WO2011004735A1 - Puzzle en trois dimensions - Google Patents

Puzzle en trois dimensions Download PDF

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Publication number
WO2011004735A1
WO2011004735A1 PCT/JP2010/061065 JP2010061065W WO2011004735A1 WO 2011004735 A1 WO2011004735 A1 WO 2011004735A1 JP 2010061065 W JP2010061065 W JP 2010061065W WO 2011004735 A1 WO2011004735 A1 WO 2011004735A1
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WO
WIPO (PCT)
Prior art keywords
convex
polyhedron
polyhedrons
tetradron
space
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Application number
PCT/JP2010/061065
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English (en)
Japanese (ja)
Inventor
仁 秋山
郁郎 佐藤
憲久 田宮
Original Assignee
学校法人 東海大学
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Application filed by 学校法人 東海大学 filed Critical 学校法人 東海大学
Publication of WO2011004735A1 publication Critical patent/WO2011004735A1/fr

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    • A—HUMAN NECESSITIES
    • A63—SPORTS; GAMES; AMUSEMENTS
    • A63F—CARD, BOARD, OR ROULETTE GAMES; INDOOR GAMES USING SMALL MOVING PLAYING BODIES; VIDEO GAMES; GAMES NOT OTHERWISE PROVIDED FOR
    • A63F9/00—Games not otherwise provided for
    • A63F9/06—Patience; Other games for self-amusement
    • A63F9/12—Three-dimensional jig-saw puzzles

