Classification of equiangular lines with fixed angle $\arccos(1/(1+2\sqrt2))$

TL;DR

This paper completely determines the maximum number of equiangular lines with a specific fixed angle in various dimensions, filling a longstanding gap in geometric line configuration knowledge.

Contribution

It provides the first complete characterization of the maximum number of equiangular lines with a fixed nontrivial angle across all dimensions, for the specific angle os(1/(1+22)).

Findings

Exact values of maximum equiangular lines for dimensions 2 to 14.

Asymptotic formula for dimensions 15 and above.

First comprehensive result for fixed nontrivial angles since 1973.

Abstract

We determine the maximum number $N_\alpha(d)$ of equiangular lines with fixed angle $\arccos\alpha$ for $\alpha = 1/(1+2\sqrt2)$ in $d$-dimensional Euclidean space: $2,3,4,6,8,10,14,15,16,17,18,20,22$ for $d \in \{2,\dots,14\}$, and $\max(24, \lfloor 3(d-1)/2 \rfloor)$ for $d \ge 15$. This appears to be the first complete determination of $N_\alpha(d)$ in all dimensions $d$ for a fixed nontrivial $\alpha$, since the work of Lemmens and Seidel for $\alpha = 1/3$ in 1973.