THREE DIMENSIONAL PUZZLE
A three-dimensional puzzle which forms a regular polyhedron and has not conventionally existed is realized. In addition, a three-dimensional puzzle which realizes a Fedrov space filling solid and has not conventionally existed is realized. According to the invention, a three-dimensional puzzle is provided having a regular polyhedron consisting of a plurality of convex polyhedrons which fill an interior of the regular polyhedron comprising the plurality of convex polyhedrons having a plurality of a pair of convex polyhedrons in a mirroring image relationship, wherein the plurality of convex polyhedrons are indivisible into two or more congruent shaped polyhedrons. In addition, the plurality of convex polyhedrons may be four convex polyhedrons and include three pairs of convex polyhedrons in a mirroring image relationship. Further the plurality of convex polyhedrons may be five convex polyhedrons and include four pairs of convex polyhedrons in a mirroring image relationship.
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This application is based upon and claims the benefit of priority from the prior Japanese Patent Application No. 2008-156057, filed on Jun. 14, 2008, Japanese Patent Application No. 2008-268221, filed on Oct. 17, 2008, Japanese Patent Application No. 2008-277198, filed on Oct. 28, 2008, and PCT Application No. PCT/JP2009/060763, filed on Jun. 12, 2009, the entire contents of which are incorporated herein by reference.
BACKGROUND OF THE INVENTION1. Technical Field
The present invention is related to a three-dimensional puzzle. In particular, the present invention is related to a regular polyhedron puzzle which includes a plurality of convex polyhedrons which fill an interior, and a three-dimensional puzzle which can realize a Fedorov space-filling three dimensional solid.
2. Description of the Related Art
A regular polyhedron puzzle is known in which the regular polyhedron is divided into several convex polyhedrons. The convex polyhedrons which form this conventional regular polyhedron puzzle do not have particular regularity or characteristics, and are appropriately divided in order to adjust the difficulty and complexity of the puzzle (e.g. Japan Laid Open Utility Model Publication No. S63-200867, Japan Laid Open Patent Publication No. S58-049168, Japan Registered Utility Model Application No. 3107739, International Laid Open Pamphlet No. 2006/075666 and Japan Laid Open Patent Publication No. H03-155891).
BRIEF SUMMARY OF THE INVENTIONThe present invention forms a regular polyhedron and realizes a three-dimensional puzzle which has not conventionally existed. In addition, it is possible to realize a Fedrov space filling solid and realize a three-dimensional puzzle which has not conventionally existed.
According to one embodiment of the present invention, a three-dimensional puzzle is provided including four types of convex polyhedrons from which a regular tetrahedron, a cube, a regular octahedron, a regular dodecahedron or a regular icosahedron are formed, wherein three types among the four types of convex polyhedrons each having a pair of convex polyhedrons in a mirroring image relationship, the four types of convex polyhedrons are indivisible into two or more congruent shaped polyhedrons, and the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are formed using only the four types of convex polyhedrons so that the interior of the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are filled.
In addition, according to one embodiment of the present invention a three-dimensional puzzle is provided including five types of convex polyhedrons from which a regular tetrahedron, a cube, a regular octahedron, a regular dodecahedron or a regular icosahedron are formed, wherein four types among the five types of convex polyhedrons each having a pair of convex polyhedrons in a mirroring image relationship, the five types of convex polyhedrons are indivisible into two or more congruent shaped polyhedrons, and the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are formed using only the five types of convex polyhedrons so that the interior of the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are filled.
According to one embodiment of the present invention, a Fedorov space-filling three-dimensional puzzle is provided including a plurality of first convex polyhedrons and a plurality of second convex polyhedrons which are in a mirroring relationship with the first convex polyhedrons from which the three-dimensional puzzle is formed, wherein the plurality of first convex polyhedrons and the plurality of second convex polyhedrons each are indivisible into two or more congruent shaped polyhedrons, and the plurality of first convex polyhedrons and the plurality of second convex polyhedrons form all the Fedorov space-filling three dimensional solids by filling the interior of all the Fedorov space-filling three dimensional solids.
The Fedorov space-filling three-dimensional puzzle may also include an elongated rhombic dodecahedron which is formed using the plurality of first convex polyhedrons and the plurality of second convex polyhedrons includes a truncated octahedron, a parallel hexahedron, a skewed hexagonal prism and a rhombic dodecahedron.
