Geometry learning tool having three-dimensional spherical ruler

Abstract

The present invention relates to a geometry learning tool having a three-dimensional spherical ruler, wherein the three-dimensional spherical ruler having a semicircular curved shape is provided to surround and fit on a portion of a sphere, thereby enabling a large circle to be accurately drawn on a spherical surface. In addition, various geometry learning tools are applied so that a three-dimensional spherical ruler is provided to a learner, thereby enabling spherical geometry to be easily taught.

Description

Geometry learning materials equipped with a three-dimensional sphere

[0001] Cross-citation with related application(s)

[0002] This application claims the benefit of priority to Korean Patent Application No. 10-2024-0035899, filed March 14, 2024, the entire contents of which are incorporated herein by reference.

[0003] The present invention relates to a teaching aid having a three-dimensional sphere.

[0004] Spatial ability significantly influences students' achievement in fields like math and science. It's widely accepted that experiential learning is more effective for developing excellent spatial abilities than formal or verbal learning. Memorizing the concepts of three-dimensional spatial geometry is not only difficult to grasp, but can also lead to significant confusion and misunderstanding.

[0005] A regular polyhedron is a basic solid in three-dimensional space, and its basic contents are included in first-year middle school mathematics textbooks.

[0006] A regular polyhedron is a solid in which all faces are congruent regular polygons and the same number of faces meet at each vertex. There are only five types: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

[0007] To deepen the understanding of regular polyhedra, there is a method of representing the vertices of a regular polyhedra on a sphere and connecting these vertices to implement a regular polyhedra on the sphere.

[0008] The two things needed to implement a regular polyhedral sphere (or spherical regular polyhedron) are summarized as follows: first, marking the vertices of the regular polyhedron on the surface of the sphere, and second, drawing the arc of the great circle connecting the two vertices. The great circle refers to the largest circle among the circles that would be cut when the sphere is cut by a plane, and refers to the cross section when the plane passes through the center of the sphere.

[0009] Marking vertices on a sphere is possible by following the mathematical theory of finding the vertices of a regular polyhedron on a sphere or by using related tools. On the other hand, accurately drawing an "arc of a great circle" is difficult because the sphere is not a flat surface.

[0010] On a sphere, it is possible to implement not only regular polyhedra, but also semi-regular polyhedra, spherical triangles, spherical lunes, and tessellations. To implement these, drawing great circles is the most basic.

[0011] Great circles must be accurately depicted on a sphere. However, if the great circle deviates from the center of the sphere, various shapes or lines implemented on the sphere will be drawn asymmetrically. While rulers exist for drawing straight or curved lines, they are designed for drawing lines on a flat surface and are difficult to apply to a sphere. Furthermore, while a tape measure can be used to draw lines on a sphere, it is difficult to secure the tape measure while drawing, making it difficult to draw great circles even with this method.

[0012] Therefore, a new means of drawing a great circle on a sphere is needed that is both accurate and easy.

[0013] The purpose of the present invention is to provide a geometry learning tool equipped with a three-dimensional spherical ruler to enable drawing a great circle on a sphere.

[0014] The present invention provides a geometry learning tool having a three-dimensional sphere having a semicircular curved shape for wrapping and fitting a part of a sphere.

[0015] The above geometry learning aid has a sphere.

[0016] The above three-dimensional sphere includes a band-shaped drawing portion extending in the longitudinal direction, and a support portion symmetrically positioned at both ends of the drawing portion, and having through holes formed at positions corresponding to both poles of the sphere.

[0017] The three-dimensional sphere according to the present invention can accurately draw a great circle on a sphere, and can be applied to various geometric learning tools to easily teach geometry to learners.

[0018] FIG. 1 is a 3D drawing showing a three-dimensional sphere according to a first embodiment of the present invention.

[0019] Figure 2 is a stereoscopic photograph that served as the source of the idea for the first embodiment.

[0020] Figure 3 is a 3D drawing showing a three-dimensional sphere in which a display portion is formed.

[0021] FIG. 4 is a 3D drawing showing a three-dimensional sphere according to a second embodiment of the present invention.

[0022] Figure 5 is a three-dimensional photograph that is the source of the idea for the second embodiment.

[0023] Figure 6 is a 3D drawing showing a three-dimensional sphere with a display portion formed thereon.

[0024] Figure 7 is a 3D drawing showing a three-dimensional sphere according to a third embodiment of the present invention.

[0025] Figure 8 is a stereoscopic photograph that serves as the source of ideas for the third embodiment.

[0026] Figure 9 is a 3D drawing showing a three-dimensional sphere with a display portion formed thereon.

[0027] Fig. 10 is an exemplary drawing showing drawing a great circle using a three-dimensional sphere and a sphere of the present invention.

[0028] Figure 11 is a drawing for explaining how to draw a regular octahedral sphere using the geometric learning teaching aid of the present invention.

[0029] Figure 12 is a drawing for explaining how to draw a regular polyhedral sphere using the geometric learning teaching aid of the present invention.

[0030] Figure 13 is a drawing for explaining how to draw a cuboctahedral sphere using the geometric learning teaching aid of the present invention.

[0031] Figure 14 is a drawing for explaining how to draw the meridians of the Earth using the geometric learning teaching aid of the present invention.

[0032] Figure 15 is a drawing showing a spherical design and tessellation using the geometric learning teaching aid of the present invention.

[0033] Figure 16 is a drawing for explaining how to draw a straight line, a spherical triangle, and a spherical moon using the geometry learning teaching aid of the present invention.

[0034] According to one embodiment of the present invention, a geometry learning tool may be provided, which comprises a three-dimensional sphere having a semicircular curved shape that is fitted around a portion of a sphere, wherein the three-dimensional sphere has one or more straight sections that can draw a great circle on the sphere of the sphere.

[0035] Hereinafter, a geometric learning tool equipped with a three-dimensional sphere according to the present invention will be described.

[0036] The 'stereoscopic sphere' referred to in this specification means a ruler capable of indicating the great circle path connecting two points (both poles) so as to accurately draw a great circle on a sphere.

