WOODEN POLYHEDRA 30




正多面体5種類、準正多面体16種類、準正多面体の双対6種類と、平行多面体から六角柱・長菱形12面体、
に切稜立方体を加えた木製多面体のセットです。

fig.1fig.2
fig.3fig.4
fig.5fig.6
fig.7fig.8
fig.9fig.10
fig.11fig.12
fig.13fig.14
fig.15fig.16
fig.17fig.18
fig.19fig.20
fig.21fig.22
fig.23fig.24
fig.25fig.26
fig.27fig.28

切稜立方体は立方体からさまざまな多面体を木工製作するさいに、しばしば原型となる立体です。
(The chamfered cube often appears to be a prototype in the hand-made modelling from the wooden cube to various polyhedra.)



Wooden solid wonder-world of Hiroshi NAKAGAWA
Ikuro SATO



Hiroshi NAKAGAWA is a special carpenter skillful for making polyhedra of wood. He planned to make polyhedron concrete so that students can begin to understand and appreciate mathematics.

The wooden polyhedra are composed of 30 pieces shown in figures, including 5 Platonic(Fig 1-5), 16 Archimedean(Fig 6-19), 6 Catalan(Fig 20-25), 2 Fedrov(Fig 26-27) and the chamfered cube(Fig 28).

About the chamfered cube(Fig 28), this is the most favorite solid of him, because the chamfered cube often appears to be a prototype in the hand-made modeling from the wood cube to various polyhedra.

He hopes and believes that the wooden polyhedra are useful for the students to entertain, inform and teach some mathematics. But, his interest is not limited in hand-made modeling of wooden polyhedra. He is interested in surprisingly wide topics. Especially, I want to introduce a recent episode of the collaborative study of NAKAGAWA and J. AKIYAMA.

In June 2008 an international congress was held in Moscow for celebrating the 100 years anniversary of birth of the Russian mathematician L. S. Pontrjagin. In this congress a Japanese mathematician J. Akiyama made a lecture on the set and the element number of parallelohedra, "New theorem about parallelohedra".

Fedorov's parallelopolyhedra are defined as polyhedra with the following conditions:

a. each face has a parallel counterpart,
b. each edge belong to a group of parallel edges,
c. repetition of the same polyhedra must fill the space to produce a translation symmetry only (neither rotation nor reflection).

These polyhedra are limited to the five members consisting of cube(Fig 2), truncated octahedron(Fig 8), rhombic dodecahedron(Fig 24), hexagonal prism(Fig 26) and elongated rhombic dodecahedron(Fig 27). These polyhedra are less popular than the Platonic bodies, but are considered equally important because they play a role of space packing.

Now, let us consider the problem "what is a space-tessellation producer?", i.e. what is an elementary body producing space filling bodies?". General proof of this problem is quite difficult, but simple examples are found as follows.

We can construct all of the parallelopolyhedra with one kind of element, where its mirror image is looked upon as the same one. Let the element be denoted by σ, then the cube is constructed by 96 elements of one kind (σ96), the hexagonal prism is σ144, the rhombic dodecahedron is σ192, the elongated rhombic dodecahedron is σ384 and the truncated octahedron is σ48.

fig.29

fig.30

Fig.29 The pentahedron σ is an atom building up all of the 5 types of Fedorov's polyhedra, whose development is shown in Fig.30.



November 1,2013

各種多面体のご注文、お問い合わせはこちらへどうぞ
ご予算におうじて製作いたします。お気軽にご相談ください。
Price list
WOODEN POLYHEDRA 5Platonic solids set(Fig 1-5) with tray ; $30US including shipping
=23euro including shipping=\2,750(トレー、送料込)
WOODEN POLYHEDRA 20(Platonic & Archmedean solids set in case including shipping);$300US=270euro=\30,000(箱入、送料込)
WOODEN POLYHEDRA 30(Platonic, Archmedean, Catalan & Fedrov solids set in case including shipping);$500US=460euro=\49,980(箱入、送料込)

One piece price($1usd=1.1euro=\115yen)
fig.1:正4面体tetrahedron=\400
fig.2:立方体cube=\100
fig.3:正8面体octahedron=\400
fig.4:正12面体dodecahedron=\500
fig.5:正20面体icosahedron=\600
fig.6:切頂4面体truncated tetrahedron=\500
fig.7:切頂立方体truncated cube=\500
fi fig.9:切頂12面体truncated dodecahedron=\1,000
fig.10:切頂20面体truncated icosahedron=\1,000
fig.11:立方8面体cub-octahedron=\500
fig.12:小菱形立方8面体rhomb-cub-octahedron=\800
fig.13:ミラーの立体Miller's solid=\800
fig.14:大菱形立方8面体great rhomb-cub-octahedron=\800
fig.15:ねじれ立方体snub cube=\1,200
fig.16:20・12面体icosi-dodecahedron=\1,000
fig.17:小菱形20・12面体rhomb-icosi-dodecahedron=\3,000
fig.18:大菱形20・12面体great rhomb-icosi-dodecahedron=\3,000
fig.19:ねじれ12面体snub dodecahedron=\5,000
fig.20:三方四面体triakis tetrahedron=\2,000
fig.21:三方八面体triakis octahedron=\2,000
fig.22:四方六面体tetrakis hexahedron=\2,000
fig.23:凧型24面体deltoidal icositetrahedron=\2,400
fig.24:菱形12面体rhombic dodecahedron=\1,000
fig.25:菱形30面体rhombic triacontahedron=\3,000
fig.26:六角柱hexagonal prism=\600
fig.27:長菱形12面体elongated rhombic dodecahedron=\800
fig.28:切稜立方体chamfered cube=\500
fig.29:ペンタドロンPentadron=\500
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