same, a teaching material, and a puzzle toy. The sheet referred to in the present application is a thin plate-like material having flexibility and bendability, and is typically a sheet of paper, plastic such as polyvinyl chloride or polypropylene.

[0002]

2. Description of the Related Art For example, flaps extending from the side surface of a corrugated cardboard box are temporarily fixed by superimposing projecting portions of sheet-like members, as in the case of closing flaps adjacent to each other in an alternating manner. The technique of partially overlapping and fixing the flaps (tongue-shaped protrusions) in a mesh-like shape is partially performed in the packaging field and the like. However, the inventor does not know an example in which this is applied to formation of a solid or a three-dimensional puzzle. The present invention provides a method for forming a polyhedron and a material thereof, a container thereof, a teaching material, a puzzle toy, and the like using this method. However, its application is not limited to this. In the present invention, a part may be dented.

[0003]

According to the present invention, a unit sheet piece in which a tongue-shaped protrusion is provided on each of the left and right halves of each side of a basic surface having the shape of each surface constituting a polyhedron is sequentially separated. (3) A method of forming a polyhedron, which comprises overlapping the respective protruding portions of the sheet piece on the front side of the basic surface of the sheet piece and combining and fixing them in a freezing state. Theoretically speaking, the above-mentioned half does not necessarily mean a true half that is divided at the midpoint of the side, but in some cases, each on-lingual protrusion should have a complementary shape at the corresponding position.

The superposed tongue-shaped projections are preferably shaped such that when superposed and assembled, they have exactly the same shape as the surface on which they are superposed. However, it is more convenient for the assembling operation to keep the tip rounded, and the damage to the seat is less. If the surface is a polygon with five or more sides, the shape of the tongue-shaped protrusion that corresponds to the surface is wider at the tip than at the base, which makes it impossible to stack and assemble it, but It can be made flat by stacking it on top and then folding it down. The tongue-shaped protruding piece can be easily folded if it is tapered or shortened to such an extent that no gap is produced when the tongue-shaped protruding pieces are stacked. (In the case where it is desired that no gap appears.) The range of the polyhedron to which the present invention can be applied is theoretically not limited as long as it includes a gap that is formed on the surface where the protruding pieces are superposed. . That is, the polyhedron according to the present invention is not limited to a convex polyhedron, and the present invention can be applied to a polyhedron in which a part is depressed.

The method for forming a polyhedron of the present invention is a regular polyhedron, a prism, a pyramid, an oblique prism, an oblique pyramid, a tetradecahedron consisting of square and equilateral triangles, and a thirty-two part consisting of regular pentagons and hexagons. It is applied to a tetrahedron, a point-symmetrical octagon and a tetradecahedron consisting of regular triangular faces, etc., but especially a regular polyhedron, a prism, a pyramid, a tetrahedron consisting of square and regular triangular faces, and a regular pentagon. It is preferably applied to a dodecahedron having a regular hexagonal surface.

The present invention also has the shape of the surface of each of the faces that make up the polyhedron, and when the left or right halves of each side of the polyhedron are overlapped with each other in a free space, the corresponding basic face is overlaid. There is provided a polyhedron-forming material comprising a set of equal number and the shape of the surface of the polyhedron of the sheet piece provided with the tongue-shaped protrusions having the shape of the above.

The present invention is also the above polyhedron forming material, which, when assembled, has a desired color coding, pattern,
Provided is a convex polyhedron forming material used as a teaching material or a puzzle in which each unit sheet piece is provided with a color, a pattern, or a figure so that the figure appears. That is, the assembled puzzle material can be obtained by applying desired colors, patterns and figures in the assembled state and disassembling.

According to the present invention, the basic surface of an equilateral triangle is defined as the base of each left (or each right) half of each side, the side extending in the direction perpendicular to the side, and the left end (or right end) of the side. 6
A unit sheet piece 20 in the shape of a triangle connected to a basic surface triangle formed of sides extending at an angle of 0 °
Provided is a regular icosahedron forming material.

The present invention also provides a puzzle toy, which is the above-mentioned icosahedron forming material, and in which, when assembled, desired colors, patterns, and figures appear on each surface or the entire surface.

The present invention is also the above-mentioned icosahedron-forming material, wherein each protruding portion of 20 unit pieces is color-coded into 3 colors selected from 5 colors, and 20 sheets include a color arrangement. We provide puzzle toys in different colors.