- 100 regular tetrahedron
- 101 prism
- 103 pentahedron
- 110 atom α
- 111 atom α′
- 121 tetrahedron LJKW
- 123 tetrahedron LHKW
- 124 mirror image of tetrahedron 123
- 151 atom ξ
- 152 atom ξ
- 200 cube
- 201 regular triangular pyramid
- 203 golden tetra (atom γ)
- 204 atom γ′
- 205 polyhedron except golden tetra from regular triangular pyramid
- 207 equihepta
- 209 mirror image of equihepta
- 210 atom β
- 211 atom β′
- 300 regular octahedron
- 301 quadrangular pyramid
- 400 regular icosahedron
- 500 regular dodecahedron
- 501 roof
- 510 atom δ
- 601 hexahedron
- 602 mirror image of hexahedron 601
- 610 atom ε
- 611 atom ε′
- 700 α2
- 701 quadrangular pyramid LHKWK′
- 800 atom θ
- 801 atom θ′
- 900 atom η
- 901 atom η′
- 1100 sphenoid
- 1101 σ
- 1103 σ′
- 1105 c-squadron
- 1200 cube
- 1201 quadrangular pyramid
- 1203 triangular pyramid (right tetrahedron)
- 1300 rhombic dodecahedron
- 1301 polyhedron which is a rhombic dodecahedron cut in half
- 1350 rhombic dodecahedron puzzle
- 1351 puzzle which is a rhombic dodecahedron cut in half
- 1353 a puzzle except half a cube from a puzzle which is a rhombic dodecahedron cut in half
- 1400 skewed hexagonal prism
- 1401 triangular prism
- 1403 mirror image symmetrical triangular prism
- 1420 skewed hexagonal prism
- 1450 skewed hexagonal prism puzzle
- 1500 elongated rhombic dodecahedron
- 1501 helmet-shaped polyhedron
- 1503 polyhedron cut away from helmet-shaped polyhedron 501
- 1505 skewed triangular prism
- 1507 mirror image of skewed triangular prism 505
- 1550 elongated rhombic dodecahedron puzzle
- 1551 helmet-shaped polyhedron puzzle
- 1553 puzzle cut away from helmet-shaped polyhedron puzzle 551
- 1600 truncated octahedron
- 1601 enneahedron (diamond)
- 1603 half of truncated octahedron
- 1650 truncated octahedron puzzle
- 1651 half of truncated octahedron puzzle
- 1700 cube
- 1701 hexahedron
- 1703 polyhedron
- 1705 cube with open hole
- 1707 polyhedron formed by cutting through cube with open hole
- 1709 polyhedron formed by cutting through half of cube
- 1750 cube puzzle
- 1751 puzzle formed by cutting through cube with open hole
- 1753 half of cube puzzle
When the interior of a regular polyhedron is filled perfectly by a plurality of convex polyhedrons, these convex polyhedrons can be called the component elements (here called “atoms”) of a regular polyhedron. In other words, this means the regular polyhedron can be divided into several atoms. Only a regular tetrahedron, a cube, a regular octahedron, a regular icosahedron and a regular dodecahedron exist as a regular polyhedron.
However, there are limitless methods for dividing a regular polyhedron into several convex polyhedrons. That is, there are limitless atoms for forming a regular polyhedron.
Thus, the inventors of the present invention keenly examined which atoms should be adopted in order to be able to fill all the regular polyhedrons with their atoms while reducing the number of different atoms. As a result, a shape of the minimum number of atoms for filling all the regular polyhedrons was discovered. The circumstances in which the inventors of the present invention discovered a shape of the minimum number of atoms for filling all the regular polyhedrons is explained below.
First, a step which excludes a self-evident atom is explained. For example, a regular tetrahedron is examined as a polyhedron. Let the center of the regular tetrahedron be the peak, and four triangles on the surface of the regular tetrahedron as the bottom surface and the regular tetrahedron is divided into four congruent triangular pyramids. Because the bottom surface of this triangular pyramid is a regular triangle, this triangular pyramid has threefold rotation symmetry and mirror image symmetry and these are further divided into six congruent triangular pyramids. Here, when the mirror image symmetrical convex parts are defined as the same atom, it is possible to perfectly fill the interior of regular tetrahedron using 24 atoms of single type.
It is possible to apply the same division method as the division method of the regular tetrahedron stated above to a regular polyhedron. That is, a cube and regular octahedron are filled by 48 atoms and a regular dodecahedron and regular icosahedron are filled by 120 atoms. However, each of these atoms is obvious and effective in filling a specific regular polyhedron. However, when each of these atoms is diverted to an atom of another regular polyhedron, it instantly becomes an atom with bad efficiency.
The condition A below is attached in order to remove this obvious atom.
[Condition A]It is possible to use any atom as at least two types of regular polyhedron.
Then, when an efficient atom is adopted for a specific polyhedron, it is necessary to create a method which can use this atom effectively in the filling of another regular polyhedron. Here, it is preferred that as many atoms as possible should be used to fill any one of the regular polyhedrons. However, a supplement is also required. This is because a cube is self-expanding, and when one filling method is discovered, it is possible to easily increase the number of atoms by 23, 32 etc. simply by lining it up down, left right. Consequently, a cube is examined restricted to a basic filling method which does not use the self expanding properties. In this way, filling all of the regular polyhedrons using as many atoms of as few varieties as possible while satisfying [condition A] becomes a problem.
