pol=Polygon[{{3,0},{1,1},{0,3},{-1,1},{-3,0},{-1,-1},{0,-3},{1,-1}}];
pol=Polygon[{{1,-1},{1,1},{-1,1},{-1,-1}}];
Graphics[{EdgeForm[Orange],Opacity[0],{pol,Translate[Rotate[pol,Pi/6],{6,0}]}}]

dt = 0.2;v=1;ω=0.75;
Manipulate[
Graphics[Append[Table[{EdgeForm[Orange],Opacity[0],Translate[Rotate[pol,-ω i dt],{v i dt,0}]},{i,1,Ceiling[3.5 Pi /(dt ω)]}],{EdgeForm[{Thick,Black}],Opacity[0],Translate[Rotate[pol,-ω k dt],{v k dt,0}]}]],
{k,1,Ceiling[3.5 Pi /(dt ω)]}]

dt = 0.05;R=2;ω=1.25;
Manipulate[
Graphics[Append[Table[{EdgeForm[Orange],Opacity[0],Translate[Rotate[pol,(1+ω)i dt],{R Cos[ i dt],R Sin[i dt]}]},{i,1,Ceiling[2 Pi/dt]}],{EdgeForm[{Thick,Black}],Opacity[0],Translate[Rotate[pol,(1+ω)k dt],{R Cos[ k dt],R Sin[k dt]}]}]],
{k,1,Ceiling[2 Pi/dt]}
]

dt = 0.05;v=1;ω=.6;Graphics[{EdgeForm[Orange],Opacity[0],Table[Translate[Rotate[pol,-ω i dt],{v i dt,0}],{i,1,Ceiling[3.5 Pi /(dt ω)]}]}]

dt = 0.01;R=6;ω=2;Graphics[{EdgeForm[Orange],Opacity[0],Table[Translate[Rotate[pol,(1+ω)i dt],{R Cos[ i dt],R Sin[i dt]}],{i,1,Ceiling[2 Pi/dt]}]}]