= Equal-length crystallographic root configurations in a plane
{title2=$|\Phi|\in\{2,4,6\}$}
Let $\Phi$ be a nonempty set of <unit vectors> in a real Euclidean plane, closed under the reflections $s_v(u)=u-2\langle u,v\rangle v$ and satisfying $2\langle u,v\rangle\in\mathbb Z$ for every pair. Then it consists of an antipodal pair, a <square>, or a <regular hexagon>. Relative to any fixed member, the inner products can only be $0,\pm1/2,\pm1$, so only eight angles are initially possible. <Reflection> gives every negative vector. If a perpendicular pair occurs, it and its negatives exhaust the set, since any extra sixty-degree direction would have forbidden <inner product> $\pm\sqrt3/2$ with one of them. Otherwise any noncollinear pair gives, after negation, a sixty-degree angle; <reflection> supplies the third direction and negation supplies the other three hexagon vertices. If no such pair occurs, only the antipodal pair remains. Each configuration directly satisfies the conditions.
These are the equal-length cases of the <classification of rank-two root systems>, with the antipodal case having rank one. The <square> is the <orthogonal> sum of two <A1 root systems>, and the hexagon is an <A2 root system>. The <root system> is considered in the span of its vectors when the ambient plane is not spanned.
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