群論應用於艾雪鑲嵌藝術之對稱構成研究－以多媒體創作為例

A Study of Symmetrical Construction in Escher’s Tessellation Art form the Perspective of Group Theory－A Case of Multimedia Design

張瑜軒(Yu-Hsuan Chang)

指導教授 ： 林珮淳

中原大學/設計學院/商業設計研究所/碩士(2002年)

https://doi.org/10.6840/cycu200200416

摘要

荷蘭版畫家－艾雪，是一位極富創造力及想像力的藝術家，他擅長利用對稱構成的數理概念，融入具象且充滿趣味性的造形中，以表現一個平面如何被規則分割，或填滿不留空隙、形狀相似而又相鄰反覆的圖案。除了從美學的角度欣賞這些美麗的鑲嵌圖案之外，圖案背後所蘊含著的龐大數理秩序與對稱原則，也是值得我們去深究與探討的，為了描述這樣的對稱現象，本研究引進「群」的數學概念，來探討平面鑲嵌圖案的對稱構成，因為群論是研究對稱性問題的數學基礎，也適用於其他領域的相關對稱性問題。
　　此外，艾雪所創作的平面規則分割圖案，其本身融合了美感形式與數理概念的雙重特性，對於腦力的激盪和創意的啟發，頗具深遠意義及價值。故本研究之創作部分，欲將圖案對稱構成的理論藉由多媒體媒材來詮釋與呈現，並應用平面規則分割圖案的特性於互動式多媒體遊戲的設計與開發，讓使用者可以在愉快的情境、活潑的引導下，認識艾雪及艾雪的鑲嵌藝術，在趣味的遊戲之中獲致學習及啟發的效果，以提供國內的藝術教育與設計領域之參考。

關鍵字

平面規則分割 ； 鑲嵌 ； 群論 ； 對稱

並列摘要

M. C. Escher, the noted Netherlands woodcut artist, is specialized at imbedding mathematical principles of symmetrical construction in his tessellation works. Through which, he demonstrates how a two-dimensional surface can be regularly divided into, or filled up with similar-shaped figures that are contiguous to one another, without leaving any open space. These works and principles are of particular value to visual designers.
　　To unveil the underlying structure of the tessellation works, this study first employs “Group Theory” on selected works. The theory is a mathematical method useful in interpreting the symmetrical features of products of Mother Nature as well as Escher’s works. Second, to illustrate how Escher’s creative works can be learned easily, the researcher develops and incorporates these features into a multimedia CD-Title. In which, one may learn to come to grips with the formation of Escher’s creations by solving jigsaw puzzles and practicing figure generations. Finally, this study obtains user evaluation of the CD-Title.

並列關鍵字

Tessellation ； Group Theory ； Symmetry ； The Regular Division of the Plane

參考文獻

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