基於阿基米德多面體所建構出球體與圓盤模型之組合性探討

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研究生：	劉家良 Liu, Jia-Liang
論文名稱：	基於阿基米德多面體所建構出球體與圓盤模型之組合性探討 A Combinatorial Study of Models of Spheres and Disks based on the Small Rhombicosidodecahedron, Truncated Dodecahedron and Truncated Icosahedron
指導教授：	全任重 Chuan, Jen-Chung 黃皓瑋 Huang, Hao-Wei
口試委員:	李明恭 Lee, Ming-Gong 李華倫 Li, Hua-Lun
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2022
畢業學年度：	110
語文別：	英文
論文頁數：	36
中文關鍵詞：	阿基米德多面體、卡塔蘭多面體、組合學
外文關鍵詞：	Archimedean solid, Catalan solid, Combinatorics
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阿基米德多面體的研究始於約2300年前的阿基米德、阿波羅尼奧斯以及帕普斯等數學家。阿基米德多面體共有13種，本文我們不討論扭棱立方體和扭棱十二面體。在11種阿基米德多面體當中，僅小斜方截半二十面體、截角二十面體、截角十二面體恰好擁有60個頂點，下面將這三種多面體合稱為60頂點多面體。本篇論文主要探討以下之組合性:
1. 基於60頂點多面體的頂點所建構之球體模型。
2.
(a)基於60頂點多面體的邊所建構之圓盤模型與
(b)基於60頂點多面體的面之內切圓所建構之圓盤模型。

The Archimedean solids were studied by Apollonius, Pappus and Archimedes 2300 years ago. Among the 11 Archimedean solids (excluding snub cube and snub dodecahedron) only Small Rhombicosidodecahedron, Truncated Dodecahedron, and Truncated Icosahedron have 60 vertices. In what follows these 3 polyhedra are collectively called 60-vertex polyhedra. The purpose of this paper is to study the combinatorial properties of:
1. 60-Sphere Models based on the 60-vertex polyhedra.
2. Disk Models based on
(a) the edges of the 60-vertex polyhedra and
(b) the incircles of all faces of the 60-vertex polyhedra.

CONTENTS
摘要...................................................i
Abstract..............................................ii
誌謝..................................................iii
Contents..............................................iv
List of Figures........................................v
Introduction........................................1
1 Research settings..................................1
2 Archimedean Solids and Catalan Solids..............1
Spheres.............................................3
1 Small Rhombicosidodecahedron.......................3
1.1 Patterns on the 60-Sphere Model..................4
2 Truncated Dodecahedron............................13
2.1 Patterns on the 60-Sphere Model.................14
3 Truncated Icosahedron.............................19
3.1 Patterns on the 60-Sphere Model.................20
4 Pattern of inverted spheres.......................25
4.1 Inversion with respect to a Sphere..............25
4.2 Animation of 60-Sphere Model....................25
Disks..............................................28
1 Small Rhombicosidodecahedron......................28
1.1 Disks based on edges............................28
1.2 Disks based on incircles of faces...............29
2 Truncated Dodecahedron............................30
2.1 Disks based on edges............................30
2.2 Disks based on incircles of faces...............32
3 Truncated Icosahedron.............................33
3.1 Disks based on edges............................33
3.2 Disks based on incircles of faces...............34
References............................................36


                                

[1] H. Martyn Cundy and A. P. Rollett, Mathematical Models, Oxford Clarendon Press 1961.
[2] John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The symmetries of Things, A K Peters/CRC Press, 2008.
[3] Weisstein, Eric W. "Archimedean Solid." From MathWorld--A Wolfram Web Resource. available at https://mathworld.wolfram.com/ArchimedeanSolid.html
[4] Weisstein, Eric W. "Catalan Solid." From MathWorld--A Wolfram Web Resource. available at https://mathworld.wolfram.com/CatalanSolid.html
[5] Weisstein, Eric W. "Inversion." From MathWorld--A Wolfram Web Resource. available at https://mathworld.wolfram.com/Inversion.html
[6] 阮賢彧 (2011)。卡塔蘭多面體的動態幾何作圖法。國立中央大學數學系碩士論文，桃園市。取自https://hdl.handle.net/11296/7q677h
[7] 黃俊晏 (2016)。對偶多面體上的內切圓形成的正交圖形。國立清華大學數學系碩士論文，新竹市。取自https://hdl.handle.net/11296/7pn3pr
[8] 顏琇婷 (2008)。對偶多面體。國立清華大學數學系碩士論文，新竹市。取自https://hdl.handle.net/11296/8cb6m8

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