Exact geometry — stereographic projection of S³ ⊂ ℂ² into 3-space. The circles are true Hopf fibers, and −p really is a half-turn along the fiber through p, so the identification p ∼ −p acts along each circle, and the Hopf base S² never moves — the identification works purely along fibers. This is the quotient route to ℝℙ³; τ, the conjugation on ℂℙ³ that fixes the same space, is the other one. The amber dots are points of ℝℙ³: the point each pair becomes. Only the cadence is schematic; the identification itself is timeless.