Chebyshev spiral
Create a button with the following script:
Execute[Join[{"T_0(x)=1","T_1(x)=x"},Sequence["T_{"+(i+1)+"}(x)=2*x*T_{"+(i)+"}(x)-T_{"+(i-1)+"}(x)",i,1,15,1]]]
Execute[Sequence["SetVisibleInView[T_{"+(i)+"},1,false]",i,0,16]]
Execute[Sequence["c_{"+(i)+"}=Curve[("+(i)+"+T_{"+(i)+"}(t/pi-1);t),t,0,2pi]",i,0,16,2]]
Execute[Sequence["ShowLabel[c_{"+(i)+"},false]",i,0,16,2]]