Rotating spiral

t=Flatten[CirclePoints[#,9]&/@Range[0,2,.1],1];
 Animate[Graphics[{Rotate[b={Blue,BezierCurve[Reverse@t]},n,{0,0}],Rotate[b,-n,{0,0}]},PlotRange->1,Axes->False],{n,0,2Pi,.1}] 

If we slightly change the code of this program, we get the following

t = Flatten[CirclePoints[#, 9] & /@ Range[0, 2, .1], 1]; 
Animate[ Graphics[{Cyan, Polygon@t, Rotate[b = {Blue, Thickness@.01, BezierCurve[Reverse@t]},n, {0, 0}], 
Rotate[b, -n, {0, 0}] }, PlotRange -> 1, Axes -> False], {n, 0, 2 Pi, .096}]

Created by Silva Torosyan in Wolfram Language