Furthermore, defining a mirror image symmetrical convex polyhedron as the same variety of atom.
- (1) because a regular polyhedron has various symmetries it is predicted that a lot of mirror image symmetrical polyhedrons will be used, (2) because it is common sense to identify a pair of reverse diagrams in a tiling problem very similar to this filling problem
Five types of regular polyhedron are known, a regular tetrahedron, a regular hexahedron (cube), a regular octahedron, a regular dodecahedron and a regular icosahedron. In order to fill these regular polyhedrons with various atoms so as to satisfy [condition A], first, it is necessary to investigate the mutual relationships between the five types of regular polyhedron.
In the cube 200 as in
Next, as is shown in
Next, focusing on the triangular pyramid 201 shown in
[formula 1]
AQ: QB=AT:TC=AU:UF=1:τ (1)
Here, τ is a golden proportion, and as is common knowledge can be obtained by the formula (2):
[formula 2]
Then, as is shown in
[formula 3]
CR:RB=FS:SB=FV:VC=1:τ (3
When a heptahedron 207 is created in the center of
Next, the relationship established in the formula (4) below will be focused on.
[Formula 4]
QBS=RBQ=SBR=90° (4)
Four pairs of this heptahedron 207 and its mirror image symmetrical heptahedron 209 are created. By attaching (gluing) these together, a regular icosahedron 400 is formed as shown in
Next, a regular dodecahedron 500 is shown in
The regular tetrahedron 100, regular octahedron 300 and regular icosahedron 400 are obtained from dividing the cube 200 as a result of the above stated examination. In addition, the cube 200 is obtained from dividing the regular dodecahedron 500. This dividing satisfies the condition [condition A].
Next, it is examined how it is possible to fill the entire regular polyhedron while satisfying condition [condition A] with as few types of atoms as possible. The case is considered where the regular tetrahedron 100 is formed from the dash line regular tetrahedron itself shown in
From the examination above, it is clear that any of four types of convex polyhedron (six types if mirror images are included) are used in forming a regular polyhedron, namely
[1] regular tetrahedron 100 (used for regular tetrahedron 100, cube 200 and regular dodecahedron 500),
[2] heptahedron (equihepta) 207 (and its mirror image 208) (used in cube 100, regular octahedron 300 and regular icosahedron 400),
[3] tetrahedron (golden tetra) 203 (and its mirror image 204) (used in cube 100 and regular octahedron 300), and
[4] pentahedron (roof) 501 (used in regular dodecahedron 500).
If the four types of convex polyhedron are each divided into several congruent convex polyhedrons, the number of atoms which form the regular polyhedron increases. Obvious dividing is removed and examined. First, in the regular tetrahedron 100 in [1], the regular tetrahedron 100 which is extracted from
Here, as is shown in
[formula 5]
JL:LK:KL:LH:HL:LJ:JK:KH:JH=2:√2:√2:√2:√2:2:√2:√2:√6 (5)
Next, when the equihepta 207 in [2] is examined, the equihepta 207 shown again in
One atom which forms a regular polyhedron with the heptahedron 210 as β is defined below. The atom β 210, as shown in
Here, a, k, k′, b, c, d satisfy the following formulas (6) to (11)
(formula 6)
a=3−√5=2(2−τ) (6)
(formula 7)
k=1/(τ+2)=(5−√5)/10 (7)
(formula 8)
k′=1−k=(5+√5)/10 (8)
(formula 9)
b=2/√3 (9)
(formula 10)
C=(2/τ2)√(τ2−2k′+3k′2) (10)
(formula 11)
d=(2/τ2)√(4/3−4k′+4k′2) (11)
At this time, atom β 210 satisfies the conditions shown in formula (12) below.
Because a regular icosahedron is formed by twenty-four atoms β 210, it is possible to describe this as β24.
Next, when the golden tetra 203 in [3] is examined, it is impossible to divide the golden tetra 203 into a plurality of congruent convex polyhedrons. Thus, one atom which forms a regular polyhedron with the golden tetra 203 as γ is defined.
(formula 13)
CR:RC:CV:VC:CT:TC:RT:RV:VT =a:a:√3a:√3a:a:a:√2a:2a: √2(√5−1) (13)
Then, because a regular octahedron is formed by twenty-four β and γ, it is possible to describe this as β24γ24. In addition, because a cube in
Next, when the roof 501 in [4] is examined, while considering that the roof 501 has up down left and right symmetry, it is divided into two congruent pentahedrons 501 as is shown in
(formula 14)
BE:ED1:D1A:AD1:D1E:EB:BC1:C1B: C1E:C1D:BA:AE=2:√(4−τ):√(4−τ):√(4−τ): √(4−τ):2:2(τ−1):2(τ−1):2(τ−1): τ−1:2:2√2 (14)
A solution for forming a regular polyhedron by a minimum number of atoms α, β, γ and δ is obtained as described above. The number of each regular polyhedron using the atoms α, β, γ, δ can be expressed as:
However, a mirror image symmetrical atom is also shown by α, β, γ, δ. If a prime symbol (′) is attached to distinguish a mirror image symmetrical atom, then they can be expressed as:
From the above, it was discovered that the minimum number of atoms for forming a regular polyhedron is four types α, β, γ, δ.