The present invention is also the above-mentioned icosahedron forming material, wherein the basic triangle of 20 unit pieces is divided by a straight line connecting the center of gravity and the apex thereof, and each part is formed by 3 colors selected from 5 colors. (EN) Provided is a puzzle toy, which is color-coded and colored so that both sides of the side have the same color when assembled in the correct position.

According to the invention of the present application, in the above-mentioned icosahedron forming material, the basic triangle of 20 unit pieces is divided by a straight line connecting the center of gravity and the apex thereof, and each part is formed by 3 colors selected from 6 colors. (EN) Provided is a puzzle toy, which is color-coded and colored so that both sides of the side have the same color when assembled in the correct position.

The present invention also provides the above-mentioned icosahedron forming material, in which a puzzle toy is drawn on each unit piece so that a world map appears when assembled at the correct position.

According to the present invention, the basic surface of an equilateral triangle, the left (or right) half of each side thereof as the base, the side extending in the direction perpendicular to the side, and the left end (or right end) of the side. 6
Provided is a regular octahedron forming material comprising eight unit sheet pieces each having a shape of a triangle connected to a basic triangle having sides extending at an angle of 0 °.

The present invention also provides the above-mentioned regular octahedron forming material, and a puzzle toy in which desired colors, figures, patterns and the like appear on each surface or the entire surface when assembled in the correct position.

The present invention is also composed of a square and a tongue-like projecting piece having a left half (or right half) of each side as a base and extending in a direction perpendicular to the side and having a shape of exactly half the square. A cube forming material composed of unit sheet pieces 6 is provided.

The present invention also provides a puzzle toy, which is the above-mentioned cube-forming material, in which desired colors, figures, patterns, etc. appear on each surface or the entire surface when assembled in the correct position.

The polyhedral assembly material and puzzle toy of the present invention are made of a flexible and bendable sheet material such as paper or plastic (for example, polyvinyl chloride, polypropylene) as described above.

[0019]

DETAILED DESCRIPTION OF THE INVENTION Specific embodiments of the present invention will be described below with reference to the drawings, but the present invention is not limited thereto.

[0020]

EXAMPLE 1 This example relates to a method of assembling an icosahedron (a geometric solid consisting of 20 equilateral triangular faces) from 20 pieces of the same shape sheet, a material and its use.

As shown in FIG. 1, an equilateral triangular basic surface (hereinafter referred to as a basic triangle) A and each left (or each right) half of each side thereof as a bottom side, and a side extending in a direction perpendicular to the side. And a triangle connected to the basic triangle (a triangle having a half shape of an equilateral triangle, hereinafter referred to as a protrusion), which is formed by a side extending at an angle of 60 ° from the left end (or right end) of the side (1a, 2a , 3a) form 20 sheet pieces (hereinafter referred to as unit pieces). The unit piece is B below
(1b, 2b, 3b), C (1c, 2c, 3c), D
(1d, 2d, 3d), E (1e, 2e, 3
e). ． ． ． ． ． Is named. The protrusions are named in the same rotation direction on each piece in the order 1, 2, 3. The protrusion of each unit piece has a crease that is folded along the side toward the same side with respect to the basic triangle during assembly. The side where the protruding portion faces upward is called the back surface, and the opposite side is called the front surface. The names A, B, C and D are not only the names of the basic triangles but also the names of the unit pieces as a whole. Hereinafter, it is mainly used to refer to a unit piece. If the capital letters are enclosed in (), it means the back side. When the lower case letter is enclosed by (), it means that the protrusion is on the lower side of another sheet. Folded lines are shown as dotted lines, and dashed-dotted lines mean that the side is below another sheet. In the drawing, when viewed from the surface, it is shown as having a protrusion on the left side of each side.

In order to assemble a regular icosahedron using this unit piece, the unit pieces A and B are viewed from the back side as shown in FIG. (Front side, abbreviated below) are combined so that 2a overlaps the lower side of the B piece. Next, the unit pieces C are combined as shown in FIG. 3 (rear surface) and FIG. 4 (front surface). That is, the protruding portion 3c of the unit piece C is overlapped and combined with the lower side of the B piece so as to cover the protruding portion 2a when viewed from the back surface. FIG. 4 is a surface view of this state.

Next, as shown in FIG. 5, the unit piece D is assembled on the lower surface of the unit piece C so that the protruding portion 3d covers the protruding portion 2b on the lower side of the basic triangle. Up to this point, they can be assembled on a flat surface. FIG. 6 is a view from the surface in this state. In the figure, x is a portion without a sheet, that is, a desk top surface, for example.