[0037] The 'great circle' mentioned in this specification means the largest circle among the circles that form a cross section when a sphere is cut by a plane, and is defined as a straight line in spherical geometry.

[0038] The geometry learning tool according to the present invention comprises a sphere and a three-dimensional sphere.

[0039] The sphere can be made of any material capable of drawing lines with a writing instrument. Examples include plastic, Styrofoam, wood, and rubber. Furthermore, transparent or translucent materials can be used to facilitate the identification of drawn lines. If necessary, the sphere's vertices can be pre-marked on its surface using a vertex-finding tool.

[0040] At this time, any writing instrument that can draw a circle on a sphere, such as a pencil, ballpoint pen, or marker pen, can be used.

[0041] When a sphere and a solid sphere are used as a teaching aid, the solid sphere has the same curvature as the sphere, and the radius of curvature of the outer surface of the sphere and the radius of curvature of the inner surface of the solid sphere are manufactured to be the same so that the solid sphere is completely in contact with the sphere.

[0042] The above three-dimensional sphere has a semicircular curved shape to enclose and fit a portion of the sphere.

[0043] Specifically, the three-dimensional sphere includes a band-shaped drawing portion extending in the longitudinal direction, and a support portion symmetrically positioned at both ends of the drawing portion, and having through holes formed at positions corresponding to both poles of the sphere.

[0044] The drawing part has a band shape extending in one direction so that a great circle can be marked on a sphere using a writing instrument, and has a semicircular curved structure and has the same curvature as the sphere.

[0045] The support member may be formed in a rectangular, polygonal, semicircular, circular, elliptical, or spiral shape to facilitate the attachment of a three-dimensional sphere to a sphere. Preferably, the support member may be semicircular, circular, or circular with one open end.

[0046] A through hole is formed at a location corresponding to the two extreme points of the sphere to mark the path of a circle. The through hole may be formed in any of the following shapes: circular, triangular, square, rectangular, or rhombic, and may be formed in a closed or open shape.

[0047] The drafting unit and support unit are manufactured as a single unit. The material of the three-dimensional sphere is not particularly limited; any material used for drafting rulers may be used, including plastic or steel. Furthermore, the main body may be made of a transparent, translucent, or opaque material.

[0048] In particular, the slits and penetration holes of the drawing part are positioned on the same straight line (great circle) so that a great circle passing through two points on the sphere can be drawn. One or more of the straight line sections can be formed depending on the shape of the drawing part, and thus, three-dimensional spheres of various shapes can be implemented.

[0049] The following is a detailed explanation with reference to the drawings.

[0050] FIG. 1 is a 3D drawing showing a three-dimensional sphere (10) according to a first embodiment of the present invention, wherein the three-dimensional sphere (10) has a semicircularly curved shape and a spherical curvature to fit a part of a sphere without any gaps on its inner side.

[0051] The drawing portion (11) is formed in the form of a band extending in the longitudinal direction, and a narrow slit (H1) is formed to form an opening area along the longitudinal direction. The left and right areas of the drawing portion (11) are symmetrical with the slit (H1) as the center, and the left and right areas have a sufficiently wide width throughout the longitudinal direction. This is because, when drawing a great circle passing through two points, the three-dimensional sphere must tightly and widely surround the sphere so that the sphere does not shake.

[0052] The drawing part (11) has one slit (H1), and the slit (H1) is a great circle line that can draw a great circle. In addition, the drawing part (11) has two outline lines (14a, 14b) that form an outer boundary, but the two outline lines (14a, 14b) are not great circle lines. The slit (H1) forming the opening area is narrow enough to be marked with a writing instrument and has a rectangular or trapezoidal shape that is extended in the longitudinal direction. Preferably, the width of the slit (H1) is formed narrowly to be less than 2 mm in a three-dimensional spherical ruler (10) manufactured to fit a medium-thickness pen, and to be less than 1 mm in a three-dimensional spherical ruler (10) manufactured to fit a thin-thickness pen.

[0053] The support portions (13a, 13b) are formed symmetrically at both ends of the drawing portion (11) and have a polygonal, semicircular, circular, oval, or spiral shape. Preferably, they may be semicircular, circular, or circular with one side open. Through holes (15a, 15b) forming an opening area are formed in one area of ​​the support portions (13a, 13b), preferably in the central area.

[0054] At this time, the through holes (15a, 15b) and slits (H1) forming the opening area are connected to each other.

[0055] The three-dimensional sphere (10) of the first embodiment has one straight section (S11).

[0056] A straight section (S11) means a section extending from one through hole (15a or 15b) through a slit (H1) to another through hole (15b or 15a).

[0057] To easily explain the three-dimensional sphere in Figure 1, let's consider the surface (sphere) of a globe. Figure 2 is a photograph of the three-dimensional object that served as the source of the idea for the first implementation example.

[0058] As shown in Fig. 2, if we cut the globe by a plane containing the latitude 13° north and then again by a plane containing the latitude 13° south, the globe is divided into three parts. Among them, let's select the middle part containing the equator and cut it in half by a plane perpendicular to the equator. In other words, if we cut it by a plane perpendicular to the equator and passing through the center of the equator, it is separated into two congruent curved figures. A curved figure that implements one of these curved figures on the inside of a solid sphere is the solid sphere of Fig. 1. Here, half of the great circle representing the equator becomes the straight section (S11) of the solid sphere of Fig. 1 and is excavated by the slit (H1) part. Part of the latitude 13° north and part of the latitude 13° south become the outer lines of both ends of the island section. Here, the latitude line of 13° north latitude and the latitude line of 13° south latitude are factors that determine the width of the three-dimensional sphere (10), and 13° can be replaced with 13.3°, 14.5°, 15°, etc.

[0059] Since the three-dimensional sphere (10) of the present invention is manufactured with the support portion (13a, 13b) and the drawing portion (11) being symmetrical on both sides, it is easy to draw a great circle passing through two points indicated in the slit (H1) or through hole (15a, 15b).

[0060] Drawing an arc of a great circle passing through two points can be considered in two cases.

