spirograph.nb

Clear[x, y, R, r]

x[\[Theta]_, r_, R_, d_: 1] := (R - r)*Cos[\[Theta]] + 
  d*Cos[(R - r)/r*\[Theta]]
y[\[Theta]_, r_, R_, d_: 1] := (R - r)*Sin[\[Theta]] - 
  d*Sin[(R - r)/r*\[Theta]]
  
  \[Theta]max[r_, R_] := 2 \[Pi]*LCM[r, R]/R;
  
  plot[iter_, r_, R_ ] :=
 
 ParametricPlot[{x[\[Theta], r, R], y[\[Theta], r, R]}, {\[Theta], 0, 
   iter*\[Theta]max[Rationalize[r], Rationalize[R]]}, 
  PlotRange -> {{-6, 6}, {-6, 6}}, 
  PlotStyle -> Hue[(2.75 - r)/2.75] , Background -> Black, 
  AxesStyle -> White, Ticks -> None, Axes -> False, Frame -> False]
  
  Clear[singlePlot];

singlePlot[r_, R_, BRix_: 0] := Module[ {plots},
  If[BRix == 0,
   Table[  
     plot [Rationalize[iter], Rationalize[r], Rationalize[R]], {iter, 
      0, 1, .05}  ][[2 ;;]],
   
   Table[   
     Show[BRs[[1 ;; i]],  
      plot [Rationalize[iter], Rationalize[r], 
       Rationalize[R]]    ], {iter, 0, 1, .05}  ][[2 ;;]]
   ]
  
  ]

generateBR[r_, R_] := plot[1, r, R];

BRs = Table[generateBR[r, 3], {r, 1, 3, .2}];



plotWBR[rmin_: .25, rmax_: 3, R_: 4, step_: .2] := Module[{anims},
  anims = {};
  i = 1;
  Table[ If[r == rmin, 
    anims = Append[anims, singlePlot[r, R, 0]]
    , anims = Append[anims, singlePlot[r, R, i] ];
    i = i + 1  ]
   , {r, rmin, rmax, step}];
  Return[Join[anims, Table[anims[[-1, -1]] , {i, 0, 1, .2} ]] ]
  ]
  
  
  Export["spiro.gif", Flatten[ plotWBR[1, 3, 3, .2]], 
 "AnimationRepetitions" -> \[Infinity] ]