KR101112163B1 - teaching aids for a regular polyhedron study - Google Patents

Abstract

The present invention relates to a regular polyhedral learning precinct, the skeleton coupled in a state in which the plurality of plate (13, 14, 15, 16) is orthogonal to each other is installed tightly inside the regular polyhedron (10).
By the skeleton there are various lines and faces connecting the respective faces of the regular polyhedron 10 in the internal space of the regular polyhedron 10, and the internal space is divided into a plurality.
Therefore, in addition to observing the appearance of the regular polyhedron, the regular polyhedron can be studied in more various ways by considering the internal space and finding the distance between arbitrary vertices, the volume of the regular polyhedron, the volume of each divided space, and the like.

Description

Teaching aids for a regular polyhedron study}

The present invention relates to a regular polyhedral learning parish, and more particularly, to a regular polyhedral learning parish that enables more diverse learning about regular polyhedrons.

A regular polyhedron is a convex polyhedron consisting of regular polygons in which all faces are congruent and having the same number of faces at each vertex.

When the above conditions are satisfied, as shown in FIG. 1, only five types of regular polyhedrons are a tetrahedron (a), a cube (b), an octahedron (c), a dodecahedron (d), and a icosahedron (e). Only exists.

Currently, the mathematics curriculum of the first year of junior high school contains information about regular polyhedrons under the section Understand the shape> Three-dimensional figure> polyhedron, which is a regular polyhedron made of paper, plastic cardboard, or a connecting member of rod I'm using the teaching tools to make them.

However, the regular polyhedrons used are simple regular ones, and are all aspects of the regular ones congruent? Are the same number of faces at each vertex? It is used only to count and confirm the contents related to the definition of the regular polyhedron, the number of faces, the number of edges, and the number of vertices.

In other words, the existing regular polyhedral learning teaching aid is used only to observe the appearance of the regular polyhedron, so its utilization was not high in actual education.

Accordingly, the present invention has been made to improve the above-described point, improve the understanding of not only the appearance of the regular polyhedron but also the internal space, and can be used for more various learning to provide a teaching aid for regular polyhedral learning improved.

The present invention for achieving the above object,

Regular polyhedron with internal view;

Skeleton made of a plate installed inside the regular polyhedron;

Including;

The tetrahedron is octahedron,

The armature is formed of two rectangular plates formed with a through slit in the longitudinal center line,

One rectangular plate formed with an open slit open at one end along a longitudinal centerline,

One of the two rectangular plates formed with the through-type slits is inserted at right angles into the through-type slits formed in the other plate, and the rectangular plate having the open slits is formed at right angles with the through-slit of the inserted rectangular plate. Inserted,

Vertices of the rectangular plates are characterized in that located at the point where the corner of the octahedron is divided by the golden ratio.

According to the present invention made of the above configuration,

Skeleton formed by combining a plurality of joint plates in a state orthogonal to each other is installed tightly inside the regular polyhedron.

It can be seen that the plurality of quadrangles jointly inside the regular polyhedron may be installed by the skeleton.

In addition, various lines and surfaces connecting the respective faces of the regular polyhedron exist in the internal space of the regular polyhedron by the skeleton, and the internal space is divided into a plurality.

Therefore, in addition to observing the appearance of the regular polyhedron, such as the number of faces, sides, vertices and the shape of the face and the number of faces gathered at the vertices, the internal space of the regular polyhedron can be considered in various ways.

That is, learning to find the distance between any vertex of the regular polyhedron, the volume of the regular polyhedron, the volume of each divided space, and the like can be further performed.

Therefore, more diverse contents can be learned during regular polyhedron education, thereby improving the understanding of the regular polyhedron.

1 is a perspective view of a regular polyhedron,
2 to 4 is a diagram illustrating the configuration and assembly of three embodiments of a tetrahedron with a skeleton,
5 is an explanatory view of the configuration and assembly of a cube with a skeleton,
6 is an explanatory view of the configuration and assembly of the octahedron with skeleton;
7 is an explanatory diagram of the configuration and assembly of a dodecahedron having a skeleton;
8 is an explanatory view of the configuration and assembly of an icosahedron with a skeleton.

Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings.

1 is a perspective view of a tetrahedron, (a) is a tetrahedron with four sides congruent, (b) a six-sided cuboid, (c) is an octahedron with eight sides congruent, and (d) is a twelve side congruent Dodecahedron, (e) is a icosahedron with congruent 20 faces.

