The silver ratio δₛ = 1 + √2 ≈ 2.41421 is the positive solution to x² = 2x + 1. It is the second member of the metallic means family: the golden ratio satisfies x² = x + 1 (all 1s in continued fraction), and the silver ratio satisfies x² = 2x + 1 (all 2s in continued fraction [2; 2, 2, 2, …]).
The Pell numbers 1, 2, 5, 12, 29, 70, 169, 408… are defined by Pₙ = 2Pₙ₋₁ + Pₙ₋₂. Their ratios converge to δₛ just as Fibonacci ratios converge to φ. The silver ratio governs the regular octagon: the ratio of a diagonal to a side is δₛ. It also appears in Ammann-Beenker quasiperiodic tilings.
The red diagonal connects vertices 3 apart (skipping 2). The green side is one edge. Their ratio is exactly 1 + √2 ≈ 2.414, the silver ratio. This is the octagon equivalent of the golden ratio diagonal in a pentagon.
The silver ratio has self-similarity: δₛ = 2 + 1/δₛ = 2 + 1/(2 + 1/(2 + ⋯)). Removing two unit squares from a δₛ × 1 rectangle leaves a smaller rectangle with the same proportions. The A-paper series uses √2 (which is δₛ - 1) so that halving a sheet preserves the aspect ratio. Value: 2.41421356237309504880168872…
A0, A1, A2… each sheet is half the previous. The ratio 1:√2 is the only ratio that survives halving. Fold a 1:√2 sheet: you get a √2:1 sheet, the same proportions rotated. √2 = δₛ - 1, linking the paper series directly to the silver ratio.