Here, it is clear that the minimum number of atoms for forming a regular polyhedron can be expressed as theorem 1 under the definitions (1)-(4) below.
- (1) polyhedrons P and Q are “congruent” means either that P and Q are identical or that P and Q have a mirror image relationship.
- (2) a polyhedron is “indecomposable” means that P is indivisible into two or more congruent polyhedrons.
- (3) Let II be a group of polyhedrons P1, P2, . . . Pn.
- i.e. II={P1, P2, . . . Pn}
- Let E be a group of indecomposable polyhedrons e1, e2, . . . em,
- i.e. E={c1, c2, . . . cm}, ∀e1 is indecomposable, ci, and cj are non-congruent (i≠j)
- At this time, E is a group E (II) of the element (atom) of II means satisfying formula (15) below.
(formula 15)
(∀i<1≦i≦n), ai is a nonnegative integer. (15)
That is, whichever polyhedron Pi belongs to II, it is possible to divide into an indecomposable polyhedron belonging to E.
- (4) The element number (atom number) e (II) of II means the minimum number of an order among various element groups with respect to II.
- That is e (II)=min|E (II)|
- Let II1={regular polyhedron group}
- e (II1)≦4, e (II1)={α, β, γ, δ}
Here, a three-dimensional puzzle of the present invention related to the present embodiment which uses the minimum number of atoms α, β, γ, δ for forming all the regular polyhedrons will be explained in detail. The three-dimensional puzzle of the present invention related to the present embodiment creates a regular polyhedron using the minimum number of atoms α, β, γ, δ, and has the excellent effects of being able to visually explain the above stated theorem 1. Consequently, the three-dimensional puzzle of the present invention related to the present embodiment is excellent as an educational material for explaining the above stated theorem 1.
(Regular Tetrahedron Three-Dimensional Puzzle)As explained above, because the three-dimensional puzzle of the present invention related to the present embodiment can be formed by a minimum number of atoms (convex polyhedrons) which fill all the regular polyhedrons, the present invention demonstrated excellent effects of being able to form all the regular polyhedrons (regular tetrahedron, cube, regular octahedron, regular dodecahedron, regular icosahedron) even with few parts. In addition, it is possible to visually prove the theorem discovered by the inventors and the present invention can also be used as an excellent educational material for explaining the above stated theorem 1.
Second EmbodimentIn the second embodiment, an example of a three-dimensional puzzle which can form a cube and a regular dodecahedron without using the atom α of embodiment one is explained.
A cube which is different to the cube explained in the first embodiment is comprised of two types of convex polyhedron, namely, six roofs 501 (δ2), four hexahedrons 601 and four mirror image symmetrical hexahedrons 602. In addition, the regular dodecahedron 500 is comprised of two types of convex polyhedron, namely, twelve roofs 501 (δ2), four hexahedrons 601 and four mirror image symmetrical hexahedrons 602.
From the above investigation it is clear that the five types of convex polyhedrons are used in the structure of any two or more regular polyhedrons:
- [1] regular tetrahedron 100 (used in a regular tetrahedron 100 and cube 200).
- [2] equihepta 207 and its mirror image 209 (used in a cube 100, regular octahedron 300 and regular icosahedron 500).
- [3] golden tetra 203 and its mirror image 204 (used in a cube 200 and regular octahedron 300).
- [4] roof 501 (used in a regular dodecahedron 500).
- [5] hexahedron 601 and its mirror image 602 (used in a cube 200 and regular dodecahedron 500).
Here, because the convex polyhedrons [1] to [4] were explained in the first embodiment, an explanation is omitted here.
When the hexahedron 601 in [5] is examined, it has three rotational symmetry as is clear from the hexahedron in
Atom ε 610 is a pentahedron (convex polyhedron) which has edges D1E1, E1F1, F1G1, G1H1, H1I1, I1J1, J1K1, K1D1, D1F1, D1I1, D1J1 and F1I1 Atom ε satisfies the following formula (16)
(formula sixteen)
D1E1:EiF1:F1G1:G1H1:H1I1:I1J1: J1K1:K1D1:D1F1:D1I1:D1J1:F1I1=√3:2−τ:2−τ:2−τ:√(18−11τ): √(18−11τ):2−τ:√3:√(4−τ):2(τ−1): √(4−τ):τ−1 (16)
Because the cube 200 is comprised of twelve atoms δ 510 and twenty-four atoms ε (twelve atoms ε and twelve mirror image atoms ε′), it is expressed as δ12ε24 (δ12ε12ε′12) and because the regular dodecahedron 500 is comprised of twenty-four δ and twenty-four ε, it is expressed as δ24ε24 (δ24ε12ε′12).