The tetrahedron, the octahedron and the icosahedron have a regular triangular face, the regular hexahedron has a square face, and the icosahedron has a regular pentagonal face.

The present invention is characterized in that the structure consisting of a rectangular plate (hereinafter referred to as a skeleton) is provided inside the regular polyhedron.

Since the skeleton should be visible from the outside, the regular polyhedron may be made of a transparent plate, or only a corner portion may be made of a linear material (consisting of a rod made of metal, plastic, or wood, and having a perforated surface).

Now look at embodiments in which the skeletons of various structures are installed inside each regular polyhedron.

2 is an embodiment in which one square plate 11 is installed inside the tetrahedron 10.

Four vertices of the square plate 11 is located at the center of each corner of the tetrahedron 10, the four sides are in close contact with each surface of the tetrahedron 10.

Since the square plate 11 is installed in the above state, the inner space of the tetrahedron 10 is divided into two spaces of a joint space that is 90 ° twisted with respect to the square plate 11.

3 is an embodiment in which two square plates 11 and 12 are embedded in the tetrahedron 10, each square plate 11 and 12 having the same width as the thickness of the plate between one vertex and the center of the square. The slits 11a and 12a having the same shape are formed. (In the present invention, all the plates have the same thickness, and all the slits have the same width as the plates so that the plates can be fitted.)

The two square plates 11 and 12 are inserted into and coupled to the slits 11a and 12a of the counterpart, respectively, and the two square plates 11 and 12 are orthogonal to each other by 90 °.

Skeleton of the above structure is embedded in the tetrahedron (10). Also in this case, the vertices of each square plate 11 and 12 are located at the center of the corner of the tetrahedron 10, and each side is in close contact with each surface of the tetrahedron 10.

The inner space of the tetrahedron 10 is divided into four by the two orthogonal square plates 11 and 12.

4 is a case where the skeleton consisting of three rectangular plates cross-linked in a state orthogonal to each other is installed inside the tetrahedron 10.

In fact, since it is impossible to cross-couple three rectangular plates at right angles to each other, one rectangular plate is completely divided along a diagonal line and used as two rectangular triangle plates.

The shape of each board | plate material is demonstrated.

One square plate 13 is formed with a line connecting two vertices facing each other, that is, a slit 13a passing through the center point of the square from one vertex along a diagonal line. The inner end of the slit 13a (the end located inside the plate) is located at an intermediate portion between the center point of the square and the other vertex.

Another square plate is completely divided along the diagonal as described above, and divided into two right triangle plates 14 and 15.

Each of the right triangular plate members 14 and 15 is partially cut from a line connecting the vertices facing the hypotenuse from the central point of the hypotenuse so that slits 14a and 15a are formed.

The inner ends of the slits 14a and 15a are located in the middle portion between the center of the hypotenuse and the corresponding vertex.

On the other hand, the last sheet of square plate 16 is formed with slits 16a, 16b, 16c cut out from three vertices toward the center of the square.

The inner ends of the slits 16a, 16b, 16c are located in the middle of the length between the vertex and the center of the square.

The two square plates 13 and 16 and the two right triangle plates 14 and 15 are combined as follows.

First, the square plate 16 having the slits 16a, 16b, and 16c formed at three vertices is horizontally disposed at the center thereof, and then the other square plate 13 of the slit 16b having no slit is formed on the opposite side. The slits 13a are fitted to each other and coupled in a 90 ° staggered state (perpendicular state) with respect to the horizontal square plate 16. That is, the square plate 13 is coupled in a vertical state.

Subsequently, the slits 14a and 15a of the right triangle plate 14 and 15 are respectively disposed in the horizontal two square plates 16 and the remaining two slits 16a and 16c located in both directions of the vertical square plate 13. Fitting in a state perpendicular to the horizontal square plate (16).

The inclined surfaces of the right triangular plates 14 and 15 abut on both sides of the vertical square plate 13 to constitute another square plate perpendicular to both the horizontal square plate 16 and the vertical square plate 13. Done.

Thus, the skeleton is consequently formed in a structure in which the three square plates are joined in a diagonal direction at right angles to each other.

As shown in (d) of FIG. 4, the skeleton may be installed inside the tetrahedron with all vertices located at the center of each corner of the tetrahedron.

Looking at the tetrahedron with the skeleton as described above, it can be seen that the skeleton consisting of a structure in which the three square plates are orthogonal to each other in a diagonal direction can be installed tightly inside the tetrahedron.

