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}]