By using the atoms α, β, γ, δ, ε, each individual regular polyhedron becomes:
However, the same symbols α, β, γ, δ, ε, also indicate mirror image symmetrical atoms. If the mirror image symmetrical atoms are attached with a prime symbol (′) then:
In the three-dimensional puzzle of the present invention related to the present embodiment, it is possible to form a cube and a regular dodecahedron without using atom α by using ε in addition to the minimum number of atoms α, β, γ, δ which form all the regular polyhedrons. Consequently, the three-dimensional puzzle of the present invention related to the present embodiment is created from regular polyhedrons using the atom ε in addition to the minimum number of atoms α, β, γ, δ, it is possible to visually explain the above stated theorem 1 and demonstrate excellent effects which can compare both by a transformation example using ε which is the fifth atom. Therefore, the three-dimensional of the present invention related to the present embodiment is also excellent as an educational material.
Third EmbodimentIn the present embodiment, an example of a three-dimensional puzzle which can form a regular tetrahedron 100, cube 200 and regular octahedron 300 without using the atom α of the first embodiment is explained.
(formula 17)
LK1:M1L1:L1M1:M1J:JM1:M1L:LL1,:LJ:L1J=2/√3:1 /√6:1/√6:√(2/3):√(2/3):√(2/3):2√3: √(3/2):√2:1/√2 (17)
In addition, with respect to the right side tetrahedron 123 in
In order to show this, first the atom γ 203 is focused on. This is a slightly long and triangular pyramid, however, as is shown in
Atom θ 800 is a tetrahedron (convex polyhedron) which has edges VO1, O1P1, P1O1, O1N1, N1O1, O1V, VP1, VN1 and N1P1 as is shown in
(formula 18)
e=√(2/5(9−4√5)) (18)
(formula 19)
VO1:O1P1:P1O1:O1N1:N1O1:O1V: VP1:VN1:N1P1=√3ak:e:e:ak:ak:√3ak√2(√5−2):2ak:√3e (19)
Atom η 900 is a pentahedron (convex polyhedron) which has edges O1N1, N1O1, O1P1, P1O1, O1C, CT, TC, CR, RC, CO1, N1P1, P1T, TR and N1R, as is shown in
(formula 20)
O1N1:N1O1:O1P1:P1O1:O1C:CT:TC:CR:RC: CO1:N1P1:P1T:TR:N1R=ak:ak:e:e;√3ak′:a:a:a:a:√3ak′: √3e:√2:2a:2ak′ (20)
By combining one of three types of atom, β 210, η 900 and θ 800, it is possible to make a quadrangular pyramid 701 shown in
By adding four atoms ξ 151 to the quadrangular pyramid 701, two atoms α 110, that is, the polyhedron 700 shown in
When a regular tetrahedron is formed by this method, the entire structure using the atom α 110 is replaced with this, and the regular tetrahedron 100, the cube 200, the regular octahedron 300 which are formed by the atoms β 210, ξ 151, η 900 and θ 801 each become:
In this way, the number of atoms which are used in the structure increases significantly, however, the type of atom only increases by one.
In the three-dimensional puzzle of the present invention related to embodiment 1, an example in which it is possible to make a structure with a minimum number of atoms α, β, γ, δ which form all the regular polyhedrons is explained. However, the three-dimensional puzzle of the present invention related to the present embodiment can form a regular tetrahedron, cube and regular dodecahedron without using the atom α. Consequently, because the three-dimensional puzzle of the present invention related to the present embodiment can form the atom α by combining different atoms and form a regular tetrahedron, cube and regular dodecahedron without using the atom α, it is possible to visually explain the above stated theorem 1 and demonstrate excellent effects which can compare both by a transformation example using ξ, θ, η which are the sixth to eighth atoms. Therefore, the three-dimensional puzzle of the present invention related to the present embodiment is also excellent as an educational material.
Fourth EmbodimentIn the fourth embodiment, an example is explained in which the three-dimensional puzzle which has the atoms explained in the first to third embodiments as structural components is applied to a Fedorov polyhedron or a Fedorov space filling solid. As stated previously, in a convex polyhedron, convex polyhedrons P and Q are congruent if either the two convex polyhedrons P and Q are the same or when they have a relationship where one of the convex polyhedrons is a mirror image of the other.