In addition, the internal space of the tetrahedron by the skeleton is divided into eight pieces consisting of four cubes (located at the vertex of the tetrahedron) and four tetrahedrons (located at the triangular portion connecting the corner centers of the tetrahedron). You can check it.

On the other hand, it can be seen that the octahedron consisting of eight equilateral triangular joints to fill all the concave parts of the skeleton, each vertex of the octahedron can be installed in a state filled tightly so as to be located at the corner center of the tetrahedron It can be confirmed.

In addition, it can be seen that the minimum distance returning to the place while passing all the faces of the tetrahedron is the same as the circumference of the square plate constituting the octahedron in the octahedron embedded inside the tetrahedron.

Figure 5 shows an embodiment in which the skeleton is installed inside the cube, the skeleton of the cube is made of three square plate cross-linked in the direction of the center line of each other.

The slits of the square plates for forming the skeleton of the cube are formed as follows.

One square plate 21 is formed with a slit 21a cut along the center line, that is, toward the center of the side facing from the center of one side, and the unopened end (right end of the drawing) of the slit 21a is It is located at the midpoint of the distance from the center of the center line to the opposite side.

Another square plate is cut along the center line (the line connecting the center points of both sides facing each other) to form two rectangular plates 22 and 23, from the center of one long side of each of the rectangular plates 22 and 23. Half of the distance between the long side and the long side is cut toward the center of the opposite long side to form respective slits 22a and 23a.

The other one of the square plate 24 is cut half of the distance between the center of the square and the side from the center of the three sides except for one side to the center of the square to form three slits 24a, 24b, and 24c.

The combination of the square and rectangular plate members 21, 22, 23 and 24 is arranged to horizontally arrange the square plate member 24 having the slits 24a, 24b and 24c formed at the centers of the three sides, and the The slit 21a of the square plate 21 in which one slit 21a is formed along a center line is aligned at a slit in which no slit is formed on the corresponding side of the slit (24b in the middle of the three slits). Join in the crossed state.

Subsequently, the slits 22a and 23a of the rectangular plates 22 and 23 are aligned at right angles to both slits 24a and 24c of the horizontal square plate 24 on both sides of the vertically arranged square plate 21. Join in the crossed state.

As a result, the skeleton assembled as described above has a shape in which three square plates are coupled in a direction along the center line in a state orthogonal to each other.

The skeleton of the cube may be embedded tightly in a state where each vertex is located at the center of each corner of the cube. Each edge of the armature is in close contact with the face of the cube.

When the skeleton is built, it can be seen that the internal space of the cube is divided into eight cubes that are congruent.

In other words, it can be seen that the cube can be divided into eight pieces into three square plates.

On the other hand, Figure 6 shows the octahedral with the skeleton embedded, the skeleton in this case is joined in a state where the three rectangular plate joints intersected at right angles to each other.

Two rectangular plates 31 and 32 are formed with slits 31a and 32a having a length through which another rectangular plate can be penetrated along a center line (a rectangular longitudinal center line) connecting the two short sides facing each other. do.

The slits 31a and 32a are formed in the same center position as the rectangular plate materials 31 and 32.

Slit 33a is formed along the longitudinal centerline of the other rectangular plate 33, and one end of the slit 33a extends in a straight line and is open toward the short side of the adjacent side.

Accordingly, the rectangular plate 32 having the through slits 32a formed at both ends thereof is horizontally disposed, and the through slits 31a having the other ends blocked at the slits 32a of the horizontal rectangular plate 32 are formed. After the rectangular plate 31 is inserted vertically, the long side of the horizontal rectangular plate 32 is inserted into the slit 33a of the other rectangular plate 33, and the rectangular plate 33 itself is also perpendicular to the vertical plate. It inserts into the slit 31a of the rectangular board 31.

As a result, the three rectangular plate members 31, 32, and 33, which are conjoined, are joined in a state where they cross at right angles to each other.

The rectangular plates 31, 32, and 33 are installed on the same plane as each of the three virtual orthogonal squares existing inside the octahedron 30. (A square is connected when the corners of the octahedron are connected to the same plane. do.)

In addition, the long sides and the short sides of the rectangular plates 31, 32, and 33 may be determined such that the vertices of the rectangular plates 31, 32, and 33 are located at the dividing points where the corners of the octahedron 30 are divided by the golden ratio. have.

