aa=1; bb=3;

ParametricPlot3D[{(bb + aa Cos[s]) Cos[t],(bb + aa Cos[s]) Sin[t],aa Sin[s]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[0.75]}]

{((2/3)Cos[t]+(1/3)Cos[2t]), ((2/3)Sin[t]-(1/3)Sin[2t])}

ParametricPlot3D[{(bb + ((2/3)Cos[s]+(1/3)Cos[2s])) Cos[t],(bb + ((2/3)Cos[s]+(1/3)Cos[2s]))Sin[t],((2/3)Sin[s]-(1/3)Sin[2s])},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[0.95]},PlotPoints->{128,128},PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}]

ParametricPlot3D[{(bb - ((2/3)Cos[s]+(1/3)Cos[2s])) Cos[t],(bb -((2/3)Cos[s]+(1/3)Cos[2s]))Sin[t],((2/3)Sin[s]-(1/3)Sin[2s])},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[0.95]},PlotPoints->{128,128},PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}]

Replace i with {Cos[α],Sin[α]}, then the new j is {-Sin[α],Cos[α]}

ParametricPlot[{((2/3)Cos[s]+(1/3)Cos[2s]), ((2/3)Sin[s]-(1/3)Sin[2s])},{s,0,2 Pi},PlotRange->{{-1,1},{-1,1}}]

Replace i with {Cos[α],Sin[α]}, then the new j is {-Sin[α],Cos[α]}

Manipulate[ParametricPlot[((2/3)Cos[s]+(1/3)Cos[2s]){Cos[α],Sin[α]}+ ((2/3)Sin[s]-(1/3)Sin[2s]){-Sin[α],Cos[α]},{s,0,2 Pi},PlotRange->{{-1,1},{-1,1}},ImageSize->200],{α,0,2 Pi}]

((2/3)Cos[s]+(1/3)Cos[2s]){Cos[α],Sin[α]}+ ((2/3)Sin[s]-(1/3)Sin[2s]){-Sin[α],Cos[α]}

In the next manipulation we change the position of the deltoid which is making the deltoidal torus.

Manipulate[ParametricPlot3D[{(bb + ((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[α]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[α])) Cos[t],(bb +((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[α]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[α]))Sin[t],Cos[α] (((2 Sin[s])/3)-(1/3) Sin[2 s])+(((2 Cos[s])/3)+(1/3) Cos[2 s]) Sin[α]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[0.95]},PlotPoints->{128,128},PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}],{α,0,2 Pi/3},ControlPlacement->Top]

In the next surface we replace α with t, so that the deltoid rotates about its center as it rotates about the z-axis. Here deltoid makes a full one rotation about its center.

ParametricPlot3D[{(bb + ((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[t])) Cos[t],(bb +((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[t]))Sin[t],Cos[t] (((2 Sin[s])/3)-(1/3) Sin[2 s])+(((2 Cos[s])/3)+(1/3) Cos[2 s]) Sin[t]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[0.95]},PlotPoints->{128,128},PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}]

In the next surface we replace α with 2 t, so that the deltoid rotates about its center as it rotates about the z-axis. Here deltoid makes two full rotations about its center.

ParametricPlot3D[{(bb + ((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[2t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[2t])) Cos[t],(bb +((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[2t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[2t]))Sin[t],Cos[2t] (((2 Sin[s])/3)-(1/3) Sin[2 s])+(((2 Cos[s])/3)+(1/3) Cos[2 s]) Sin[2t]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[0.95]},PlotPoints->{128,128},PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}]

In the next surface we replace α with (1/3) t, so that the deltoid rotates about its center as it rotates about the z-axis. Here deltoid makes 1/3 of a full rotation about its center.

mm=(1/3); ParametricPlot3D[{(bb + ((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t])) Cos[t],(bb +((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t]))Sin[t],Cos[mm t] (((2 Sin[s])/3)-(1/3) Sin[2 s])+(((2 Cos[s])/3)+(1/3) Cos[2 s]) Sin[mm t]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[1]},PlotPoints->{128,128},Mesh->False,PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}]

In the next surface we replace α with (2/3) t, so that the deltoid rotates about its center as it rotates about the z-axis. Here deltoid makes 2/3 of a full rotation about its center.

mm=(2/3); ParametricPlot3D[{(bb + ((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t])) Cos[t],(bb +((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t]))Sin[t],Cos[mm t] (((2 Sin[s])/3)-(1/3) Sin[2 s])+(((2 Cos[s])/3)+(1/3) Cos[2 s]) Sin[mm t]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[1]},PlotPoints->{128,128},Mesh->False,PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1},ImageSize->500]

In the next surface we replace α with (4/3) t, so that the deltoid rotates about its center as it rotates about the z-axis. Here deltoid makes 4/3 of a full rotation about its center.

mm=(4/3); ParametricPlot3D[{(bb + ((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t])) Cos[t],(bb +((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t]))Sin[t],Cos[mm t] (((2 Sin[s])/3)-(1/3) Sin[2 s])+(((2 Cos[s])/3)+(1/3) Cos[2 s]) Sin[mm t]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[1]},PlotPoints->{128,128},Mesh->False,PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}]

mm=(5/3); ParametricPlot3D[{(bb + ((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t])) Cos[t],(bb +((((2 Cos[s])/3)+(1/3) Cos[2 s]) Cos[mm t]-(((2 Sin[s])/3)-(1/3) Sin[2 s]) Sin[mm t]))Sin[t],Cos[mm t] (((2 Sin[s])/3)-(1/3) Sin[2 s])+(((2 Cos[s])/3)+(1/3) Cos[2 s]) Sin[mm t]},{t,0,2 Pi},{s,0,2 Pi},PlotStyle->{Opacity[1]},PlotPoints->{128,128},Mesh->False,PlotRange->{{-4,4},{-4,4},{-1,1}},
BoxRatios->{4,4,1}]