JPH03205082A - Method for forming rhombic polyhedron - Google Patents

Abstract

PURPOSE:To obtain a rhombic polyhedron represented by n(n-1) by excluding the specific polyhedrons among the rhombic polyhedrons which are obtained as the locus of the lines by synthesizing a plurality of vectors which are directed to the apex from the center of the polyhedron. CONSTITUTION:A vector is generated towards the apex from the center of a regular polyhedron or quasi-regular polyhedron, and the truncated octahedron (Kelvine's 14-hedron) of the quasi-regular polyhedron, 4.6.8-hedrons, truncated 12,20-hedrons among the rhombic polyhedrons obtained as the locus of the lines through the synthesis of a plurality of vectors are excluded, and further the rhombic 12-hedron, rhombic 30-hedron and the truncated rhombic 12-hedron are excluded. As the result, the rhombic polyhedron is obtained which is represented by n(n-1), n=10 or more. The rhombic polyhedron which is represented by n(n-1) is prepared by arranging n-pieces of strips made of paper, plastics, metal plate, etc., and weaving like woven fabric or forming a puzzle form by printing a world map, drawing, or design, on the strip or forming an unevenness.

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention relates to a method for creating a rhombic polyhedron, and a method for creating a rhombic polyhedron by applying the characteristics of the shape realized by the method for creating a rhombic polyhedron.

A method for creating a rhombic polyhedron according to the present invention will be explained with reference to the drawings.

In Figure 1, when vectors are generated from the center O of a regular tetrahedron toward each of the four vertices and combined, the outer shape becomes a rhombic dodecahedron. In other words, a point moves from the center 0 toward the vertex to form a line, the line moves in parallel and becomes a rhombic surface, the rhombic surface moves in parallel to become a rhombic hexahedron, and the rhombic hexahedron is parallel It moves and becomes a rhombic dodecahedron. Here, internal lines and surfaces are not considered as a problem in the present invention, but only the shape of the outer circumferential surface of the trajectory of the line is taken up. And X,
A rhombic polyhedron is obtained because the length of the vector from the center to each vertex of a regular polyhedron (Plato's solid) or quasi-regular polyhedron (Archimedes' solid) that looks the same when viewed from the triangular direction of Y and z is the same. All sides of are of equal length.

Figure 2 shows that when a vector is generated from the center of the cube toward each vertex, it becomes a rhombic dodecahedron with 2 sides.The cube has 8 vertices, but in the figure P O, Pt, -
Since they are lined up in a straight line, the combination of the two vectors does not become a rhombic surface but degenerates into a straight line. And P+, O, P-
Also, since they are lined up in a straight line, from the middle school 7th grade O to the apex Pl,
For the first time, a rhombic surface with 2 sides is created using four vectors directed toward P,, P+, and 'P4. The same goes for the other vertices, so it becomes a rhombic dodecahedron, just like the regular tetrahedron in Figure 1.

Similarly, in FIG. 3, there are six vertices of the regular octahedron, and there is another vertex on a straight line from each vertex passing through the center 0. Therefore 1
It becomes a cube with 2 sides.

Viewed from a plane, FIG. 4(A) shows that when vectors are generated from the center of a triangle toward the apex, the combination of vectors becomes a hexagon. In the case of the hexagon (D), there is another Tr1 point on the straight line passing through each vertex and the center, so one side is 2
becomes a hexagon. From the pentagon in (C), there is a decagon, (F
) makes a decagon with side 2.

In general, if a regular polygon is an odd-numbered polygon, if a C vector is generated from the center toward the vertex, the shape of the outer periphery becomes an even-numbered polygon, and if it is an even-numbered polygon, it becomes a similar shape with 2 sides.

Since one of the purposes of the present invention is to find shapes, only shapes with one side are found. For example, (B) in Figure 4, (DJ
. (IE), (In order to find the shape of the hatched area of Fl, it is only necessary to pick up one side of the line that indicates the cross section.

The figure at the end of the arrow in Figure 4 is made into a polygonal pyramid by moving the point 0, which generates a vector, from the polygon's face upwards, making it a parallelepiped from a triangle, a parallelepiped from a quadrilateral, and a parallelidodecahedron from a pentagon. obtains a parallel icosahedron, but if point 0 is on a surface of the polygon, the parallel polyhedron degenerates into a surface. In a two-dimensional regular polygon, there are odd and even numbers at every other corner, but in a three-dimensional regular polyhedron, there are no other vertices on the straight line passing through the vertex and the center 0, which means it is cut by a regular tetrahedron. It has only a regular tetrahedron. All other regular polyhedra and quasi-regular polyhedra have other vertices on a straight line passing through the vertex and the center 0. Therefore, the regular tetrahedron and truncated regular tetrahedron are odd-numbered polyhedra.

All others can be said to be even-numbered polyhedra.

An even-numbered polyhedron can be divided into two by a plane that passes through the center D and does not pass through the vertices to obtain the shape of a rhombic polyhedron, although the regular octahedron in Figure 3 has six vertices.

Three vectors are needed to create a cube.