[0061] First, if the two points are not the endpoints of the diameter, the arc of the great circle passing through the two points is drawn as follows.

[0062] Align the three-dimensional sphere (10) with the sphere so that two points are visible in the slit (H1), and ensure that the three-dimensional sphere (10) and the sphere are in perfect contact with each other. Next, draw a line by inserting a pen into the slit (H1) to draw an arc of a great circle. The reason is as follows. The width of the symmetrical three-dimensional sphere (10) is sufficiently wide, and the curvature and curvature of the three-dimensional sphere (10) match those of the sphere, so the three-dimensional sphere (10) and the sphere are in perfect contact with each other. Therefore, the plane of symmetry of the three-dimensional sphere (10) becomes the plane of symmetry of the sphere, and the line drawn in the slit (H1) is an arc of a great circle.

[0063] By using the three-dimensional sphere (10) once, an arc of a great circle can be drawn up to half the length of the great circle. This is because a straight section (S11) including the slit (H1) portion is formed up to the through holes (15a, 15b) on both sides.

[0064] Second, if two points are the endpoints of a diameter, there are infinitely many arcs of a great circle passing through the endpoints. Therefore, another condition is needed to determine the diameter. That is, one more point is needed. This leaves three points. Since two of the three points are not the endpoints of the diameter, we end up with the first case.

[0065] Drawing a complete great circle passing through two points can be done by using a solid ruler (10) at least three times. Using the solid ruler (10) twice yields two halves of a great circle, but these two halves may not be halves of the same great circle. This is because each may represent a half of a different great circle. Therefore, even if you draw an arc of a great circle passing through two points by using the solid ruler (10) once to halfway through the arc, and then extend this arc by using it a second time, you will never reach a complete great circle. Ultimately, a third use is necessary.

[0066] Additionally, the three-dimensional sphere (10) of the present invention is formed with a display section so as to measure the length of an arc of a great circle or the size of a central angle corresponding to the arc.

[0067] Figure 3 is a 3D drawing showing that a scale-shaped marking portion (17a, 17b) is formed on the surface of the drawing portion (11).

[0068] The marking portions (17a, 17b) are formed to have equal intervals in the longitudinal direction of the straight section (S11), and the marking portions (17a, 17b) are formed at either position of the two surfaces of the slit (H1) at the center of the drawing portion (11), and may be formed on both sides as shown in FIG. 3. At this time, the marking portions (17a, 17b) are formed on the outer side of the three-dimensional spherical ruler (10). The shape may be any one of a protrusion, a groove, or a scale shape, and for convenience, it is shown as a protrusion shape in FIG. 3, but the present invention is not particularly limited thereto. However, the depth (or thickness) or size of the marking portions (17a, 17b) is adjusted so as not to interfere with the display of the straight section (S11).

[0069] In addition, the number of display portions (17a, 17b) is not particularly limited in the present invention, and can be any number that can be displayed at regular intervals. For example, since the central angle corresponding to the arc of a great circle from one pole to the other pole is 180°, the number of display portions (17a, 17b) can be 5, 8, and 11. If 5 display portions (17a, 17b) are formed, they can be formed at 30° intervals, if 8, they can be formed at 20° intervals, and if 11, they can be formed at 15° intervals. If necessary, more display portions can be formed.

[0070] Figure 3 shows 11 marks (17a, 17b) spaced at 15° intervals. The details are as follows.

[0071] In Fig. 3, if an imaginary line segment connecting the centers of both through holes (15a, 15b) is drawn and the midpoint is “O”, the point O is the center of a great circle extending the straight section (S11) and the center of the sphere. In addition, the intersection points of three arbitrary consecutive marks and the slit (H1) are designated as R1, R0, and R2. To be more precise, three arbitrary consecutive points among the 11 centers of the parts where each mark and slit intersect are designated as R1, R0, and R2.

[0072] At this time, the expression “marks forming a 15° interval” mentioned above means ∠R0OR1= 15°, ∠R0OR2= 15°. That is, it means that the center mark (R0) and the marks on both sides (R1, R2) are formed with a 15° interval.

[0073] FIG. 4 is a 3D drawing showing a three-dimensional sphere (20) according to a second embodiment of the present invention, wherein the three-dimensional sphere (20) has a semicircular curved shape to wrap and fit a part of a sphere on the inside thereof.

[0074] The drawing part (21) is in the form of a band extending in the longitudinal direction, and a narrow slit (H2) forming an opening area along the longitudinal direction is formed. The left and right regions of the drawing part (21) are symmetrical with respect to the slit (H2), and the left and right regions have a shape in which the width is widest in the central region in the longitudinal direction and the width gradually decreases toward the end region. Accordingly, the inner surface of the drawing part (21) in contact with the slit (H2) forms a great circle, and the outer surface of the drawing part (21) that does not contact the slit (H2) has a curved shape. The drawing part (21) has one slit (H2) and two outer lines (24a, 24b) that can be drawn along the outer surface of the drawing part (21) that does not contact the slit (H2). At this time, the slit (H2) and the two outer lines (24a, 24b) are all great circle lines that can draw a great circle.

[0075] The slit (H2) forming the opening area is narrow enough to be marked with a writing instrument and has a rectangular or trapezoidal shape that extends lengthwise. Preferably, the width of the slit (H2) is formed narrowly, less than 2 mm in a three-dimensional sphere (20) manufactured to fit a medium-thickness pen, and less than 1 mm in a three-dimensional sphere (20) manufactured to fit a thin-thickness pen.

[0076] The support portions (23a, 23b) are formed symmetrically at both ends of the drawing portion (21) and have a shape of a circle, an oval, a square, or a combination thereof. The support portions (23a, 23b) of Fig. 4 have a shape of a circle and a square combined. Through holes (25a, 25b) forming an opening area are formed in one side region of the support portions (23a, 23b), preferably in the central region. At this time, the through holes (25a, 25b) forming the opening area and the slit (H2) are connected to each other.

