In case you had not known TaP (Tweet a Program) now supports animation. You can check that code wrapped in Animation - like this one:

Animate[VoronoiMesh[AnglePath[Table[{Sqrt[k], GoldenAngle}, {k, t}]], PlotTheme -> "Lines"], {t, 10, 800}]

will make something like THISTHIS. This is from @Michael Trott blog on Aspect Ratios in Art.

In case you want to see straight at the "root" of the code look a bit beyond (forgive me the pun - there is Sqrt in the code ;) ) - look at GoldenAngle and neatest AnglePath. Those along with Sqrt line up nicely 2D points in phyllotaxis pattern and you can hang the Voronoi mesh on them.

And for those who likes complete control and own GIFs:

test = ParallelTable[VoronoiMesh[AnglePath[Table[{Sqrt[k], GoldenAngle}, {k, t}]], 
PlotTheme -> "Lines", ImageSize -> 300 {1, 1}, PlotRangePadding -> 0, AspectRatio -> 1], 
{t, 10, 800, 20}];

Export["test.gif", test~Join~Reverse[test]]