Next, between the unit pieces D and A, there is a fifth unit piece E.
Combine. The combination is the same as before, seeing from the back side, 3
Stack so that e covers 2c under D. In this case, the shape is three-dimensional, and the umbrella is upside down. FIG. 7 is a rear view of this state, and FIG. 8 is a front view of this state.

FIG. 9 is an approximate side elevational view of a regular icosahedron viewed from a certain direction. Approximate means that it is not a three-dimensionally accurate elevation view, for example, the surface visible in the center is drawn as an equilateral triangle, but this triangle is actually a vertical Its orthographic projection is shorter than an equilateral triangle because it is tilted rearward with respect to the central axis, but this schematic is sufficient for understanding the invention. For convenience of explanation, it is assumed that the regular icosahedron has a bottom portion Y, a side wall portion X, and a top portion Z.

By the description of FIG. 8, the bottom Y is formed. When the state of FIG. 8 is viewed from the same direction as FIG. 9, it becomes as shown in FIG. However, only the A unit piece, the B unit piece, a part of the E unit piece, and a part of the C unit appearing on the front side are shown, and those on the opposite side are omitted.
Now, the sixth unit piece F is combined with this. The method is as shown in FIG. 11, when viewed on the surface, if is superposed on 3b, and the tip of the winning 1f is inserted under 2e. With this, the state of "three freezing" is achieved and one surface is completed. The substrate on this overlapping surface is A, and 1a is projected. F is overlapped on the back side of 1a, and 3f and 2f project. In this case, 2f and 3f are shown in a size in a state of being parallel to the paper surface. In other words, it is shown in the same size as in FIGS.

As shown in FIG. 9, the side wall portion X has five triangular faces 5 with their vertices on the top and triangular faces 5 with their vertices on the bottom.
It is formed of individual pieces. Finished bottom (Fig. 7, 8, 1)
For 0), as described with reference to FIG. 11, first, five triangular faces with vertices on top are formed, and then, equilateral triangular faces with vertices on bottom are formed between them. It is a convenient way to assemble. In any case, if you assemble so that the state of "three freezing" is formed on each side,
Finally, a regular icosahedron as shown in FIG. 12 is formed. In each of the surfaces, what is called a basic triangle is a substrate on which the three freezing portions of the projecting piece are formed.

So far, the formation of the regular icosahedron from the unit pieces has been described, which is relatively easy, but may itself be a puzzle. However, in order to make the above structure more interesting as a puzzle, it is necessary to color-code, pattern, and figure each surface so that a desired color-coding, pattern, and figure appear when assembled. According to the viewpoint, the regular icosahedron is 5 shown in FIG.
It can be considered that there are 12 units (that is, the number of vertices) of overlapping plane units (that is, the bottom portions described with reference to FIGS. 7, 8 and 9), but any 5 planes have one color of the same color. , 5 surfaces with different colors can appear. That is, as shown in FIG. 14, it can be painted in five colors.

Five colors are required, but the unit piece has only three protrusions that appear on the surface. That is, as shown in FIG.
Only one color, p, q, or r, can be arranged on one unit piece. The color scheme should be nCr (n = 5, r = 3) = 10, which is the number of combinations that select three out of five.
Since the left-hand color scheme and the right-hand color scheme are different, they are doubled to form 20 kinds of unit pieces. That is, no one is the same. Five colors, for example, 1, 2, 1, 2, 3
If you name them 3, 4, 5, the color scheme of each unit piece is (1, 2,
3), (1, 2, 4), (1, 2, 5); (1, 3,
2), (1, 3, 4), (1, 3, 5); (1, 4,
2), (1, 4, 3), (1, 4, 5); (1, 5,
2), (1, 5, 3), (1, 5, 4); (2, 3,
4), (2, 3, 5); (2, 4, 3), (2, 4,
5); (2,5,3) (2,5,4); (3,4
5) and (3, 5, 4). In this case, there is only one solution in which each face of the regular icosahedron has one color and each face has a different color.

In this icosahedron, a convex surface consisting of five triangles gathering at one vertex may be inverted inward to form a concave polyhedron. That is, in the present invention, the polyhedron is not limited to the convex polyhedron.