When the lengths of the long sides and short sides of the rectangular plates 31, 32, and 33 are determined as described above, when all the vertices of the skeleton formed by crossing the rectangular plates 31, 32, and 33 at right angles to each other are icosahedron. It can be seen that.

In other words, if you divide the corners of the octahedron 30 by the golden ratio and connect all of the points, it can be seen that the icosahedron, and thus it can be seen that the icosahedron 30 can be installed tightly inside the octahedron 30. .

7 and 8 are embodiments in which the skeletons coupled in a state in which three rectangular plate members conjoined to each other are orthogonal to each other inside the dodecahedron 40 and the icosahedron 50, respectively.

In the case of the skeleton embedded in the dodecahedron 40 and the icosahedron 50, the shape of the slit formed in each plate in order to combine three rectangular plates to be orthogonal to each other is shown in the octahedron 30 of FIG. It is the same as the case of the rectangular plate materials 31,32,33 which comprise skeleton.

That is, two rectangular plate members 41, 42; 51, 52 having through slits 41a, 42a; 51a, 52a formed in the center line, and one side of the through slits 41a, 42a; 51a, 52a It consists of one rectangular plate 43 and 53 in which the open slit 43a; 53a extended and opened toward the short side.

Since the assembly method is also the same as the case of FIG. 6 described above, repeated description is omitted.

In the case of the dodecahedron 40 shown in FIG. 7, it is composed of 12 regular pentagons, and the shared corners of two pentagons adjacent to each other have corners under the same conditions (the shared corners of two adjacent pentagons adjacent to each other).

Therefore, the length of the short side of the rectangular plate | board material 41, 42, 43 which is congruent is made to be equal to the length of the edge of the icosahedron 40, ie, the length of the one side of the pentagon which comprises the dodecahedron 40, When the length of is equal to the length between the edges facing each other in the dodecahedron 40, the skeleton consisting of three rectangular plates (41, 42, 43) conjoined inside the dodecahedron 40 is filled up tightly You can see that it can be installed.

Meanwhile, the icosahedron 50 illustrated in FIG. 8 is composed of 20 equilateral triangles, and the number of vertices is 12, and it can be seen that the same edge exists on the opposite side with respect to any corner.

Therefore, when a joint is formed by combining rectangular plates 51, 52, and 53 which are mutually orthogonal to each other, the skeleton also has 12 vertices (4 vertices of a rectangle 3 × 3), and corners must face each other. Since there is an edge to be seen that it can be installed tightly the skeleton consisting of three rectangular plate (51, 52, 53) joint even inside the icosahedron 50.

On the other hand, the short sides of the rectangular plate 51, 52, 53 can be equal to the length of one side (edge) of the icosahedron 50. In this case, it can be seen that the lines connecting all the vertices of the skeleton consisting of the three rectangular plates 51, 52, 53 correspond to all the corners of the icosahedron 50, respectively. That is, the corners of the three-dimensional figure formed by connecting the vertices of the skeleton and the corners of the icosahedron remain unpaired or missing, and the two sides can be exactly matched.

That is, it can be seen that the icosahedron can be made by connecting the vertices of the rectangular plate materials 51, 52, and 53 which are congruent with each other.

In addition, it has already been described in the description of FIG. 6 that the icosahedron made as described above may be installed tightly inside the octahedron.

10: tetrahedron 11, 12, 13, 16: square plate
14,15: right triangle plate 11a, 12a, 13a, 14a, 15a, 16a, 16b, 16c: slit
20: cube 21,24: square plate
22, 23: rectangular plate 21a, 22a, 23a, 24a, 24b, 24c: slit
30: octahedron 40: dodecahedron
50: icosahedron 31,32,33,41,42,43,51,52,53: rectangular plate
31a, 32a, 33a, 41a, 42a, 43a, 51a, 52a, 53a: slit

Claims (1)

Regular polyhedron with internal view;
Skeleton made of a plate installed inside the regular polyhedron;
Including;
The tetrahedron is octahedron,
The armature is formed of two rectangular plates formed with a through slit in the longitudinal center line,
One rectangular plate formed with an open slit open at one end along a longitudinal centerline,
One of the two rectangular plates formed with the through-type slits is inserted at right angles into the through-type slits formed in the other plate, and the rectangular plate having the open slits is formed at right angles with the through-slit of the inserted rectangular plate. Inserted,
Vertices of the rectangular plate material is a tetrahedral learning parish, characterized in that located at the point where the corner of the octahedron divided by the golden ratio.

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