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Pell numbers converge to the silver ratio
s d δ_S = d/s = 1 + √2 ≈ 2.4142... (skip-3 diagonal)
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The silver ratio diagonal in a regular octagon
diagonal = δₛ ≈ 2.414 side = 1 diagonal / side = 1 + √2 = δₛ

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.

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A-paper series: fold in half, keep the same proportions
A0 841 × 1189mm A1 A2 A3 A4 Why ratio 1 : √2? Fold A0 in half →A1 New dims: w×(w√2/2) New ratio: 1 : √2/2 = √2 : 1 (rotated) Aspect ratio stays 1:√2 after every fold! √2 = δₛ - 1 = silver ratio - 1 connects paper to δₛ

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.

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