[0031]

[Embodiment 2] With the same configuration, as shown in FIG. 15, a basic triangle is divided into three by s, t,
It is possible to form a color-coded surface so that the rhombus portions formed by adjacent triangles have the same color as shown in FIG. In this case, the protruding portion of the unit piece needs to be transparent. Usually, using a unit piece of transparent material (plastic),
It may be convenient to attach a piece of paper in which the basic triangle is painted separately as shown in FIG. 15 to the back side with the color-coding side facing the front side. According to the same principle as in Example 1, 20 kinds of unit pieces are prepared. The difference is that the color coding is applied not to the protruding portion but to the basic triangular portion. In this case also there is only one solution.

[0032]

Third Embodiment With the configuration of the second embodiment, it is possible to classify six colors. There are 40 possible color classification methods for selecting 3 colors from 6 colors by the combination calculation as described above, but 20 colors are appropriately selected from them. The method of selection can be theoretically possible, but in practice, the regular icosahedron can be assembled with white paper unit pieces, desired color coding can be performed, and then disassembled. For example, if 6 colors are named 1, 2, 3, 4, 5, 6, then (1, 2, 3), (1, 4, 2), (1,
2,5), (1,6,2), (1,3,4), (1,
5,3), (1,3,6), (1,4,5), (1,
6,4), (1, 5, 6), (2, 4, 3), (2
3,5), (2,6,3), (2,5,4), (2
4,6), (2,6,5), (3,4,5), (3
6,4), (3,5,6), (4,6,5). In this case, there are multiple solutions.

[0033]

[Embodiment 4] A globe can be constructed by imitating an icosahedron as a sphere. FIG. 17 shows the set of the unit pieces. Assemble exactly as in Example 1. It should be called a three-dimensional jigsaw puzzle.

[0034]

Fifth Embodiment A regular tetrahedron is assembled with four unit pieces shown in FIG. It can be easily formed by forming three freezing engagements on each surface.

[0035]

Sixth Embodiment Similarly, a regular octahedron is assembled with eight unit pieces shown in FIG. In this case, it is considered that four pyramid surfaces (four surfaces excluding the bottom surface of the pyramid) are overlapped by six (the number of vertices). Therefore, in order to color the surface into four colors, a three-dimensional puzzle can be created by making nCr × 2 = 8 different color-coding unit pieces for each unit piece. In this case, the upper half pyramid and the lower half pyramid are color-coded in contrast to each other with respect to the symmetric rotational axis connecting the vertices. This is an easy puzzle for younger children.

[0036]

[Embodiment 7] A cube is assembled with six unit pieces shown in FIG. In this case, a puzzle can be made by applying colors, patterns, and figures.

[0037]

[Embodiment 8] A regular square pyramid is assembled using one unit piece shown in FIG. 19 and four unit pieces shown in FIG. In this case, the shape of the side protruding portion is, conceptually, a shape that overlaps the basic triangle when folded back. The side wall may be an isosceles triangle if the side wall is formed so as to match the side wall by extending the protruding portion of the unit piece on the bottom surface of FIG. 19 as shown in FIG.

[0038]

[Embodiment 9] A regular pentagonal prism is formed by two unit pieces shown in FIG. 22 and five unit pieces shown in FIG. In this case,
The unit piece shown in FIG. 3 has a shape in which the protruding piece to be stacked on the bottom surface matches the regular pentagon. The unit piece shown in FIG. 23 has a square basic portion, but this portion may be rectangular. It is more convenient to round the tip of the protrusion, rather than making it perfectly pentagonal.

[0039]

[Embodiment 10] A dodecahedron consisting of 6 square faces and 8 equilateral triangle faces as shown in FIG. 25 is assembled with 6 unit pieces shown in FIG. 19 and 8 unit pieces shown in FIG.

[0040]

[Embodiment 11] A dodecahedron is formed with 12 unit pieces shown in FIG.

[0041]

[Embodiment 12] A tetrahedron shown in FIG. 25 is a solid obtained by cutting a triangular pyramid determined from the six vertices of a cube and the midpoints of the sides gathering at the vertices. Similarly, if the pentagonal pyramid is cut from the 12 vertices of the regular icosahedron so that the regular hexagon remains on the surface of the original triangle, a dodecahedron is formed. A soccer ball is formed from such face units. All of the pentagonal faces are surrounded by hexagonal faces, but every other hexagonal face is in contact with the hexagonal faces and the pentagonal faces. In the same manner as above, according to the method of the present invention, this trihedron can be formed using the unit pieces shown in FIGS. 27 and 28.

[0042]

[Embodiment 13] A slanting column is assembled with two unit pieces having the shape shown in FIG. 35, two unit pieces shown in FIG. 29 and two unit pieces shown in FIG. By the same principle, an oblique pyramid can be formed.

