(* Define Stereographic projection from the north pole of S3, as well as its inverse*)

StereoN[x_] := {((x[[1]])/(1-x[[4]])),((x[[2]])/(1-x[[4]])),((x[[3]])/(1-x[[4]]))};

IStereoN[x_] := {2((x[[1]])/(Norm[x]^2+1)),2((x[[2]])/(Norm[x]^2+1)),2((x[[3]])/(Norm[x]^2+1)),((Norm[x]^2-1)/(Norm[x]^2+1))};

(*Here is our torus, stereographically projected from the north pole of S3*)

tor[th1_,th2_]:= {((Cos[2 Pi th1])/(Sqrt[2])),((Sin[2Pi th1])/(Sqrt[2])),((Cos[2 Pi th2])/(Sqrt[2])),((Sin[2Pi th2])/(Sqrt[2]))};
ParametricPlot3D[StereoN[tor[th1,th2]],{th1,0,1},{th2,0,1}]

(* The point {1+Sqrt[2],0,0} is on the surface of this torus. I intend to do a stereographic projection from the corresponding point in S3*)

projpoint= IStereoN[{1+Sqrt[2],0,0}];

(* Define rotation matrices for stereographic projections. R is a rotation matrix sending projpoint to the north pole of S3*)

Rot[th_] := {{Cos[th],0,0,-Sin[th]},{0,1,0,0},{0,0,1,0},{Sin[th],0,0,Cos[th]}}
R = {Rot[Pi/2].projpoint,{0,1,0,0},{0,0,1,0},projpoint};

(* Plot the resulting projection of the torus*)

ParametricPlot3D[StereoN[R.tor[th1,th2]],{th1,0,1},{th2,0,1},PlotRange->{-10,10},PlotStyle->{{Opacity[0.8],FaceForm[Hue[0.59,0.73,0.79],Hue[0.11,0.73,0.88]]}},AxesLabel->{x,y,z}, ViewPoint->{1.721792493400969`,-0.8671937374601687`,2.78090014768203`},ViewVertical->{-0.4669018501380959`,0.2603456483843665`,-0.9471894444621316`}]

(* This is a picture of the torus going through the point at infinity. The fourth order symmetry is a rotation Pi/2 radians CCW about the x axis, followed by a reflection in the xy plane.