F₂ inside SO(3) ≅ ℝP³

Generators A = rotation by arccos(1/3) about x-axis, B = same about z-axis. Each reduced word w becomes a rotation in SO(3); we plot it as the point θn̂ in the ball of radius π, where θ ∈ [0, π] is the rotation angle and n̂ is its unit axis. Edges are true SO(3)-geodesics — some wrap through ∂B and reappear at the antipodal point (the ℝP³ identification).

6
0.018
0.00
starts with A
starts with A⁻¹
starts with B
starts with B⁻¹
identity
— points
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