[0043]

[Embodiment 14] A dodecahedron consisting of six octagonal faces and eight equilateral triangles as shown in FIG. 33 is assembled with six unit pieces shown in FIG. 31 and eight unit pieces shown in FIG. The theoretical shape is shown by the dotted line on the left side of FIG. If it does not matter if there is a gap, it may be shaped like a broken line.

In the fifth to fourteenth embodiments, even if not otherwise specified, a puzzle can be formed by constructing color codes, patterns, figures and the like to appear in the assembled state. The figures associated with the above description are often drawn in a theoretical position, rather than considering the actual assembly (superposition). In an actual product, the tongue-shaped protrusions may be rounded or shortened to facilitate the superposition of the meshes.

[0045]

INDUSTRIAL APPLICABILITY The present invention can form a solid three-dimensional space without using an adhesive, and has an interesting application useful as a container for candy, prizes, puzzles, and teaching materials for three-dimensional geometry.

[Brief description of drawings]

1 is a plan view of a unit piece for forming a regular icosahedron according to an embodiment of the present invention. 2-12: It is a figure which shows the procedure of assembling a regular body using the unit piece of FIG. FIG. 13 is a diagram showing color coding of the unit pieces. FIG. 14 is a diagram showing the color coding that appears in an icosahedron assembled using the color coding unit pieces. FIG. 15 is a diagram showing another color coding of the unit pieces. FIG. 16: is a diagram showing the color coding that appears on the regular icosahedron assembled using the color coding unit pieces of FIG. 15. 17 and 18: A plan view of a unit piece of an embodiment in which an icosahedron is likened to a globe. 19 and 20: Plan views of a unit piece for forming a quadrangular pyramid having side surfaces of an equilateral triangle. FIG. 21 is a plan view of a side unit piece for forming a quadrangular pyramid having an isosceles triangular side surface. 22 and 23: Plan views of a unit piece for forming a regular pentagonal prism. FIG. 24: A plan view of a unit piece that constitutes the equilateral triangular portion of the polyhedron shown in FIG. 25. FIG. 25 is a perspective view of a polyhedron composed of 6 square faces and 8 equilateral triangle faces. FIG. 26 is a plan view of a unit piece forming a regular dodecahedron. 27 and 28: Plan views of a unit piece forming a trihedron composed of a square pentagonal meter and a regular hexagonal meter. 29 and 30: Plan views of a unit piece forming an oblique column having a square bottom surface. FIG. 31: 6-point octagonal octagon and 8 equilateral triangles shown in FIG.
It is a top view of a unit piece which constitutes the octagonal side of the polyhedron which consists of faces. FIG. 32: 6-sided octagonal octagon and equilateral triangle 8 shown in FIG.
It is a top view of a unit piece which constitutes a triangular face of a polyhedron which consists of faces. FIG. 33: is a perspective view of a polyhedron composed of six octagonal planes and eight equilateral triangles which are point-symmetrical. FIG. 34: A plan view of a unit piece for forming a cube. FIG. 35 is a plan view of a base unit piece for forming a square base having a square shape.

Claims (4)

[Claims] 1. A unit sheet piece having a basic surface having the shape of each surface constituting a polyhedron and a tongue-shaped protrusion on each left or right half of each side of the basic surface, and another similar sheet sequentially. A method for forming a polyhedron, comprising stacking the respective projecting portions of one piece on the front side of the basic surface of the sheet piece and combining and fixing them in a freezing state. 2. The method for forming a polyhedron according to claim 1, wherein the polyhedron is a regular polyhedron, a prism, a pyramid, an oblique prism, an oblique pyramid, a tetradecahedron composed of square and equilateral triangular faces, and a regular pentagon. And a hexahedron that is a regular hexagonal surface, and a tetrahedron that is a regular triangle and a point-symmetric octagonal surface. 3. A basic surface having the shape of each surface constituting a polyhedron, when it is overlapped with each left half or each right half of each side of the basic surface in a freezing manner, it is overlaid thereon. A polyhedron-forming material comprising a set of sheet pieces provided with tongue-shaped projections each having a shape that forms a basic surface, the number of which is equal to the number and shape of the convex polyhedron surfaces. 4. The polyhedron-forming material according to claim 3, wherein each unit sheet piece is provided with a color, pattern, or figure so that a desired color code, pattern, or figure appears when assembled. material.