[0077] Points P1 and P2 represent points where the outer surface of the drafting portion (21) meets the support portion (23a or 23b). The outer surface of the drafting portion (21) has a curved (or curved) structure from point P1 to point P2, and meets the support portion (23a or 23b) in a bent state at points P1 and P2. Therefore, it is impossible for the outer surface of the drafting portion (21) to have a curved structure up to the center point of the through hole (25a or 25b). Therefore, in order to make the outer surface of the drafting portion (21) as long as possible, the points P1 and P2 must be positioned in an area close to the through hole (25a or 25b).

[0078] The three-dimensional sphere (20) of the second embodiment has three straight sections (S21, S22, S23).

[0079] One straight section (S21) means a section extending from one through hole (25a or 25b) through a slit (H2) to another through hole (25b or 25a).

[0080] Another straight section (S22) refers to a section extending from one through hole (25a or 25b) along the outer line (24a) of the left area of ​​the drawing portion (21) that is not in contact with the slit (H2) to another through hole (25b or 25a). However, since there is a part that is not indicated by the support portion (23a, 23b), the section excluding this part is the straight section (S22). In Fig. 4, the straight section (S22) corresponds to point P1 to point P2.

[0081] Another straight section (S23) refers to a section extending from one through hole (25a or 25b) along the outer line (24b) of the right area of ​​the drawing portion (21) that is not in contact with the slit (H2) to another through hole (25b or 25a). However, since there is a part that is not indicated by the support portion (23a, 23b), the section excluding this part is a straight section (S23).

[0082] To easily explain the three-dimensional sphere (20) of Fig. 4, let's consider the surface (sphere) of a globe and consider a case where the angle between three straight sections (S21, S22, S23) is 15°. Fig. 5 is a three-dimensional photograph that is the source of the idea for the second embodiment. As in Fig. 5, draw the prime meridian on the globe, and draw the meridian of 15° east longitude and the meridian of 15° west longitude. Now, if the North Pole is N and the South Pole is S, a lune is created with the two end points being N and S, and the meridian of 15° east longitude and the meridian of 15° west longitude as the boundary. The lune with the two red meridians as the boundary is the lune created from the meridian of 15° west longitude to the meridian of 15° east longitude. The three-dimensional sphere that embodies this shape inside is the three-dimensional sphere (20) of Fig. 4, but with some reinforcement added to create supports (23a, 23b). Here, the meridian representing the prime meridian becomes the straight section (S21) of the three-dimensional sphere (20) of Fig. 4, and is dug into the slit (H2) portion. A part of the meridian of 15° west longitude becomes the straight section (S22), and a part of the meridian of 15° east longitude becomes the straight section (S23).

[0083] Let's discuss Figure 4 again.

[0084] In the two straight sections (S22, S23), the outlines (24a, 24b) of the drawing section (21) and the through holes (25a and 25b) are not directly connected, but when connecting an imaginary straight line, they are located on the same straight line, so they are additionally connected to complete the section. Specifically, a line is drawn along the left outline (24a) of the drawing section (21) that is not in contact with the slit (H2) to point P1 or point P2, and a point is marked at the through hole (25a or 25b). Then, by connecting the line drawn to point P1 or point P2 and the point indicated by the through hole (25a or 25b), one straight section (S22) is drawn. Another straight section (S23) is completed in the same manner.

[0085] The three straight sections (S21, S22, and S23) have a certain angle with respect to each other. At this time, since the two straight sections (S22, S23) are formed symmetrically on the left and right of the drawing section (21) centered on the slit (H2), the angles formed by S21 and S22 and the angles formed by S21 and S23 are equal to each other. Considering that the central angle of the arc corresponding to half of the equator is 180°, the angles formed by S21 and S22 and the angles formed by S21 and S23 may be 10°, 15°, or 30°, and preferably 15°. When the angle is 15°, the angles between the three great circles drawn through the three straight sections (S21, S22, and S23) become 15°. The longitudes of 15° west, 0°, and 15° east in FIG. 5 are examples of this. This makes it easy to mark meridians at 15° intervals when making a globe out of a sphere.

[0086] The above angle can be adjusted by changing the width of the left and right regions of the drawing portion (21). That is, when the angle is x°, a lunar shape created from the meridian of west longitude x° to the meridian of east longitude x° of the globe is implemented on the inside of the drawing portion (21) and the support portion (23a or 23b) is reinforced. In that shape, the left and right regions have a wider width in the central region in the longitudinal direction than in the end region, as shown in Fig. 4, and the outer lines (24a, 24b) of the drawing portion (21) that do not contact the slit (H2) have a curved shape. As the width at the center of the left and right regions is widened, the angle x between the three straight sections (S21, S22, and S23) increases, and as the width is narrowed, the angle x decreases.

[0087] Since the three-dimensional sphere (20) of the present invention is manufactured with the support portions (23a, 23b) being symmetrical left and right with respect to the drawing portion (21), it is easy to draw a great circle passing through two points drawn in the through holes (25a, 25b). In particular, since it has three straight sections (S21, S22, S23), it is possible to draw up to three arcs of a great circle with the same diameter at a time.

[0088] Drawing an arc of a great circle passing through two points can be considered in two cases.

[0089] First, if the two points are not the endpoints of the diameter, the arc of the great circle passing through the two points is drawn as follows.

[0090] Draw an arc of a great circle by aligning the two points above with any of the three straight sections S21, S22, and S23. It is preferable to align S21, which is formed by slit (H2). If aligned with S21, the pen will enter the fine part of slit (H2) and will not shake. On the other hand, if aligned with S22 or S23, the pen may shake.

[0091] Align the three-dimensional sphere (20) with the sphere so that two points are visible in the slit (H2), and ensure that the three-dimensional sphere (20) and the sphere are in perfect contact with each other. Next, draw a line by inserting a pen into the slit (H2) to draw an arc of a great circle. The reason is as follows. The width of the symmetrical three-dimensional sphere (20) is sufficiently wide, and the curvature and curvature of the three-dimensional sphere (20) match those of the sphere, so the three-dimensional sphere (20) and the sphere are in perfect contact with each other. Therefore, the plane of symmetry of the three-dimensional sphere (20) becomes the plane of symmetry of the sphere, and the line drawn in the slit (H2) is an arc of a great circle.