When a non-congruent convex polyhedron Pi is 1≦i≦n, then II={P1, P2, . . . , Pn}. If at least one element P which is included in Pi, is formed by face to face joining of congruent convex polyhedrons of convex polyhedron σ, then convex polyhedron σ is called an atom of II.
In addition, a parellelohedron is a convex polyhedron which tiles three-dimensional space using a face to face joining of its translates. Minkowski obtained the following results for a general d-dimensional parallelohedron.
(Minkowski Theorem A)
- If P is a d-dimensional parallelohedron, then
- 1) P is centrally symmetric,
- 2) All faces of P are centrally symmetric,
- 3) The projection of P along any of its (d-2) faces onto the complementary 2-plane is either a parallelogram or a centrally symmetric hexagon.
The number fd-1 (P) of faces of a d-parallelohedron P does not exceed 2(2d−1) and there is a parallelohedron P with fd-1=2(2d−1).
Dolbilin extended Minkowski's theorems for non-face to face tilings of space. There are also numerous studies on parallelohedra discussed by Alexandrov and Gruber.
In 1890, a Russian crystallographer, Evgraf Fedorov, established that there are exactly five types of parallelohedra, namely, parallelopiped (cube), rhombic dodecahedron, skewed hexagonal prism (parallel hexagonal prism), elongated rhombic dodecahedron and truncated octahedron shown in
The characteristics which are common to these space filling solids is not only that any one type of solid can fill space without any gaps, but also that any two solids can be overlapped by only translation of it as filling the space of each solid.
Let II1 be a group consisting of a cube, skewed hexagonal prism, rhombic dodecahedron, elongated rhombic dodecahedron and truncated octahedron. The inventors keenly examined which atom should be adopted for filling all the space filling solids included in II1 by the minimum number of atoms. As a result, the shape of the minimum number of atoms for filling all the space filling solids was discovered. Next, the circumstances in which the inventors discovered the shape of the minimum number of atoms for filling all the space filling solids is explained. In addition, in the present embodiment, a three-dimensional puzzle having convex polyhedrons which form five types of Fedrov space filling solids is explained.
(Cube and Rhombic Dodecahedron)The cube 1200 in
The quadrangular pyramid 1201 is further divided into four congruent triangular pyramids 1203 as shown in
In addition, as is shown in
As is shown in
It is known that the sphenoid 1100 has noteworthy characteristics which are called self-expanding. This is because when eight sphenoid 1100 are combined, the length of each edge expands to exactly twice that of a similar triangular pyramid which makes it easy to understand that the sphenoid is a space filling solid.
Moreover, when three sphenoid 1100 are combined it is possible to form a triangular prism 1401 with a regular triangle as a perpendicular cross section shown in
Next,
The sphenoid 1100 has another important property. As is shown in
By arranging these four c-squadrons in an appropriate way, it is possible to form an enneahedron (diamond) 1601 as shown in
From the above, it can be seen that the cube (parallel hexagram) 1200 can be formed by twenty-four right tetras 1203, the rhombic dodecahedron 1300 can be formed by forty-eight right tetras 1203, the skewed hexagonal prism 1400 can be formed by eighteen sphenoids 1100, the elongated rhombic dodecahedron 1500 can be formed by ninety-six tetras and the truncated octahedron can be formed by twenty-four c-squadrons 1105.
Next, an examination is made whether common constituent elements exist between the right tetra 1203, sphenoid 1100 and c-squadron 1105.
Here, one among the four c-squadrons 1105 shown in
In addition, as is shown in
It is not possible to divide the pentahedrons 1101 and 1103 into further congruent solids. The pentahedrons 1101 and 1103 are atoms of the five types of Fedrov space filling solids, and the pentahedron 1101 is defined as a and its mirror image 1103 is defined as σ′. A development including dimensions of the atom σ 1101 and atom σ′ 1103 is shown in
(formula 21)
ab, bc, ca, cf, fc, ca, ad, da, ab, be, eb, de, ef, fd=2:√2:√2:√6:√6:√2:√2:√2:2:4:4:3√2: 2√3:√6 (21)
By using the atom σ 1101 and atom σ′ 1103, it is clear that the cube (parallel hexahedron) 1200 can be formed by ninety-six atoms σ (forty-eight atoms σ 1101 and forty-eight atoms σ′ 1103), the rhombic dodecahedron 1300 can be formed by one hundred and ninety-two atoms σ (ninety-six atoms σ 1101 and ninety-six atoms σ′ 1103), the truncated octahedron 1600 can be formed by forty-eight atoms σ (twenty-four atoms σ 1101 and twenty-four atoms σ′ 1103), the skewed hexagonal prism 1400 can be formed by one hundred and forty-four atoms σ (seventy-two atoms σ 1101 and seventy-two atoms σ′ 1103) and the elongated rhombic dodecahedron 1500 can be formed by three-hundred and eighty-four atoms σ (one-hundred and ninety-two atoms σ 1101 one-hundred and ninety-two atoms σ′ 1103).
