Numberphile:

"The Golden Ratio is especially irrational...." What???!!!

Andy Long
March, 2021

This animation was inspired by an episode of Numberphile on YouTube. 
Thanks to Blake Weimer for bringing it to my attention.

This simulation emulates the production of seeds by a flower. We generate sequences of coordinates of “seeds”, ((Subscript[x, k]),(Subscript[y, k])), which expand about the seed-producing center of a flower.

We also generate a sequence of distances from the seed-producing center, a sequence which diverges, and pushes those coordinates to diverge, as well 
(Divergent sequences can be useful, too!).

Manipulate[
dist[ratio_,n_]:=
Module[{seedsPerTurn=2Pi/ratio},
n*seedsPerTurn
];
ListPlot[
Table[dist[ratio, k]{Cos[2Pi ratio k],Sin[2Pi ratio k]},{k,1,npts}]
,PlotMarkers->{"☺"}
,AspectRatio->1
]
(* 
ratio = Golden, npts=30 -> 8/5 spiral
ratio = Golden, npts=100 -> 13/8 spiral
ratio = Golden, npts=200 -> 21/13 spiral
ratio = Golden, npts=400 -> 34/21 spiral
ratio = Golden, npts=800 -> 55/34 spiral
*)
,{{ratio,GoldenRatio},1,2Pi,.0001}
,{{npts,200},25,10000}
]

The sequence of successive Fibonacci number, ((Subscript[F, k+1])/(Subscript[F, k])), converges to the Golden Ratio:
		ϕ = ((1+(Sqrt[5]))/2)
and even small successive Fibonacci numbers do a really good job of approximating ϕ:

N[GoldenRatio]
55.0/34.0