[0092] By using the three-dimensional sphere (20) once, an arc of a great circle can be drawn up to half the length of the great circle. This is because a straight section (S21) including the slit (H2) portion is formed up to the through holes (25a, 25b) on both sides.

[0093] Second, if two points are the endpoints of a diameter, there are infinitely many arcs of a great circle passing through the endpoints. Therefore, another condition is needed to determine the diameter. One more point is needed. If there are three points, two of them are not the endpoints of the diameter, so we end up with the first case.

[0094] Drawing a complete circle passing through two points is done by using a solid sphere (20) at least three times.

[0095] Using the sphere ruler (20) twice yields two halves of a great circle. However, these two halves may not be halves of the same great circle. This is because each may represent a half of a different great circle. Therefore, even if you draw an arc of a great circle passing through two points by using the sphere ruler (20) once and extend the arc of this great circle by using it a second time, you will never reach a complete great circle. Consequently, a third use is necessary.

[0096] Additionally, the three-dimensional sphere (20) of the present invention is formed with a display section so as to measure the length of an arc of a great circle or the size of a central angle corresponding to the arc.

[0097] Figure 6 is a 3D drawing showing that a scale-shaped marking portion (27a, 27b) is formed on the surface of the drawing portion (21).

[0098] The marking portions (27a, 27b) are formed to have equal intervals in the longitudinal direction of the straight section (S21), and the marking portions (27a, 27b) are formed at either position of the two surfaces of the slit (H2) which is the center of the drawing portion (21), and may be formed on both sides as shown in Fig. 6. In this case, the marking portions (27a, 27b) are formed on the outer side of the three-dimensional sphere (20). Other detailed descriptions follow what is mentioned in Fig. 3. However, Fig. 6 shows nine of the eleven marking portions (27a, 27b) that form 15° intervals.

[0099] To explain in detail, it is as follows.

[0100] In Fig. 6, if a virtual line segment connecting the centers of both through holes (25a, 25b) is drawn and the midpoint is O, point O is the center of a great circle extending the straight section (S21) and the center of a sphere. In addition, the intersections of three arbitrary consecutive marks and the slit (H2) are designated as R1, R0, and R2. To be more precise, three arbitrary consecutive points among the nine centers of the points where each mark and slit intersect are designated as R1, R0, and R2.

[0101] At this time, the expression “marks forming a 15° interval” mentioned above means ∠R0OR1= 15°, ∠R0OR2= 15°. That is, it means that the center mark (R0) and the marks on both sides (R1, R2) are formed with a 15° interval.

[0102] FIG. 7 is a 3D drawing showing a three-dimensional sphere (30) according to a third embodiment of the present invention, wherein the three-dimensional sphere (30) has a semicircular curved shape to wrap and fit a part of a sphere on the inside thereof.

[0103] The drawing part (31) is formed in the form of a band extending in the longitudinal direction, and a slit-shaped aperture (Q) forming an opening area in the longitudinal direction is formed. The left and right areas of the drawing part (31) with the aperture (Q) as the center should be symmetrical and sufficiently wide. They should be sufficiently wide so that the spherical surface of the three-dimensional sphere (30) and the sphere are in perfect contact and do not shake. The left and right areas may have the same width throughout the longitudinal direction, or may have a shape in which the central area has a wide width and the width gradually decreases toward the end area, or may have a shape in which the central area has a narrow width and the width gradually increases toward the end area. The opening area of ​​the aperture (Q) is formed sufficiently wide so that two straight sections can be drawn. The drawing part (31) has one aperture (Q), two inner boundary lines (38a, 38b), and two outer lines (34a, 34b). The two inner boundary lines (38a, 38b) are in contact with the opening (Q) and form the inner boundary of the drawing portion (31), and the two outer lines (34a, 34b) are not in contact with the opening (Q) and form the outer boundary of the drawing portion (31). Here, the two inner boundary lines (38a, 38b) in contact with the opening (Q) are great circle lines that can draw a great circle, but the two outer lines (34a, 34b) are not great circle lines.

[0104] Supporting portions (33a, 33b) are formed symmetrically at both ends of the drafting portion (31) and have a semicircular shape. Through holes (35a, 35b) are formed in one side region of the supporting portions (33a, 33b), preferably in the central region, forming an opening region. At this time, the through holes (35a, 35b) are connected to the through holes (Q) of the drafting portion (31).

[0105] The three-dimensional sphere (30) of the third embodiment has two straight sections (S31, S32).

[0106] One straight section (S31) means a section extending from one through hole (35a or 35b) along the inner boundary line (38a) on the right side of the drawing portion (31) that contacts the through hole (Q) to another through hole (35b or 35a).

[0107] Another straight section (S32) means a section extending from one through hole (35a or 35b) along the inner boundary line (38b) on the left side of the drawing portion (31) that contacts the through hole (Q) to another through hole (35b or 35a).

[0108] The two straight sections (S31, S32) form a certain angle with respect to each other. Considering that the central angle of an arc is 180° when the length of the arc is half of a great circle, the angle formed by S31 and S32 can be 10°, 15°, or 30°. Fig. 7 is a drawing designed for 15°. The two straight sections (S31, S32) of Fig. 7 can be viewed as representing the meridians of longitude 7.5° west and 7.5° east of Fig. 8.

[0109] The above angle is related to the width of the aperture (Q). The larger the angle between the two straight sections (S31, S32), the wider the width of the aperture (Q), and the smaller the angle, the narrower the width.

[0110] Figure 8 is a three-dimensional photograph that is the source of the idea for the third embodiment.