Definitions

  • the present invention relates to a three-dimensional puzzle.
  • the present invention relates to a three-dimensional puzzle that can realize Fjordoff's space-filling solid.
  • This invention makes it a subject to implement
  • a three-dimensional puzzle composed of a plurality of first convex polyhedrons and a plurality of second convex polyhedrons that are mirrored with the first convex polyhedrons, and the first convex polyhedrons.
  • each of the second convex polyhedrons cannot be divided into two or more congruent convex polyhedrons, and the two first convex polyhedrons form a third convex polyhedron
  • the second convex polyhedron forms a fourth convex polyhedron
  • the third convex polyhedron and the fourth convex polyhedron form a space-filling solid of all Fedrovs by filling them inside.
  • Fedorov's space-filled three-dimensional puzzle is provided.
  • the third convex polyhedron and the fourth convex polyhedron may be mirror images of each other.
  • the Fedorov space-filled three-dimensional puzzle has a plurality of third convex polyhedrons that form a convex polyhedron similar to the third convex polyhedron, and a plurality of the fourth convex polyhedrons is the fourth convex polyhedron.
  • a convex polyhedron similar to that may be formed.
  • an unprecedented three-dimensional puzzle capable of realizing a Feodorov space-filling solid.
  • (A) shows tetrahedrons 103 and 105, and (b) shows their developed views.
  • (A) to (e) show five types of Fedoroff space-filled solids, (a) shows parallelepipeds, (b) shows rhomboid dodecahedrons, (c) shows oblique hexagonal columns, ( d) shows a rhomboid dodecahedron, (e) shows a truncated octahedron, and is a diagram of a three-dimensional puzzle according to an embodiment of the present invention.
  • FIG. 1 It is a schematic diagram of atoms ⁇ 11 and ⁇ ′13 according to an embodiment of the present invention.
  • FIG. 6 is a development view of ⁇ 11 and ⁇ ′13 according to an embodiment of the present invention.
  • (A) shows the cube 200 of unit length
  • (b) is a figure which shows the two triangular prisms 210 by the cutting of the cube 200.
  • FIG. (A) shows the triangular prism 210
  • (b) is a diagram in which the triangular prism 210 is cut into three congruent tetrahedrons.
  • FIG. 3 It is a figure which comprises an oblique hexagonal prism using tetradron, (a) shows a mode that the oblique triangular prism 310 is formed from the tetradron 103, (b) shows the mirror 330 of the oblique triangular prism 310, (c ) Shows the oblique hexagonal prism 300.
  • a method of constructing a rhomboid dodecahedron using tetradron is shown, (a) showing tetradrons 103 and 105, (b) showing a right-angled tetrahedron 410, and (c) showing a regular pyramid 450.
  • (A) shows how each face of the cube is covered with a regular quadrangular pyramid 450, and (b) shows a rhomboid dodecahedron 400.
  • (A) shows a state in which the rhomboid dodecahedron 400 is divided into a nine-sided body 490
  • (b) shows a state in which a rectangular solid of 2 ⁇ 2 ⁇ 2 ⁇ 2 ⁇ ⁇ 2 is inserted between two nine-sided bodies 490
  • (C) shows the long rhomboid dodecahedron 500.
  • the tetradron is a tetrahedral reptile, (a) shows a tetradron 113 that is three times larger, and (b) shows a tetradron 111 that is twice as large as the same shape. , (C) shows tetradron 103.
  • (A) shows regular tetragonal pyramid 610
  • (b) is a figure which shows a mode that the octahedron 600 is obtained from two regular tetragonal pyramids 610.
  • FIG. (A) shows the octahedron 600
  • (b) shows a state where a part of the upper regular pyramid 610 is removed
  • (c) shows the truncated octahedron 700.
  • (A) shows two ⁇ ′ 13, and (b) shows the tetradron 103. It is the figure which reposted ⁇ '13. It is the figure which showed a mode that (sigma) '13 was cut
  • (A) is a figure which shows the cube 200, (b) is the figure cut into six congruent square pyramids 1201.
  • convex polyhedrons P and Q when two convex polyhedrons P and Q are the same, or one of the convex polyhedrons is in a mirror relationship with the other convex polyhedron, the convex polyhedrons P and Q are congruent.
  • the parallel polyhedron is a convex polyhedron that fills a three-dimensional space with a parallel moving body that contacts the surfaces.
  • Minkowski has obtained the following conclusion about a general d-dimensional parallel polyhedron.
  • P is a d-dimensional parallel polyhedron, 1) P is symmetric with respect to the center 2) All planes of P are symmetric with respect to the center 3) Along one of the (d-2) planes on the two complementary planes
  • the projection of P is either a parallelogram or a hexagon that is symmetrical about the center.
  • FIGS. 3A to 3E are diagrams showing the space filling characteristics of each parallel polyhedron.
  • FIG. 5 is a development view of the atoms ⁇ 11 and ⁇ ′13. Details of these are disclosed in Japanese Patent Application No. 2008-268221.
  • the present inventors further examined the possibility of existence of an atom having a shape different from ⁇ 11 and ⁇ ′13.
  • the background of how the present inventors have found a new configuration for the minimum number of atoms filling all the Fedrov space filling solids will be described.
  • a three-dimensional puzzle having convex polyhedrons constituting the five kinds of Fedrov space filling solids will be described.
  • a cube 200 having a unit length is shown again in FIG.
  • the cube 200 is cut by a plane that passes through the center of the cube 200 and includes two opposite sides, two congruent triangular prisms 210 are formed (FIG. 6B).
  • FIG. 7B One of the triangular prisms 210 having apexes A, B, C, D, E and F is shown in FIG.
  • this triangular prism 210 is divided into three parts using planes BCD and CDE, three congruent tetrahedrons (FIG. 7B) are obtained in which all faces are right triangles.
  • the tetrahedron ABCD 101 and the CDEF 105 are identical, whereas the tetrahedron BCDE 103 is a reflection of both of the two described above, and therefore these tetrahedrons are congruent. Both the tetrahedron such as ABCD101 and its mirror are called tetradron. Let us show that Tetradron is another atom for the assembly trap.
  • FIG. 8A The construction of a diagonal hexagonal column using a congruent copy of a congruent tetradron is shown in FIG.
  • FIG. 8B shows the mirror 330 of the oblique triangular prism 310, which can be configured by the same adhesion of the tetradron 105, which is the mirror described above.
  • the diagonal hexagonal prisms 300 and 330 are bonded together to form a diagonal hexagonal prism 300 (FIG. 8C).
  • a pair of tetradrons 103 and 105 one of which is a mirror of the other, has a right angle of unit height that forms a bottom surface whose hypotenuse is a right-angled isosceles triangle with a unit length of 2 units. It joins so that the tetrahedron 410 may be formed (FIG.9 (b)).
