本論文將會整理離散幾何中一個有趣的領域: 等角直線組（equiangular lines)的歷史演進與發展。以1973年 Lemmens-Seidel 的文章為主體，並加入後續的進展，例如 Barg-Yu 證明了24維度之後的半正定規劃的上界，Lin-Yu 對Neumann定理的推廣, Greaves et al 對於14和16維度決定最大條數的結果。我們整理這些相關文獻，把等角直線組的故事與發展說明得更完整，並且詳細寫下相關的例子或構造。本文以Lemmens-Seidel第四節和第五節為重，第四節說明柱(pillar)是甚麼和相關定理證明，第五節討論當角度固定在arccos(1/5)時，會說明上下界會如何變化。

This paper dives into an intriguing realm of discrete geometry: the historical evolution and development of equiangular lines. It primarily builds upon the 1973 Lemmens-Seidel paper, incorporating subsequent advancements. For instance, Barg-Yu proved upper bounds for semidefinite programming beyond 24 dimensions, Lin-Yu extended Neumann's theorem, and Greaves et al revealed results on determining the maximum number of lines in 14 and 16 dimensions. We'll organize these relevant works, providing a more comprehensive narrative of the equiangular lines' story and development, while delving into specific examples or constructions. The focus of this paper lies in Lemmens-Seidel's fourth and fifth sections, the fourth section what a "pillar" is and proves associated theorems, while the fifth section how upper and lower bounds shift when the angle is fixed at arccos(1/5).

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