[0111] To easily explain the three-dimensional sphere (30) of Fig. 7, let us consider the surface (sphere) of a globe and consider a case where the angle between two straight sections (S31, S32) is 15°. As in Fig. 8(a), draw a meridian of 7.5° west longitude and a meridian of 7.5° east longitude on the globe. Now, if the North Pole is N and the South Pole is S, a lune is created with the two endpoints being N and S and the meridian of 7.5° west longitude and the meridian of 7.5° east longitude as the boundaries. In the figure, the lune with the two blue meridians as the boundaries is the lune created from the meridian of 7.5° west longitude to the meridian of 7.5° east longitude. Next, draw auxiliary lines sufficiently wide on both sides of the two blue meridians so as to be symmetrical about the meridian of 0° longitude (prime meridian). The red auxiliary line in Fig. 8(a) is symmetrical to the meridian of 0° longitude (prime meridian), and the gap with the blue line is wide enough for the following explanation. That is, if the meridian of 0° longitude (prime meridian) is rotated so that it becomes the equator, Fig. 8(a) becomes Fig. 8(b), and the red auxiliary lines become the latitude lines of 20° north and 20° south.

[0112] Next, the shape from red line to red line is implemented inside the solid sphere, and the blue moon is made into a hole (Q). In this way, the meridian of 7.5° east longitude becomes a straight section S31, and the meridian of 7.5° west longitude becomes a straight section S32. In addition, the two red lines become the outer lines (34a, 34b) of the solid sphere (30).

[0113] In addition, the three-dimensional sphere (30) of the third embodiment can draw the arcs of up to two great circles at a time. Here, the two great circle arcs mean two straight sections (S31, S32), each of which has a maximum length of half of the great circle and a diameter that is the same. The angle between the two straight sections (S31, S32) can be 10°, 15°, or 30°, and in each case, it is easy to draw a meridian on the globe. For example, if the angle between the two straight sections (S31, S32) is 15°, it is easy to draw a meridian so that the angle between the meridians on the globe becomes 15°. That is, it is easy to draw meridians of 165° west longitude, 150° west longitude, … , 15° west longitude, 0° longitude, 15° east longitude, 30° east longitude, … , 180° east longitude.

[0114] In each case, the red auxiliary lines in Figures 8(a) and (b) are made sufficiently wide compared to the blue lines.

[0115] If the angle between the straight sections (S31, S32) is 10°, the two blue lines in Figures 8(a) and (b) represent the meridians of 5°W and 5°E, respectively, and the two red lines represent the latitudes of 18°N and 18°S, respectively. Instead of 18°, any other angle that provides a sufficiently wide angle with respect to the blue line may be selected.

[0116] If the angle between the straight sections (S31, S32) is 15°, the two blue lines in Figures 8(a) and (b) represent the meridians of 7.5°W and 7.5°E, respectively, and the two red lines represent the latitudes of 20°N and 20°S, respectively. Instead of 20°, any other angle that provides a sufficiently wide gap with respect to the blue line may be selected.

[0117] If the angle between the straight sections (S31, S32) is 30°, the two blue lines in Figures 8(a) and (b) represent the meridians of 15°W and 15°E, respectively, and the two red lines represent the latitudes of 28°N and 28°S, respectively. Instead of 28°, any other angle that provides a sufficiently wide angle with respect to the blue line may be selected.

[0118] When the two points are not the endpoints of the diameter, the arc of the great circle passing through the two points is drawn as follows.

[0119] Align two points on a straight section (S31 or S32) so that the solid sphere (30) and the sphere are in perfect contact. Next, draw a line along this straight section using a pen to draw an arc of a great circle. At this moment, the straight section (S31 or S32) drawn with the pen and the center of symmetry of the solid sphere (30) are different. That is, in Fig. 8(a), the center of symmetry of the sphere (30) is the meridian of longitude 0°, and the straight section (S31 or S32) is the meridian of longitude 7.5° west or east 7.5°. For this reason, one may doubt that the straight section drawn with the pen becomes an arc of a great circle, but since the width of the solid sphere (30) is sufficiently wide and the curvatures of the sphere and the solid sphere (30) match, the straight section (S31 or S32) of the solid sphere (30) becomes a straight section of the sphere.

[0120] By using the three-dimensional sphere (30) once, an arc of a great circle can be drawn up to half the length of the great circle. This is because a straight section (S31 or S32) is formed up to the through holes on both sides (35a, 35b).

[0121] Further detailed description follows what is mentioned in Fig. 1.

[0122] Additionally, the three-dimensional sphere (30) of the present invention is formed with a display section so as to measure the length of an arc of a great circle or the size of a central angle corresponding to the arc.

[0123] Figure 9 is a 3D drawing showing that a scale-shaped marking portion (37a, 37b) is formed on the surface of the drawing portion (31).

[0124] The display portion (37a) is formed to have equal intervals in the longitudinal direction of the straight section (S31), and the display portion (37b) is formed to have equal intervals in the longitudinal direction of the straight section (S32). At this time, the display portions (37a, 37b) are formed on the outer side of the three-dimensional spherical ruler (30). The shape may be any one of a protrusion, a groove, or a scale shape, and is not particularly limited in the present invention. However, the depth (or thickness) or size of the display portions (37a, 37b) is adjusted so as not to interfere with the display of the straight section (S31 or S32).

[0125] In addition, the number of display units (37a) and display units (37b) is not particularly limited in the present invention, and can be any number that can be displayed at a certain interval, and the numbers can be different from each other. For example, when the number of two display units (37a, 37b) is the same, it can be 5, 8, 11, and 17. If necessary, it can be formed with a different number. For example, as shown in FIG. 9, the number of one display unit (37a) can be 8, and the number of the other display unit (37b) can be 11. In this case, the interval between 8 display units means 20°, and when 11 display units are formed, the interval means 15°.

[0126] The sphere and solid sphere described above are used as tools for learning geometry by drawing arcs of great circles, arcs of length corresponding to specific angles, and equal points of arcs on a sphere.

[0127] As shown in (a) of Fig. 10, a sphere and a three-dimensional sphere are prepared, and as shown in Fig. 10(b), the three-dimensional sphere is tightly attached to the sphere, and then a great circle can be easily drawn using a pen. Additionally, the equal points of the arc can be indicated through the scale of the display unit, and an arc of a length corresponding to a specific angle can be indicated.

