The Fifty-Year Angle

How many lines through the origin of d-dimensional space can be mutually equiangular — every pair meeting at the same angle? The answer depends on both the dimension and the angle, and for most angles, it is unknown in most dimensions.

In 1973, Lemmens and Seidel completely determined the maximum number of equiangular lines at angle arccos(1/5) in every dimension. For fifty-three years, this was the only nontrivial angle with a complete classification. Partial results accumulated for other angles — bounds, constructions, sporadic exact values — but no second angle yielded to complete determination across all dimensions.

Gossett, Jiang, Teets, and Wellner close this gap for the angle arccos(1/(1+2√2)). They determine the exact maximum in dimensions 2 through 14, then establish a formula for all higher dimensions. The proof combines spectral graph theory, semidefinite programming bounds, and explicit constructions — methods that didn’t exist in 1973.

The through-claim: the fifty-year gap between complete classifications wasn’t caused by difficulty in the usual sense. The problem didn’t get harder; the toolkit was inadequate. Lemmens and Seidel’s 1973 methods sufficed for their angle but couldn’t reach others. The new classification required semidefinite programming (developed in optimization theory), spectral bounds from algebraic graph theory, and computational verification — tools from three different fields that had to mature independently before they could be combined. The bottleneck wasn’t a single hard step but the convergence of separately developing techniques. Sometimes a problem waits not for a breakthrough but for a toolkit.