CITE THIS NOTEBOOK: Dancing or juggling patterns of complex polynomial roots? by Vitaliy Kaurov. Wolfram Community JUN 2016.
I’ve seen Henry Segerman just posted a link to a nice visualization (i think original is actually here). I recognized Mathematica color schemes and right away wanted to recreated it. You can see the result below, and the WL code under the animation. Animation is showing the roots of the polynomials
​
5
x
+tx+1<>
5
x
+t
2
x
+1<>
5
x
+t
3
x
+1
​
as t varies along a circle around 0 in the complex plane (with radius 1 and 2).Code could be a bit shorter, but I was in a rush. But why to use the same color function, if we can improvise, right?

Define a function that plots f[x,y] in a specific style with ContourPlot:

In[]:=
plfnC[f_,R_,c_]:=ContourPlot[f,{x,-R,R},{y,-R,R},​​PlotPoints->50,Contours->25,PlotRange->{{-R,R},{-R,R},5{-1,1}},​​ImageSize->250,ColorFunction->c,Frame->False]

Choose nice different Color Schemes:

In[]:=
color={"SouthwestColors","IslandColors","BrightBands","DarkBands","RedGreenSplit","Pastel"};

Plot range:

In[]:=
range={1.5,1.5,1.7,1.7,1.8,1.8};

Pick 6 different equations to plot:

In[]:=
eqs=​​Abs[1+
 a

(x+ y)+
5
(x+ y)
],Abs[1+2
 a

(x+ y)+
5
(x+ y)
],​​Abs[1+
 a

2
(x+ y)
+
5
(x+ y)
],Abs[1+2
 a

2
(x+ y)
+
5
(x+ y)
],​​Abs[1+
 a

3
(x+ y)
+
5
(x+ y)
],Abs[1+2
 a

3
(x+ y)
+
5
(x+ y)
];

Build frames for animated GIF by changing a parameter in equations:

In[]:=
frames=ParallelTable[Grid[Partition[MapThread[plfnC,{eqs,range,color}],2],Spacings->{0,0}],{a,0,2 Pi,.07}];

Export to animated GIF:

In[]:=
Export["cellsplit.gif",frames,"DisplayDurations"->.08,ImageSize->500]
Out[]=
cellsplit.gif