Similarly, the cube in Figure 2 has eight vertices, but the center
Find the shape of the rhombic polyhedron by dividing it into two along a plane that passes through 0 and does not pass through the vertices.

The number of faces of a rhombic polyhedron is represented by n(n-1). Regular polyhedra and quasi-regular polyhedra other than regular tetrahedrons and truncated regular tetrahedrons all have a vertex and a straight upper lip passing through the center O and other vertices, so when calculating the shape of a rhombic polyhedron, predict by halving the number of vertices. I can do it.

The number of vertices of the regular icosahedron in Figure 5 is 12, so halve it to n=6 and substitute this to n(n-1) to get 6x
5 = 30, that is, a rhombic 30-hedron is obtained.

In Figure 6, the number of vertices of the cubic hexahedron is also 12, but it can be found by halving it and using 6 vectors, but since there are 4 combinations where 3 vectors are the same - plane, it degenerates into a plane. A hexagon appears on the surface, but the hexagon is divided into three diamonds.

8x3+6=30 This is a Kelvin tetradecahedron.

Figure 7 shows a 24-hedron (a projection of a concentric cube and a regular octahedron onto a spherical surface), which has 14 vertices and 7 vectors, making it a truncated n-rhombic dodecahedron. Since the hexagon is divided into 3III rhombuses, it is also a rhomboid tetrahedron.

Figure 8 shows a cubic octahedron and a regular octahedron projected onto a concentric sphere! 4-, 6-, and 8-sided pairs are required.

The hexagon is divided into three rhombuses, and the octagon is divided into 61 [1 of 1
! Since it is divided into shapes, there are 722 rhombuses.

FIG. 9 shows a rhombic 90-hedron obtained from a regular 12Fli body and a regular dodecahedron. The rhombic 90-hedron has two types of rhombuses.The wide rhombus has the same shape as the rhombic dodecahedron, and the ratio of the short diagonal to the long diagonal is l + 72, whereas the sharp rhombus has the ratio of the short diagonal to the long diagonal (5). is 1:1:', and if the short diagonal of the wide rhombus is 1, then the short diagonal of the sharper rhombus is τ, and the long diagonal is d, and the rhombuses maintain the golden ratio.

The figure on the left in Figure 10 is a convex ray-hedron formed by connecting the vertices of a cube inscribed in concentric spheres and a cuboctahedron, and from this shape another rhombic 90-hedron, shown in the figure on the right, was determined.

In Figure 1), the left side is a rhombic octahedron and the right side is a truncated tetrahedron. The two shown here are the rhombic octahedron, which is an even-numbered polyhedron with 24 vertices, which is halved to 12, and the truncated tetrahedron, which is an odd-numbered polyhedron, has 12 vertices. From these two polyhedra.

The rhombic 132-hedron shown in FIG. 12 is obtained. However, the truncated tetrahedron on the right is not a quasi-regular polyhedron, but a hexagon in which the ratio of the length of the side shared with the neighboring hexagon to the length of the side shared with the triangular surface is l·J2.

Figure 13 is obtained after the ratio of the length of the side shared by the hexagon of the truncated octahedron in Figure 6 with the neighboring quadrilateral and the length of the side shared with the hexagon is corrected to l J2. The three rings, x, y, and z, pass through the center of the hexagon.

Figure 14 was obtained from a polyhedron created by projecting a cube, regular octahedron, and cubo-octahedron onto concentric spherical surfaces.If the octagon is divided into rhombuses, it becomes a rhombic 156-hedron.

Figure 15 shows a truncated 12-icosahedron and a truncated 12-icosahedron made using 12-icosahedrons.

Since the IO internal square surface can be divided into 10 rhombuses, rhombus 2
It is also a decahedron.

Figure 16 shows a 60-hedron created by projecting a rhombic 30-hedron onto a concentric spherical surface, and a 90-hedron created using it. When 30 octagons are divided into rhombuses, it can be said to be a rhombic 240hedron.

The method of creating a rhombic polyhedron according to the present invention is to generate a vector from the center of a regular polyhedron toward the vertex to find a rhombic polyhedron corresponding to the original regular polyhedron. To get closer to the sphere, one after another, newly created rhombic polyhedra are concentrically projected onto the spherical surface, p+ be.

The rhomboid polyhedron of the present invention can be used in architecture, such as domes, etc.
Possible items include lighting equipment, globes, puzzles, etc.

As for domes, there are the Šideştsk dome and the lamellar dome developed by Palakminster Hula, but they are monotonous and do not vary in type.

A geodesic dome is made by projecting a regular icosahedron onto a concentric spherical surface and then repeatedly dividing the surface into triangles and hexagons. According to the method of the present invention, the length of the side is different from the side of the projected area, and the length of the side is constant, and it is excellent in decoration and has many variations. Also, by reinforcing the framework inconspicuously on the short diagonals of the rhombus, the same strength as a triangular structure can be obtained.

For example, I tried making a diamond-shaped 132-hedron using stained glass as a lighting fixture, and it has come in handy as a luxurious ornament with a very nice atmosphere.

