本工作缘起于李永乐老师公开科普视频中对挂谷(Kakeya)猜想的讲解及王虹(Hong Wang)与约书亚·扎尔(Joshua Zahl)在 ℝ³ 中 Kakeya 维数相关工作成就。从挂谷(Kakeya)猜想中,鲍尔重叠,佩龙树面积趋零等问题出发,思考面积产生消失与维度跨越机制,建立动态欧氏几何雏形。以此框架重释经典欧氏几何五大公设,给出经典欧氏几何与动态欧氏几何之完整对比。本文为探讨性注记,不声称证明任何猜想,抛砖引玉。 This work originates from the public science popularization video by Mr. Li Yongle explaining the Kakeya conjecture, and the achievements of Hong Wang and Joshua Zahl regarding Kakeya dimension-related work in ℝ³. Starting from the Kakeya conjecture, issues such as Borel overlap and Perron tree area tending to zero, we reflect on the mechanisms of area generation, disappearance, and dimensional transition, thereby establishing the rudiments of dynamic Euclidean geometry. Using this framework to reinterpret the five postulates of classical Euclidean geometry, we present a complete comparison between classical Euclidean geometry and dynamic Euclidean geometry. This paper is an exploratory note; it does not claim to prove any conjecture, but rather serves as a modest spark to provoke deeper inquiry. 注: 作者仅初步了解鲍尔重叠与佩龙树等 Kakeya 集基础构造,对 Kakeya 猜想的深层理论未作深入掌握。本文仅以此为出发点建立动态欧氏几何框架,不声称对该猜想有任何推进。 本文为探讨性注记,不声称证明任何猜想。核心思想、论证及全文中文均为作者原创。英文翻译因作者数学英语能力有限,借助了AI语言工具辅助完成。英文表达如有不准确之处,责任由作者本人承担。 Note: The author has only a preliminary understanding of basic constructions such as Borel overlap and the Perron tree, and has not mastered the deeper theories of the Kakeya conjecture. This paper merely uses these as a starting point to build the dynamic Euclidean geometry framework, and does not claim any advancement on the conjecture. The core ideas, arguments, and all Chinese text of this paper are solely the author's original work. Due to the author's limited proficiency in English, particularly in mathematical English, the English translations throughout this paper were produced with the assistance of AI language tools. Any inaccuracies or infelicities in the English expression remain the author's responsibility.
zhang et al. (Sun,) studied this question.