アナウンスをうっかり忘れていたが，本年６月

　　Hiroshi NAKAGAWA「Wooden Polyhedra」

が刊行された．

　この小冊子は

　　中川宏「多面体木工」ＮＰＯ科学協力学際センター

からの抜粋英訳本であるが，小生がその内容について巻頭言を書いたので，ここに掲げる．

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　　Some treasures of Hiroshi Nakagawa with respect to his masterpieces

Ikuro SATO

Hiroshi NAKAGAWA is an expert craft-worker of wooden polyhedra and this booklet is an extract from his works, including the essence of his masterpieces.

He started his first step as a craft-worker in 2000 at his age of 42. He was soon interested in the magic and mystery of wooden polyhedra. As far as I know, he seems not to have a ‘teacher’ who had instructed him in wood-craft. He had to discover for himself most of the cutting techniques.

Fortunately, he skillfully established safety methods for making wooden polyhedra. Imagine how dangerous a rolling-saw is if the wood is not stable. Firm support is fundamental to any cut. Cutting direction is classified to chamfer and truncate. The former is to cut edges and the latter is to cut vertices. The chamfered cube is not only effective to stabilize wood, but also to develop the variety of wooden polyhedra. ‘Chamfering at first and truncation at second’ is a manner of his making polyhedra.

In this booklet, he introduces some polyhedra, including Platonic, Archimedean, Catalan, Fedorov solids and ultimately 4-dimensional polytopes. The article about the regular dodecahedron　is his favorite because twelve pentagons appear simultaneously as one chamfers the 12 edges of wood-cube to make the dodecahedron.

His interest is not limited in making of his works. He is interested in a surprisingly wide range of topics. I want to introduce an episode of the collaborative study of Hiroshi and J. AKIYAMA. In June 2008, an international congress was held in Moscow for celebrating the 100 years anniversary of the birth of the Russian mathematician L. S. Pontrjagin. In this congress a Japanese mathematician J. Akiyama made a lecture on the special pentahedron as an element constructing all of the Fedorov’s solids. It was a new and epoch-making theorem about parallelohedra.

In the next step, he planned to provide wooden polyhedra to both junior high-school and the supportive-school for visual handicapped so that the students can begin to understand and appreciate concrete geometry. He believes that the wooden polyhedra are useful for students to entertain, teach some mathematics and enrich their lives. The motivation of this kind is often a simple curiosity. But, in his case, it is usually a strong motive force and the trigger of his work.

In such a pleasant time, the chamfered cube, the regular dodecahedron and the special pentahedron (Pentadron) became three treasures of him. I hope this booklet will be a ‘treasure island’ in a sense for many craft-workers. At the end of this preface, I would like to express our deepest thanks to Hiroshi for helping in many ways to complete our works.

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　来月，京都大学数理解析研究所で行われる発表

　　「木工多面体の数理」

も概ねこの線に沿って話すつもりである．

木工多面体の数理：Geometry for wood-modeling of polyhedra

中川宏、積み木インテリアギャラリー：Hiroshi NAKAGAWA, Gallery of Wooden Polypedra

佐藤郁郎、宮城県立がんセンター：Ikuro SATO, Miyagi Cancer Center

Hiroshi NAKAGAWA is an expert craft-worker of wooden-polyhedra. He planned to make wooden-polyhedron concrete so that students can begin to understand, entertain, inform and teach some mathematics. After introducing fundamentals of wood-modeling, I’m going to report a few results on tessellations of the space;

(a) with a special pentahedron only,

(b) with copies of cube, regular dodecahedron and Johnson 91.

　中川宏さんは山口県在住の木工職人であるが，多面体の申し子といってよい人である．偶然できた切稜立方体に関心を抱き，それ以降，木工多面体にすっかり心を奪われた．

　木工製作法の基本は「切頂法」と「切稜法」である．たとえば折り紙で多面体を作る場合とは違って，多面体の木工製作では安定した足場が必要となる．頂点立ちや辺立ちさせた姿勢で切削することはできす，面でしっかり支えなければならない．しっかりした足場を確保しつつ切頂と切稜の順番をうまく組み合わせること，そして切頂と切稜で削る深さをうまく調節することによってこれらを多面体を作り出す，そこに木工の難しさと面白さがある．

　これまで中川さんが製作した木工作品には，分子模型や空間充填模型，古代中国・朝鮮サイコロのレプリカ，数学マジックを体現するための各種玩具，４次元正１２０胞体の３次元空間への投影模型などがあげられる．中川さんの切稜立方体を組み上げた積み木は分子模型を形成し，刺激的な作品となった。

　また，空間充填に関する以下の２つの結果を紹介する.

(1) ５種類ある平行多面体を作ることができる単独空間充填五面体が存在する。この多面体は，pentadronと命名された．pentadronは数学的な意味のみならず，面心立方格子と対心立方格子をつなぐ多面体として物理的に重要な意味をもっている多面体である．

(2)ジョンソン・ザルガラー多面体J91を介することによって立方体と正１２面体による空間充填が可能となる．正１２面体が加わった上述の空間充填形は現在までに知られている空間充填形とは異なる．見方を変えれば，この空間充填形は，空間の分割において，すべての連結切断面が正多角形（この場合，正三角形，正方形，正五角形）であるような非自明な分割の存在を示唆していることになる．

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