(*
Simple app to visualise any configuration in the (Subscript[AB, 4]) T-4 E-mode configuration space given the configuration space spherical coordinates (θ, ϕ)
*)
(*
Analytic form for the (Subscript[AB, 4]) T-4 E-mode B-atom unit vectors
*)
AB4Emode[θ_,ϕ_]:={{(((Sqrt[2]) Cos[θ]-Sin[θ] Sin[ϕ])/(Sqrt[3])),Cos[ϕ] Sin[θ],((Cos[θ]+(Sqrt[2]) Sin[θ] Sin[ϕ])/(Sqrt[3]))},{((-(Sqrt[2]) Cos[θ]+Sin[θ] Sin[ϕ])/(Sqrt[3])),-Cos[ϕ] Sin[θ],((Cos[θ]+(Sqrt[2]) Sin[θ] Sin[ϕ])/(Sqrt[3]))},{Cos[ϕ] Sin[θ],(((Sqrt[2]) Cos[θ]-Sin[θ] Sin[ϕ])/(Sqrt[3])),-((Cos[θ]+(Sqrt[2]) Sin[θ] Sin[ϕ])/(Sqrt[3]))},{-Cos[ϕ] Sin[θ],-(((Sqrt[2]) Cos[θ]-Sin[θ] Sin[ϕ])/(Sqrt[3])),((-Cos[θ]-(Sqrt[2]) Sin[θ] Sin[ϕ])/(Sqrt[3]))}};
Manipulate[
Block[{l0={0,0,0},p1,p2,p3,p4},
{p1,p2,p3,p4}=AB4Emode[θ,ϕ];

Graphics3D[{
White,Ball[l0,.2],
Red,Ball[p1,.12],
Yellow,Ball[p2,.12],
Green,Ball[p3,.12],
Blue,Ball[p4,.12],
,Tube[{l0,p1},.06],Tube[{l0,p2},.06],Tube[{l0,p3},.06],Tube[{l0,p4},.06],
{,Point[Tuples[{-1,1},3]]},
,Thickness[0.008],
Arrow[{l0,{1.15,0,0}}],
Text[Style["x",30,Italic],{1.25,0,0}],
Arrow[{l0,{0,1.15,0}}],
Text[Style["y",30,Italic],{0,1.25,0}],
Arrow[{l0,{0,0,1.15}}],
Text[Style["z",30,Italic],{0,0,1.25}]
},SphericalRegion->True,Boxed->False,ImageSize->400  ]],
{{ϕ,0},0,360 °,1°,Appearance->"Open"},
{{θ,0},0,180 °,1°,Appearance->"Open"},
ControlPlacement->Top,SaveDefinitions->True
]