In addition, a cube is formed by ninety-six atoms σ 1101 and atoms σ′ 1103, however, it is possible to form a cube with as little as twenty-four atoms σ (twelve atoms δ 1101 and twelve atoms σ′ 1103).
From the above, it was discovered that the minimum number of atoms for forming a Fedrov space filling solid is one type of atom σ (and its mirror image σ′).
Here, it is clear that it is possible to express the minimum number of atoms which form the Fedrov space filling solids as theorem 2 under the definitions (1) to (4) below.
- (1) The polyhedrons P and Q are “congruent” means that P and Q are either exactly the same shape or are in a mirror image relationship.
- (2) A polyhedron is “indecomposable” means P are indivisible into two or more congruent polyhedrons.
- (3) Let II be a group of polyhedrons P1, P2, . . . Pn.
- i.e. II={P1, P2, . . . Pn}
- Let E be a group of indecomposable polyhedrons e1, e2, . . . em,
- i.e. E={e1, e2, . . . em}, ∀e1 is indecomposable, ei and ej are non-congruent (i≠j)
- At this time, E is a group E (II) of the element (atom) of II means satisfying formula (15) below.
That is, whichever polyhedron Pi belongs to II, it is possible to divide into an indecomposable polyhedron belonging to E.
- (4) The element number (atom number) e (II) of II means the minimum number of an order among various element groups with respect to II.
- That is e (II)=min|E (II)|
- Let II2={a group of Fedrov space filling solids}
- e (II2)≦1, E (II2)={σ}
Here, the three-dimensional puzzle of the present invention related to the present embodiment which uses a minimum number of atoms σ to form all the Fedrov space filling solids will be explained in detail. The three-dimensional puzzle of the present invention related to the present embodiment creates Fedrov space filling solids using a minimum number of atoms σ and demonstrates excellent effects of being able to visually explain the above stated theorem 2. Consequently, the three-dimensional puzzle of the present invention related to the present embodiment is excellent as an educational material for explaining the above stated theorem 2.
(Regular Tetrahedron Three-Dimensional Puzzle)The parallel hexagram three-dimensional puzzle of the present invention related to the present embodiment can be formed using forty-eight pieces which have a shape of an atom σ 1101 and forty-eight pieces which have a shape of the atom σ 1101 mirror image σ′ 1103 as explained above in detail.
(Skewed Hexagonal Prism Three-Dimensional Puzzle)The skewed hexagonal three-dimensional puzzle of the present invention related to the present embodiment can be formed using seventy-two pieces which have a shape of an atom σ 1101 and seventy-two which have a shape of the atom σ 1101 mirror image σ′ 1103 as explained above in detail.
(Truncated Octahedron Three-Dimensional Puzzle)The truncated octahedron three-dimensional puzzle of the present invention related to the present embodiment can be formed using twenty-four pieces which have a shape of an atom σ 1101 and twenty-four which have a shape of the atom σ 1101 mirror image σ′ 1103 as explained above in detail.
(Rhombic Dodecahedron Three-Dimensional Puzzle)The rhombic dodecahedron three-dimensional puzzle of the present invention related to the present embodiment can be formed using ninety-six pieces which have a shape of an atom σ 1101 and ninety-six which have a shape of the atom σ 1101 mirror image σ′ 1103 as explained above in detail.
(Elongated Rhombic Dodecahedron Three-Dimensional Puzzle)The elongated rhombic dodecahedron three-dimensional puzzle of the present invention related to the present embodiment can be formed using one hundred and ninety-two pieces which have a shape of an atom σ 1101 and one hundred and ninety-two which have a shape of the atom σ 1101 mirror image σ′ 1103 as explained above in detail.
The three-dimensional puzzle of the present invention related to the present embodiment which has the shape of a Fedrov space filling solid comprised of five types of polyhedron, namely a parallel hexagram, skewed hexagonal prism, truncated octahedron, rhombic dodecahedron and elongated rhombic dodecahedron, can be assembled from a piece which has a shape of one atom σ 1101 and a piece which has the shape of one atom σ′ 1103 mirror image. As a result, it is possible to visually explain the theorem 2 stated above and the three-dimensional puzzle of the present invention related to the present embodiment is excellent as an educational material.
Fifth Embodiment (Elongated Rhombic Dodecahedron Including a Space Filling Solid)The inventors discovered that the five types of Fedrov space filling solids each require a multiple of twenty-four c-squadrons, that is, a multiple of twenty-four atoms σ 1101 and σ′ 1103.
From the proof of the elongated rhombic dodecahedron in theorem 2 and
The truncated octahedron 1600 includes the cube 1200 by a special method. When atom σ 1101 and atom σ′ 1103 which are congruent pentahedrons shown in
This hexahedron 1701 has a face comprised of three triangles (one right isosceles triangle and two congruent right triangles), two quadrilaterals and one pentagon. This polyhedron is called a tripenquadron.