[0128] In this way, the geometry learning tool of the present invention can accurately and easily draw great circles on a sphere using a three-dimensional sphere ruler. This allows students to enhance their understanding of geometry classes by understanding spatial concepts, rather than flat surfaces.

[0129] a. Learning regular polyhedra

[0130] The geometry learning tool of the present invention can be used to draw regular polyhedra, including a regular octahedron, on a sphere.

[0131] Consider a sphere circumscribed by a regular octahedron. The octahedron's vertices lie on the sphere. Now, let's connect the vertices on the sphere. The curve connecting the vertices on the sphere is as follows.

[0132] When light is shone from the center of a regular octahedron, shadows of the octahedron's corners are cast on a sphere. The solid created from these shadows is a solid that represents the octahedron on a sphere.

[0133] This solid is called an octahedral sphere or spherical octahedron.

[0134] Figure 11 is a drawing showing how to display a great circle on a regular octahedral sphere using the geometric learning teaching aid of the present invention.

[0135] Considering the circumsphere of a regular octahedron A-BCDE-F as shown in Fig. 11(a), points A, B, C, D, E, and F all lie on a sphere. The centers of the octahedron and the circumsphere coincide and are denoted O. When a light source (bulb) shines at the center O, a shadow of the corner AB is cast on the sphere. In Fig. 11(b), the curve AB is the shadow of the line segment AB. Drawing the shadows of all corners in this way results in Fig. 11(c) and Fig. 11(d).

[0136] Consider the curve AB corresponding to the shadow in Fig. 11(b). Curve AB lies on a sphere and is contained within the plane OAB. Therefore, curve AB is a part of the circle, the cross-section created when the plane OAB cuts the sphere. Since it passes through the center O, it is a part of a great circle. In other words, curve AB is the arc AB of the great circle passing through the two points A and B. For the same reason, the shadow of every corner is an arc of the great circle.

[0137] Applying this method to the tetrahedron, cube, dodecahedron, and icosahedron, we can obtain the tetrahedral sphere, cubic sphere, dodecahedral sphere, and icosahedral sphere. These are also referred to by the following terms: spherical tetrahedron, spherical cube, spherical dodecahedron, and spherical icosahedron. That is, the tetrahedral sphere can be obtained by connecting the vertices of the tetrahedron with arcs of great circles on the circumscribed sphere of the regular tetrahedron, and the cubic sphere can be obtained by connecting the vertices of the cube with arcs of great circles on the circumscribed sphere of the regular cube. You can do it in the same way for a regular dodecahedral sphere or an icosahedral sphere.

[0138] Figure 12 shows a representation of a regular polyhedral sphere using a three-dimensional sphere of the present invention on a transparent sphere, showing (a) a regular tetrahedral sphere, (b) a regular hexahedral sphere, (c) a regular octahedral sphere, (d) a regular dodecahedral sphere, and (e) a regular icosahedral sphere. The three-dimensional sphere used here can be any of the three-dimensional spheres described above.

[0139] As a result, in implementing a regular polyhedral sphere, the arc of a great circle connecting two vertices of the regular polyhedron can be easily drawn using a three-dimensional sphere ruler according to the present invention.

[0140] b. Learning semi-regular polyhedra

[0141] Unlike regular polyhedra, semi-regular polyhedra are convex polyhedra that are composed of only two or more regular polygons and have the same configuration of faces at each vertex.

[0142] For example, we can draw a cuboctahedron, a sphere that embodies the cuboctahedron on a sphere. The cuboctahedron is a quasi-regular polyhedron that can be created by connecting the midpoints of the corners of a cube or octahedron. Here, it is created by connecting the midpoints of the corners of a regular octahedron.

[0143] Figure 13 is a drawing showing a cuboctahedral sphere using the geometric learning teaching aid of the present invention.

[0144] As shown in Figure 13, if we draw a regular octahedron A-BCDE-F and its circumscribed sphere, twelve great circle arcs corresponding to the shadows of the twelve corners of the octahedron are drawn. If we select the midpoint of each great circle arc, there are twelve in total, and these twelve points become the vertices of the cuboctahedron.

[0145] In the regular octahedron A-BCDE-F, if the midpoints of each edge are G1, G2, G3, G4, H1, H2, H3, H4, I1, I2, I3, and I4 as shown in Fig. 13(a) and Fig. 13(b), then these 12 points are the vertices of the cuboctahedron. If the center of the circumscribed sphere of the regular octahedron A-BCDE-F is O, then the center of the cuboctahedron above is also O. If the line segment OA = 1, then the line segment OG1 = √2 / 2, so the distances of the 12 points G1, G2, G3, G4, H1, H2, H3, H4, I1, I2, I3, and I4 from the point O are all √2 / 2.

[0146] Now, as shown in Fig. 13(c) and Fig. 13(d), if the 12 points are enlarged by √2 times with respect to the center O and are respectively G1', G2', G3', G4', H1', ​​H2', H3', H4', I1', I2', I3', and I4', then all 12 enlarged points will lie on the sphere O. In addition, point G1' is the midpoint of the arc AB of the great circle, point G2' is the midpoint of the arc AC of the great circle, …, point I4' is the midpoint of the arc FE of the great circle.

[0147] When the twelve vertices of a cuboctahedron are connected by arcs of great circles, a cuboctahedral sphere is completed, as shown in Figures 13(e) and 13(f). The points marked separately on the sphere are the vertices of the regular octahedron.

[0148] As a result, the arc of the great circle connecting the vertices in implementing a semi-regular polyhedral sphere can be easily drawn using a three-dimensional sphere ruler according to the present invention.

[0149] c. Earth globe production class

[0150] Social studies and geography textbooks often feature globes. Drawing the globe's meridians and latitudes on a practice ball, such as a ball pit, and marking oceans and land areas, or marking each country's city with its longitude and latitude, is a very useful way to understand the Earth's structure and world geography.

[0151] Figure 14 is a drawing showing the meridians of the Earth using the geometric learning teaching aid of the present invention.

