Rolling a square

corner[x_] :=
 Module[{\[Theta] = N@SawtoothWave[{0, 2 Pi}, x/(2 Pi)]}, Piecewise[{
     {{-1 - Cos[\[Theta]], Sin[\[Theta]]}, 0 <= \[Theta] < Pi/2},
     {{-Sqrt[2] Cos[\[Theta] - Pi/4], Sqrt[2] Sin[\[Theta] - Pi/4]},
      Pi/2 <= \[Theta] < Pi},
     {{1 + Sin[\[Theta] - Pi], Cos[\[Theta] - Pi]},
      Pi <= \[Theta] < 3 Pi/2},
     {{2, 0}, \[Theta] >= 3 Pi/2}
     }] + {4 Floor[x/(2 Pi)], 0}
  ];

frame[\[Theta]_] := Module[{t},
  t = If[\[Theta] <= 3.5 Pi, \[Theta], 7 Pi - \[Theta]];
  Show[Graphics[{
     RGBColor[0.8, 0.3, 0.2],
     Polygon[{corner[t], corner[t + Pi/2] - {1, 0},
       corner[t + Pi] - {2, 0}, corner[t + 3 Pi/2] - {3, 0}}
      ], Black,
     PointSize[Medium], Point[corner[t]]
     }
    , PlotRange -> {{-2, 5}, {-0.1, 1.5}}],
   If[\[Theta] >=  3.5 Pi,
    ParametricPlot[corner[x], {x, t, 3.5 Pi + 0.001},
     PlotStyle -> Black]
    ,
    Graphics[]]
   ]];

Manipulate[frame[\[Theta]], {\[Theta], 0, 7 Pi}]