The Fibonacci spiral: an approximation of the generated golden spiral drawing circular arcs connecting the opposite corners of squares adjusted to the values of the succession. 
Mathematica code:
Manipulate[
With[{tf = AffineTransform[{RotationMatrix[-2θ]/GoldenRatio, {1, 1}}]},
Graphics[{
{NestList[tf, RegionBoundary@Rectangle[], n]},
Orange, NestList[tf,
Circle[(Cot[θ] {1, -1} + 1)/2, Csc[θ]/√2, {-θ, θ} + 3 π/4], n]
},
PlotRange -> {{-0.1, 3}, {-0.1, 3}}, ImageSize -> 450
]],
{θ, 0.001, π/4}, {{n, 5}, 0, 9, 1}]