The polyhedron 1703 show in
When eight of these polyhedrons 1703 are glued along the right isosceles triangular faces, a cube 1705 is obtained having six holes at the center of each face as shown in
Finally, it is shown that a skewed hexagonal prism is included in an elongated rhombic dodecahedron in a particular way. The elongated rhombic dodecahedron 1500 comprised of the rhombic dodecahedron 1300 and helmet shaped polyhedron 1501 is again considered.
First, as is shown in
If the V-shaped polyhedron 1503 is glued to the lower half of the rhombic dodecahedron 1301, the skewed hexagonal prism 1420 shown in
The three-dimensional puzzle of the present invention related to the present embodiment which has the shape of a Fedrov space filling solid comprised of five types of polyhedron, namely a cube, skewed hexagonal prism, rhombic dodecahedron, truncated octahedron, and elongated rhombic dodecahedron, can be assembled from a piece which has a shape of one atom σ 1101 and a piece which has the shape of one atom σ′ 1103 of its mirror image. In addition, it is possible to visually explain the inclusion of the cube, skewed hexagonal prism, rhombic dodecahedron and truncated octahedron in the elongated rhombic dodecahedron and the three-dimensional puzzle of the present invention related to the present embodiment is excellent as an educational material.
(Truncated Octahedron Three-Dimensional Puzzle)The three-dimensional puzzle of the present invention related to the present embodiment which has the shape of a Fedrov space filling solid comprised of five types of polyhedron, namely a cube, skewed hexagonal prism, rhombic dodecahedron, truncated octahedron and elongated rhombic dodecahedron, can be assembled from a piece which has a shape of one atom σ 1101 and a piece which has the shape of one atom σ′ 1103 of its mirror image. In addition, it is possible to visually explain the theorem 2 stated above and the three-dimensional puzzle of the present invention related to the present embodiment is excellent as an educational material.
Because the three-dimensional puzzle of the present invention is formed with the minimum number of atoms (convex polyhedrons) for filing all of the regular polyhedrons, it is possible to form all of the regular polyhedrons (regular tetrahedron, cube, regular octahedron, regular dodecahedron and regular icosahedron) even if the number of parts is small. In addition, it is possible to visually prove the novel theorem discovered by the inventors and the present invention can be used as an excellent educational material for explaining this theorem.
According to the three-dimensional puzzle of the present invention, by using a different atom in addition to the minimum number of atoms for constructing all of the regular polyhedrons, it is possible to form a cube and regular dodecahedron without using one of the minimum number of atoms. In addition, it is possible to visually explain the novel theorem discovered by the inventors, compare both puzzles by a transformation example which uses a different atom, and the present invention can be used as an excellent educational material for explaining this theorem.
According to the three-dimensional puzzle of the present invention, even if one of the minimum number of atoms for constructing all of the regular polyhedrons is not used, it is possible to form a regular polyhedron, cube and regular dodecahedron by using a different atom. In addition, it is possible to visually explain the novel theorem discovered by the inventors, compare both puzzles by a transformation example which uses a different atom, and the present invention can be used as an excellent educational material for explaining this theorem.
According to the present invention, it is possible to realize a Fedorov space-filling solid and provide a three-dimensional puzzle which has not existed until now.
Claims
1. A three-dimensional puzzle comprising:
- four types of convex polyhedrons from which a regular tetrahedron, a cube, a regular octahedron, a regular dodecahedron or a regular icosahedron are formed;
- wherein
- three types among the four types of convex polyhedrons each having a pair of convex polyhedrons in a mirroring image relationship;
- the four type of convex polyhedrons are indivisible into two or more congruent shaped polyhedrons; and
- the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are formed using only the four types of convex polyhedrons so that the interior of the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are filled.
2. A three-dimensional puzzle comprising:
- five types of convex polyhedrons from which a regular tetrahedron, a cube, a regular octahedron, a regular dodecahedron or a regular icosahedron are formed;
- wherein
- four types among the five types of convex polyhedrons each having a pair of convex polyhedrons in a mirroring image relationship;
- the five type of convex polyhedrons are indivisible into two or more congruent shaped polyhedrons; and
- the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are formed using only the five types of convex polyhedrons so that the interior of the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron are filled.
Type: Application
Filed: Mar 19, 2010
Publication Date: Sep 9, 2010
Applicant: Tokai University Educational System ( Tokyo)
Inventors: Hitoshi Akiyama (Tokyo), Gisaku Nakamura ( Tokyo), Ikuro Sato (Sendai-shi), Norihasa Tamiya (Houfu-shi)
Application Number: 12/727,390
International Classification: A63F 9/12 (20060101);