全新视角：人字形刺绣的对偶对称性
A Whole New Side: The Duality Symmetries of Hitomezashi
AI总结：
本文研究广义人字形图案（GHPs）的对偶与翻转对称性，证明存在4种玫瑰花形双色对称类型，且46种双色墙纸群中17种与GHPs兼容，拓展了相关数学研究。
AI中文摘要：
人字形刺绣（Hitomezashi）是一种特定风格的刺子绣，以交叉的绣线为特征。某些人字形刺绣图案可由两个二进制字符串编码，已成为大量数学研究的对象。Sen与Martinez将这类图案命名为广义人字形图案（Generalized Hitomezashi Patterns, GHPs），这类图案可逆且可具有“自对偶性”。本文研究GHPs中可能存在的“对偶-”或“翻转对称性”，这类对称性的引入等价于考虑双色对称群。Sen与Martinez曾找出与GHPs兼容的墙纸对称群，本文在此基础上研究与GHPs兼容的可能双色对称性。我们证明存在4种可能的玫瑰花形双色对称类型，完全描述了产生这些对称性的二进制字符串的性质，并证明46种双色墙纸群中恰好有17种与GHPs兼容。
英文摘要：
Hitomezashi is a particular style of sashiko stitching characterized by crossing lines of stitches. Certain hitomezashi designs can be encoded with two binary strings and have been the subject of a wide-array of mathematical research. Dubbed generalized hitomezashi patterns (GHPs) by Sen & Martinez, these designs are reversible and can be 'self-dual.' We investigate the possible 'dual-' or 'flip-symmetries' in GHPs; the inclusion of these symmetries is equivalent to considering two-colour symmetry groups. Following the work of Sen & Martinez, who found the wallpaper symmetry groups that are compatible with GHPs, this paper investigates the possible two-colour symmetries compatible with GHPs. We prove that there are four possible rosette two-colour symmetry types, completely describe the properties on binary strings that create these symmetries, and prove that exactly 17 of the 46 two-colour wallpaper groups are compatible with GHPs.