Next, Figure 17 shows Parachminster Fuller's 1954
This is Taimakujon Sky Ocean, which was copyrighted in 2010, and Figure 18 is an expanded view of it. This is really cool, and it's designed to make each continent as connected as possible when laid out on a flat surface.

There seem to be many examples of world maps being projected onto polyhedrons.

In 1913, Cahill patented his octahedral world map.

Also in 1944, Fischer named his icosahedral world map the Leica Globe.

Created by the method of creating a rhombic polyhedron of the present invention
How to make a rhomboid polyhedron is also interesting as a puzzle.

Figure 19 is a world map printed using a hexagonal projection so that the north and south poles are at the vertices of the four-fold symmetry of a rhombic dodecahedron, but Figure 21 is an expanded view of the world map, which is broken up into pieces. Cut this horizontally and make four zigzag strips, making a mountain fold along the dotted lines. Figure 20A
Align the four strips with the map of the North Pole as shown, then fold the white strips downwards and fold them so that the map comes out like a fabric, as shown in Figure 20B.
Finally, insert the blank space at the bottom of each other's map to create a globe as shown in Figure 19.

Figure 22 shows the 3-fold symmetry with the vertices as poles.

Although it is the same belt, the weaving method is different from the developed view in Figure 21. At first, three strips are woven to match the map of the North Pole, then the equator strip is woven with other strips along the way, and finally the blank space is inserted under the other strip at the South Pole.

Figure 23 is a developed diagram of a rhombic 30-sided globe.
You can make a globe with six strips.

If there are 12 strips, the flower-shaped dodecahedron shown in Figure 26 can also be created.

FIG. 24 is a developed view of seven strips from which the truncated rhombic dodecahedron of FIG. 7 can be woven. There are two types of obi, perhaps because the symmetry is not good.

FIG. 25 is a developed view of a rhombic 90-hedron.

Cut it into 10 strips, start weaving from the 5-fold symmetrical tips in the center of the 5 strips, shift the 5 strips in the middle, finish weaving, and then use the margins of the first 5 strips to separate each other. Insert it below and you're done.

Figure 26 shows a flower-shaped dodecahedron that can be made by weaving two zigzag bands that can be made by weaving a rhombic tricosahedron.

Figure 27 shows a vector generated from the center of a regular dodecahedron by applying it to each vertex, and in the process of creating a rhombic 90-hedron with 2 sides, the sides of the regular dodecahedron are replaced with rhombic dodecahedrons. A shape like this has appeared. It is expected that if it is made of stained glass, it will become a luxurious Jumpria.

As described above, the present invention provides 1. Learn the shape of a rhomboid polyhedron by creating it. 2. Prepare multiple strips of paper, plastic, metal plates, etc., and weave the rhomboid polyhedron like textiles. 3. Weave world maps, pictures, etc. onto the strips. You can also make puzzles by printing patterns or adding unevenness.Also, since the length of the sides is constant and there are not many types of diamond shapes (there are not many types), it is possible to make a puzzle by printing a pattern or adding unevenness. If you prepare a shifter, you can create lighting fixtures with shapes that were previously unknown.

(r+=4) Rhombic dodecahedron Rhombic dodecahedron with 2 sides (n=3+ Cube with 2 sides Figure 5 Regular icosahedron Rhombic 30sahedron Figure 9 Rhombic 90sahedron (5-fold symmetry) Figure ■ Cube inside Cubic octahedron rhombic 90-hedron (4-fold symmetry) Figure truncated rhombic dodecahedron Figure 4 (cubic octahedron dodecahedron) Figure 4゜6゜octahedron Figure rhombic cubic octahedron truncated tetrahedron Figure 13 rhombic 132-hedron Rhombic 132-hedron No. 14 drawing cut l1jI12 16-Fig. Icosahedron (Rhombic 210-hedron) (Regular dodecahedron Idecahedron) Rhombic 240-hedron Rhombic 156-hedron Drawing engraving (no change in content) 8 Fig. Dymaxionda 624- No. 26 Flower-shaped dodecahedral rhombus 12 woven with a band for two rhombic tricosahedrons
'Jump Priest Procedural Amendment Form Made of 30 Face Pieces (Method) 1. Display of the Case 1999 Patent Application No. 344875 2. Name of the Invention How to Create a Rhombic Polyhedron 3. Person Making the Amendment Relationship with the Case

Claims (3)

[Claims] (1) A truncated octahedron (Kelvin ), 4, 6, and octahedrons, truncated 12, and icosahedrons, and further excluded rhombic dodecahedrons, rhombic 30hedrons, and truncated rhombic dodecahedrons, and n(n-1
) A rhombic polyhedron with n=10 or more. (2) A method of making a rhombic polyhedron represented by n(n-1) by weaving n zigzag bands like a textile. However, the cube with n=3 is excluded. (3) A globe made of n zigzag bands with maps, illustrations, color patterns, etc. placed on every other band and woven like textiles.
A puzzle that matches illustrations, colors, and horizontal patterns. However, the cube with n=3 is excluded.