[0152] Most spheres are formed by joining two hemispheres. That is, as shown in Figure 14(a), the North and South Poles and the equator are marked. Also, meridians and parallels are marked.

[0153] First, the meridian must be drawn as an arc of a great circle with the North and South Poles as its two ends. The arc of the great circle can be easily drawn using the three-dimensional sphere of the present invention.

[0154] Fig. 14(b) is a photograph of longitude lines drawn at 30° intervals on a ball pit. Let's mark the longitude lines. Let the intersection of the prime meridian and the equator be G, and the North and South Poles be N and S. Let's mark the longitude lines of the Northern Hemisphere. As shown in Fig. 14(c), the arc NG of the great circle is 1 / 4 of the length of the great circle. As shown in Fig. 14(c), on the arc GN of the great circle, select a point P such that [arc GP = (x / 90) * arc GN], fix one end of a compass at point N, and start the other end from point P and turn the compass, a circle will be made on the surface of the ball. This circle is the longitude line at latitude x' north.

[0155] As a result, the arc of a great circle can be easily drawn using a three-dimensional sphere according to the present invention in drawing the meridians of the Earth.

[0156] d. Spherical Design and Tessellation Class

[0157] Tessellation is the process of forming a repeating pattern on a sphere by representing lines or surfaces, such as curves, based on their vertices.

[0158] The lines used in the design of tessellations include straight lines (arcs of great circles), circular arcs (arcs of circles other than great circles), and other curves.

[0159] Figure 15 is a drawing showing a spherical design and tessellation using the geometric learning teaching aid of the present invention.

[0160] Figures 15(a) and 15(b) are pool balls designed only with straight lines (arcs of great circles), and Figure 15(c) is a pool ball designed with straight lines (arcs of great circles) and other curved lines.

[0161] The same applies to spherical tessellations. Straight lines (arcs of great circles) are among the basic lines used in tessellations. Figures 15(d) and 15(e) show photographs of white puddle balls tessellated with straight lines.

[0162] Sports balls, such as soccer balls and volleyballs, feature a variety of patterns. For example, through spherical design classes, students can create various balls used in ball sports, such as the official World Cup ball, the Telstar, Teamgeist, Jabulani, Brazuca, and Al Rihla.

[0163] When designing or tessellating a sphere in this way, the arc of a great circle can be easily drawn using the three-dimensional sphere of the present invention.

[0164] e. Spherical Geometry Class

[0165] In spherical geometry, which studies geometry on a sphere, the basic elements are points and great circles.

[0166] Figure 16 is a drawing showing a city of a straight line, a spherical triangle, and a spherical moon using the geometry learning teaching aid of the present invention.

[0167] As shown in Fig. 16(a), the shortest path connecting two points on a sphere is when the two points are connected by a great circle. In this sense, a great circle is called a straight line. In spherical geometry, a triangle formed by three points on a sphere, i.e. a spherical triangle, is defined as a triangle that connects each of the two points by an arc of a great circle. In the spherical triangle ABC of Fig. 16(b), arcs AB, BC, and CA are all arcs of a great circle. Fig. 16(c) is a spherical lune formed by two anticircles (half of a great circle). Here, the two points A and B are opposite endpoints of the diameter of the sphere.

[0168] In this way, in a class to help understand spherical geometry, the arc of a great circle can be easily drawn using the three-dimensional sphere of the present invention.

[0169] In order to draw a shape on a sphere, it can be said that drawing a great circle is the starting point, and the three-dimensional sphere ruler according to the present invention can accurately draw a great circle on a sphere, so that it can be applied to various geometric learning tools to easily teach geometry to learners.

[0170] (Explanation of symbols)

[0171] 10, 20, 30: Stereoscopic sphere

[0172] 11, 21, 31: Institutional Department

[0173] 13a, 13b, 23a, 23b, 33a, 33b: Support

[0174] 14a, 14b, 24a, 24b, 34a, 34b: Outline

[0175] 15a, 15b, 25a, 25b, 35a, 35b: Through holes

[0176] 17a, 17b, 27a, 27b, 37a, 37b: Display

[0177] 38a, 38b: Medial border

[0178] H1, H2: slits

[0179] Q: Public

[0180] S11, S21, S22, S23, S31, S32: Straight sections

[0181] The present invention can be applied to a geometry learning tool for learners.

Claims

1. A three-dimensional sphere having a semicircular curved shape that wraps around and fits a part of a sphere, A geometric learning tool, wherein the above three-dimensional sphere has one or more straight line segments that can draw a great circle on the spherical surface of the sphere.

2. In paragraph 1, The above geometry learning aid is a geometry learning aid having a sphere.

3. In paragraph 1, The above three-dimensional sphere is a geometry learning tool in which the inner surface has a radius of curvature equal to the radius of curvature of the outer surface of the sphere.

4. In paragraph 1, A geometric learning tool comprising a band-shaped drawing part extending in the longitudinal direction and a support part symmetrically positioned at both ends of the drawing part, and having through holes formed at positions corresponding to both poles of the sphere.

5. In paragraph 4, A geometry learning tool in which the above-mentioned system and support part are formed as one piece.

6. In paragraph 4, A geometry learning tool in which the above-mentioned system and the through hole are in the same straight line and form a straight section.

7. In paragraph 4, The above-mentioned system is a band-shaped structure extending in the longitudinal direction and having slits formed in the same direction. A geometry learning tool in which the above slit and through hole are connected to form a single straight section.

8. In paragraph 4, A geometric learning tool in which the outline and through holes of each drawing section that does not touch the above slit are connected to form three straight sections.

9. In paragraph 4, The above-mentioned system is a band-shaped structure that extends in the longitudinal direction and has a slit-shaped opening formed in the same direction. A geometric learning tool in which the inner boundary line and the through hole of each of the parts in contact with the above-mentioned opening are connected to form two straight sections.

10. In paragraph 1, A geometric learning tool in which a marking section for marking scales is formed at regular intervals on at least one portion of the surface or side of the above-mentioned drawing section.

Images