Rational Points on the Unit Circle: A Discrete Dense Subgroup via A₁ Lattice — E8 Intelligence Research

FINDING: Rational points on the unit circle form a discrete, dense subgroup isomorphic to the additive group of rationals mod 2π, directly tied to the root lattice A1 and Pythagorean triples. | MATH: Rational points on S¹: (x,y) = ((1−t²)/(1+t²), 2t/(1+t²)), t ∈ ℚ. This is the stereographic projection from (−1,0). The group law: (t₁,t₂) ↦ (t₁+t₂)/(1−t₁t₂) — the tangent addition formula. The set of such points is ℚ/ℤ under the angle map θ ↦ eiθ, θ ∈ 2πℚ. The A1 root lattice: A₁ ≅ ℤ with roots {±1}, norm² = 2. The unit circle's rational points correspond to vectors (a,b,c) ∈ ℤ³ with a²+b²=c², i.e., primitive Pythagorean triples — these are exactly the integer points on the cone x²+y²=z², whose projection to z=1 gives the rational circle. | CONNECTION: The A1 root system (rank 1, Weyl group ℤ/2) is the simplest crystallographic root system. Its lattice ℤ embeds into the circle group via t ↦ e2πit. The rational points on S¹ are the torsion subgroup of the circle group — precisely the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com