  • a regular quadrangular pyramid 450 shown in FIG. 9C is obtained by combining four right-angled tetrahedrons 410.
  • the regular quadrangular pyramid 450 is centrally symmetric and has a unit height.
  • the rhomboid dodecahedron can be configured by first dividing the rhombus dodecahedron 400 of FIG. 10 (b) into two congruent octahedrons 490 as shown in FIG. 11 (a).
  • a rectangular solid 510 of 2 ⁇ 2 ⁇ 2 ⁇ 2 ⁇ ⁇ 2 is inserted between two ninehedrons 490 to obtain a long rhomboid dodecahedron 500 as shown in FIG.
  • a 2 ⁇ 2 ⁇ 2 ⁇ 2 ⁇ k ⁇ 2 rectangular parallelepiped of any positive integer k can be inserted, that is, the dodecahedron can be extended to an arbitrary length.
  • a tetradron is a tetrahedral reptile, that is, formed by repetition of its own shape. For example, if eight tetradrons consisting of two identical tetradrons 103 and three sets of mirrors (three each of 103 and 105) are used, the isomorphic shape is twice as large as the original tetradron. Tetradron 111 can be formed (FIG. 12B). Similarly, the tetradron 113 having three times the size can be constituted by nine tetradrons 103 and nine sets of mirrors (9 each of 103 and 105) (FIG. 12A).
  • a regular quadrangular pyramid 610 that is three times larger than the regular quadrangular pyramid 450 shown in FIG.
  • two such regular tetragonal pyramids 610 are bonded to obtain an octahedron 600.
  • FIG. 14 (a) shows the same octahedron 600 composed of tetradrone.
  • 1/3 unit is removed from a part and apex of the upper regular quadrangular pyramid 610.
  • a truncated octahedron 700 of FIG. 14C is obtained.
  • ⁇ be a polyhedron containing e ′.
  • ⁇ b′c′a ′ is cut exactly into two surfaces. Further dividing into three cases determined by the sides of ⁇ b′c′a ′ where ⁇ intersects.
  • the first case means that ⁇ cannot be congruent with ⁇ because ⁇ cannot include any side with a length of 3 ⁇ 2. Therefore, consider the second case. It can be seen that both ⁇ and ⁇ include one pentagonal surface different from the previous cut surface. One of the diagonals of the ⁇ pentagonal surface has a length of 3 ⁇ 2, while the ⁇ pentagonal surface cannot contain any diagonal of 3 ⁇ 2 length.
  • ⁇ intersects sides b′c ′ and b′a ′ (FIG. 19).
  • the hexahedron ⁇ includes a side b′e ′ of length 4 as a side of a pentagonal surface, but ⁇ does not have a pentagonal surface of the same length side. Therefore, ⁇ is not congruent with ⁇ .
  • each piece which comprises the three-dimensional puzzle of this invention which concerns on this Embodiment 3 is not necessarily limited to the example shown below.
  • the cubic three-dimensional puzzle according to the third embodiment is formed of, for example, four tetradron 103-shaped pieces and two mirror-shaped 105 shaped pieces. The That is, it can be configured using eight pieces having the shape of the atom ⁇ 11 and four pieces having the shape of the mirror ⁇ ′13 of the atom ⁇ 11.
  • the cubic three-dimensional puzzle according to the third embodiment of the present invention has two tetradron 103-shaped pieces and four mirror-shaped 105 shaped pieces because of the tetradron reflection. It can also be formed. That is, it is also possible to use four pieces having the shape of the atom ⁇ 11 and eight pieces having the shape of the mirror ⁇ ′13 of the atom ⁇ 11.
  • the oblique hexagonal three-dimensional puzzle of the present invention includes, for example, nine pieces of the tetradron 103 shape and nine pieces of the shape of the mirror 105. It is formed. That is, it can be configured using 18 pieces having the shape of the atom ⁇ 11 and 18 pieces having the shape of the mirror ⁇ ′13 of the atom ⁇ 11.
  • the truncated octahedron three-dimensional puzzle of the present invention according to Embodiment 3 has, for example, 192 tetradron 103-shaped pieces and 192 mirror-shaped 105 pieces. Formed with. That is, it can be configured using 384 pieces having the shape of the atom ⁇ 11 and 384 pieces having the shape of the mirror ⁇ ′13 of the atom ⁇ 11.
  • the rhombic dodecahedron three-dimensional puzzle according to the third embodiment includes, for example, 56 tetradron 103-shaped pieces and 40 mirror-shaped 105 shaped pieces. Formed with. That is, it can be configured using 112 pieces having the shape of the atom ⁇ 11 and 80 pieces having the shape of the mirror ⁇ ′13 of the atom ⁇ 11.
  • the rhomboid dodecahedron includes a 2 ⁇ 2 ⁇ 2 cube. Further, as described above, since there are a plurality of combinations of cubic solid puzzles, the rhombic dodecahedron solid puzzle of the present invention according to the third embodiment is converted to a tetradron 103-shaped piece and its mirror 105. There are also a plurality of combinations formed of pieces of the shape.
  • the number of pieces in the shape of the atom ⁇ 11 and the shape of the mirror ⁇ ′13 of the atom ⁇ 11 necessary for forming the rhombic dodecahedron three-dimensional puzzle is also the shape of the tetradron 103 and its pieces It changes according to the number of pieces of the shape of the mirror 105.
  • the long rhomboid dodecahedron three-dimensional puzzle of the present invention has, for example, 80 pieces of tetradron 103 shape and 64 pieces of shape of the mirror 105 as described in detail above. Formed in pieces. That is, it can be configured using 160 pieces having the shape of the atom ⁇ 11 and 128 pieces having the shape of the mirror ⁇ ′13 of the atom ⁇ 11.
  • the long rhombus dodecahedron of the present invention according to the third embodiment is formed using a rhombus dodecahedron including a 2 ⁇ 2 ⁇ 2 cube, and 2 ⁇ 2 ⁇ 2 Include a square of ⁇ 2 ⁇ ⁇ 2. Further, since a rectangular solid of 2 ⁇ 2 ⁇ 2 ⁇ 2 ⁇ k ⁇ 2 of any positive integer k can be inserted, that is, the dodecahedron can be extended to an arbitrary length, a piece of tetradron 103 shape.
  • the number of pieces in the shape of the atom ⁇ 11 and the number of pieces in the shape of the mirror ⁇ ′13 of the atom ⁇ 11 necessary to form the long rhomboid dodecahedron solid puzzle of the present invention according to the third embodiment. Is also changed according to the number of pieces of the shape of the tetradron 103 and its shape of the mirror 105.
  • the three-dimensional puzzle according to the third embodiment of the present invention is a three-dimensional solid in the shape of a Fedrov space-filled solid consisting of five types: parallelepiped, oblique hexagonal prism, rhomboid dodecahedron, truncated octahedron, and rhomboid dodecahedron.
  • the puzzle can be assembled from a piece with the shape of the atom ⁇ 11 and a piece with the shape of the mirror ⁇ ′13 in a new combination.
  • theorem 2 can be explained visually, and the three-dimensional puzzle of the present invention according to this embodiment is also excellent as an educational material.
  • the tetradron which is a tetrahedron
  • the atom ⁇ which is a pentahedron
  • the atom ⁇ As an atom, it is also excellent as an educational material that can be mathematically compared with the atom ⁇ .

