The algebra proving why "halving long side H and flipping maintains aspect ratio" implies a ratio of 1:sqrt(2):

    H/W = W/(H/2)       (halving long side H and flipping maintains aspect ratio)
        = 2W/H          (multiply numerator and denominator by 2)
    H²  = 2W²           (multiply both sides by HW)
    H   = √2·W ≈ 1.41 W (take the positive square root; H, W > 0)