Hypotrochoids

Spirograph-like curves arise from simple sums of rotating vectors; a tiny tweak in frequency or phase yields striking patterns.

Compact parametric form:

$$x(t)=R\cos t+r\cos(kt+\varphi),\\ y(t)=R\sin t+r\sin(kt+\varphi),$$

where (k) is the frequency ratio. Rational (k) ⇒ closed patterns; irrational (k) ⇒ long, nonrepeating traces. ℂ numbers allow to write such expressions much shorter, for instance:

ParametricPlot[ReIm[(*SpB[*)Power[I(*|*),(*|*)-t](*]SpB*) + 3 (*SpB[*)Power[I(*|*),(*|*)t/3](*]SpB*)], {t, 0, 4Pi}] (*VB[*)(FrontEndRef["0b2a2ce6-db3b-46a8-9cfb-03bf3f69a7f7"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKGyQZJRolp5rppiQZJ+mamCVa6FompyXpGhgnpRmnmVkmmqeZAwCSjhZi"*)(*]VB*)

We can nicely decompose it back into 2 components and animate it using Animate expression and bunch of arrows:

Animate[ ParametricPlot[ReIm[(*SpB[*)Power[I(*|*),(*|*)-t](*]SpB*) + 3 (*SpB[*)Power[I(*|*),(*|*)t/3](*]SpB*)], {t,0,u}, Epilog->{ Arrow[{{0,0}, ReIm[(*SpB[*)Power[I(*|*),(*|*)-u](*]SpB*)]}], Arrow[{ ReIm[(*SpB[*)Power[I(*|*),(*|*)-u](*]SpB*)], ReIm[(*SpB[*)Power[I(*|*),(*|*)-u](*]SpB*) + 3 (*SpB[*)Power[I(*|*),(*|*)u/3](*]SpB*)]}] }, PlotRange->{{-4,4}, {-4,4}}, GridLines->Automatic, PlotStyle->(*VB[*)(RGBColor[1, 0, 0])(*,*)(*"1:eJxTTMoPSmNiYGAo5gUSYZmp5S6pyflFiSX5RcEsQBHn4PCQNGaQPAeQCHJ3cs7PyS8qYgCDD/ZQBgMDnAEA4iUPRg=="*)(*]VB*) ] , {u, 0.001, 4Pi}, RefreshRate->30, Appearance->None] (*VB[*)(FrontEndRef["c8997936-9d6d-4090-b476-7436bc1c096d"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKJ1tYWppbGpvpWqaYpeiaGFga6CaZmJvpmpsYmyUlGyYbWJqlAAB1/xUA"*)(*]VB*)

Bonus: Rosetta

Now we can try and more complex variants - to add the filling:

Animate[ParametricPlot[{ (*SpB[*)Power[I(*|*),(*|*)-(*FB[*)((20t)(*,*)/(*,*)(tt))(*]FB*)](*]SpB*)+3 (*SpB[*)Power[I(*|*),(*|*)(*FB[*)((20t)(*,*)/(*,*)(tt))(*]FB*)](*]SpB*)+(*SpB[*)Power[I(*|*),(*|*)(*FB[*)((20t)(*,*)/(*,*)(tt))(*]FB*)](*]SpB*)Sin[8 Pi (*FB[*)((20t)(*,*)/(*,*)(tt))(*]FB*)/5], (*SpB[*)Power[I(*|*),(*|*)-t](*]SpB*)+3 (*SpB[*)Power[I(*|*),(*|*)t](*]SpB*), (*SpB[*)Power[I(*|*),(*|*)-t](*]SpB*)+3 (*SpB[*)Power[I(*|*),(*|*)t](*]SpB*)+(*SpB[*)Power[I(*|*),(*|*)t](*]SpB*)Sin[8 Pi t/5] }//ReIm//Evaluate, {t,0,tt}, PlotRange -> {-5,5}, PlotPoints -> 200, PlotStyle -> {LightGray,Default,(*VB[*)(Hue[0.35000000000000003, 0.78, 0.49])(*,*)(*"1:eJxTTMoPSmNiYGAo5gUSYZmp5S6pyflFiSX5RcEsQBHn4PCQNGaQPAeQCHJ3cs7PyS8q2ubQ9Oj4jN32RTH9h75qxNy3L/r/xr3rxIZD9gCIyhxD"*)(*]VB*)}, Axes->False, Epilog -> { {Thick,Red,Line[{ReIm[(*SpB[*)Power[I(*|*),(*|*)-tt](*]SpB*)+3 (*SpB[*)Power[I(*|*),(*|*)tt](*]SpB*)-(*SpB[*)Power[I(*|*),(*|*)tt](*]SpB*)],ReIm[(*SpB[*)Power[I(*|*),(*|*)-tt](*]SpB*)+3 (*SpB[*)Power[I(*|*),(*|*)tt](*]SpB*)+(*SpB[*)Power[I(*|*),(*|*)tt](*]SpB*)]}]}, {Thick,Circle[ReIm[(*SpB[*)Power[I(*|*),(*|*)-tt](*]SpB*)+3 (*SpB[*)Power[I(*|*),(*|*)tt](*]SpB*)+(*SpB[*)Power[I(*|*),(*|*)tt](*]SpB*)Sin[8 Pi tt/5]],0.1]} }, PlotPoints -> 10 ],{{tt,19.9},0.01,4 5, 0.1}, Appearance -> None]