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Abstract

L'invention porte sur un puzzle en trois dimensions innovant, permettant de réaliser des solides de remplissage d'espace de Fyodorov. Il est proposé un puzzle en trois dimensions de remplissage d'espace de Fyodorov qui est un puzzle en trois dimensions comprenant une pluralité de premiers polyèdres convexes et une pluralité de deuxièmes polyèdres convexes qui présentent une relation de miroir avec les premiers polyèdres convexes, chacun des premiers polyèdres convexes et des deuxièmes polyèdres convexes ne pouvant pas être divisés en deux ou plusieurs polyèdres convexes congrus, et deux premiers polyèdres convexes formant un troisième polyèdre convexe, deux deuxièmes polyèdres convexes formant un quatrième polyèdre convexe, et le troisième polyèdre convexe et le quatrième polyèdre convexe pouvant être formés par remplissage de l'intérieur de tous les solides de remplissage d'espace de Fyodorov.
PCT/JP2010/061065 2009-07-07 2010-06-29 Puzzle en trois dimensions WO2011004735A1 (fr)

Applications Claiming Priority (2)

Application Number Priority Date Filing Date Title
JP2009-160732 2009-07-07
JP2009160732A JP2011015730A (ja) 2009-07-07 2009-07-07 立体パズル

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WO2011004735A1 true WO2011004735A1 (fr) 2011-01-13

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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3925959A (en) * 1967-09-22 1975-12-16 Gen Foods Corp Tetrahedral packaging means and method of making same
JPH078628A (ja) * 1993-06-14 1995-01-13 Richard Stuart Boserich 球体パズル
JPH09136365A (ja) * 1995-11-15 1997-05-27 Hajime Kawakami 中空構造体およびその製造方法

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US3925959A (en) * 1967-09-22 1975-12-16 Gen Foods Corp Tetrahedral packaging means and method of making same
JPH078628A (ja) * 1993-06-14 1995-01-13 Richard Stuart Boserich 球体パズル
JPH09136365A (ja) * 1995-11-15 1997-05-27 Hajime Kawakami 中空構造体およびその製造方法

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