從32面三角面多邊形到32面三角面多面體之摺黏

紙積木具有可拆開重覆使用的特性，不僅經濟、環保，而且可以讓所有的使用者在摺黏與拆開的過程中促進思考，啓發想像。近年來，隨著組合數學理論的引進，使得紙積木的建構更有系統，且產生更多元的發展。首屆沃爾夫數學獎得主、俄國數學家蓋爾芳德曾預言，組合數學與幾何學將是二十一世紀數學研究的前沿領域，由於個人電腦軟體的開發與進步，使得組合數學成為數學的主流分支。在日常生活中我們常會遇到組合數學的問題，例如大型的專案計劃中，怎樣安排各種人員的工作，以及各種工序之間的銜接，從而使整個工作時程儘可能縮短？還有城市中的交通規劃，哪些地方應該建橋樑？哪些地方該設單行道？凡此種種，皆涉及到組合數學。本論文之研究是以拿破崙三角面多邊形為例，探索32面三角面多邊形所有的合法摺黏模型，研究的第一個步驟是透過動態規劃演算，協助研究者完整搜尋32面三角面多邊形的可摺黏路徑。雖說動態規劃並不是尋求合法摺黏的唯一方法，任何人皆能隨意用拿破崙紙積木摺出模型，但是動態規劃卻是求解此問題的最佳方案。研究者運用動態規劃中的遞迴關係與列表式運算技巧，共搜尋得拿破崙三角面多邊形的可摺黏路徑652個，然後為這些路徑做了初步的篩選：在剔除了相同的路徑333個、同形的模型156個、不合法路徑17個後，只餘146個路徑。接著研究者依照146個路徑將拿破崙紙積木摺黏成模型後，又剔除同胚的模型58個、具有多重邊線的模型38個，最終得到50個合法且具獨特性的模型，而其中屬於對稱的模型則有15個。在整個研究的過程中，以動態規劃的演算最關鍵也最複雜，假若未來能夠運用計算機程式運算，將可大大的減少研究的時間與研究的複雜度。

關鍵字

三角面多邊形；拿破崙紙積木；三角面多面體；動態規劃；黏接樹；秩；簡單圖形；流形與非流形；同胚；同形

並列摘要

Paper blocks are reusable and foldable. Their unique characteristic is not only economical and recyclable, but also promotes user’s critical thinking or imagination by manipulating them. Recently, a widespread theory of combinatorics makes paper blocks developing models more constructive and various. Russian mathematician I.M. Gelfand, the 1978 winner of the Wolf Prize in Mathematics, said that combinatorial geometry makes our mathematical research the mainstream in the 21st century. However, a well-developed and a great deal amount of improvement of computer software, combinatorics became not so unpopular any more. In our daily life we may utilize the concepts of combinatorics, such as an official proposal that relates to how to assign jobs to different workers? How to set tasks in order? How to shorten the working hours for the whole program? How to design the effective system of transportation in a city? Where should we build a new bridge? Where should we set the single way of road? These are just parts of combinatorics. The research is based on the idea of Napoleon polyiamonds, furthermore exploring diverse models of 32-polyiamonds. The first step of research is to help researchers in search for variously foldable ways of 32-polyiamonds by dynamic programming approach. Although dynamic programming is not the only way to fold the model of Napoleon polyiamonds, it might be one of the best solutions. Researcher utilized the recurrence relation of dynamic programming approach and tabularized the computation to find a total of 652 ways to fold Napoleon polyiamonds. As a result of initial screening, we found out that there exist 333 identical legal paths, 156 isomorphic models, and 17 illegal paths, therefore a total of 146 official models left. By edge-to-edge gluing the 146 models of Napoleon paper blocks, researcher deducted 58 isomorphic models and 38 multi-edge models. Finally researcher obtained a total of 50 official models and 15 symmetrical models.

並列關鍵字

Polyiamonds ； Napoleom paper building block ； Deltahedra ； Dynamic Programming ； Gluing tree ； degree ； Simple Graph ； manifold and non-Manifold ； Homeomorphic ； Isomorphic Model

參考文獻

[4]Chen, T. and Zhang, Z. (2007), Simulation Studies of the Self-Assembly of Cone-Shaped Particles, Langmuir, 2007, 23 (12), pp 6598–6605

[5]Eades, P., Foulds, L. and Giffin, J. (1982), "An Efficient Heuristic for Identifying Maximum Weight Planar Subgraph." In Lecture Notes in Mathematics No. 952 (Combinatorial Mathematics IX). Springer-Verlag, Berlin.

[6]Foulds, L. R. and Robinson, D. F. (1978) “Graph Theoretic Heuristic for the Plant Layout Problems, Int. J. Prod. Res. 16, P27-37.

[8]Gautier, R., Halet, J. F., and Saillard, J. Y., (2009), Computational Methods: Transition Metal Clusters, Encyclopedia of Inorganic Chemistry, 2009

[9]Janner, A. (2006), Crystallographic structural organization of human rhinovirus serotype 16, 14, 3, 2 and 1A, Acta Cryst. (2006). A62, 270-286

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從32面三角面多邊形到32面三角面多面體之摺黏

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