(({{(2/φ)-3, 1, 1, 1}, {1, (2/φ)-3, 1, 1}, {1, 1, (2/φ)-3, 1}, {1, 1, 1, (2/φ)-3}, {2/φ, 2/φ, 2/φ, 2/φ}}))/((1/φ)-2);
%/.φ->((Sqrt[5])+1)/2//MatrixForm
N@%//MatrixForm

HyperPyramid Dims {1, 2, 3} as pyramid with equilateral tetrahedron (orange) with 4 of 5 vertices meeting its 3D circumradius and the 5th far outside it vs. Simplest translated to origin (blue) with same 3D shape in all 4 dim subsets and 3 of 5 vertices meeting 3D circumradius

Show[(* Origin Point *)
Graphics3D[{PointSize@.03,Red,Point@{0,0,0}},
Axes->True,AxesLabel->{"x","y","z"},AxesOrigin->{0,0,0},Boxed->True,PlotRange->{{-1,1},{-1,1},{-1,1}}],
(* Translated Simplist **)
(**),
(* HyperPyramid points *)
,(**)
Graphics3D[{PointSize@.03,
(* Origin Point *)
Red,Point@{0,0,0},
(* Translated Simplest *)
Blue,Point/@test[[All,;;3]]/.φ->((Sqrt[5])-1)/2,
(* HyperPyramid points *)
Orange,Point/@cell5Alt[[All,;;3]]/.φ->((Sqrt[5])-1)/2,
(* CircumCircle for HyperPyramid **)
White,Opacity@.4,Sphere[{0,0,0},Norm@cell5Alt[[4,;;3]]/.φ->((Sqrt[5])-1)/2],
(* CircumCircle for Translated Simplest **)
White,Opacity@.2,Sphere[{0,0,0},Norm@test[[1,;;3]]/.φ->((Sqrt[5])-1)/2]
},Axes->True,AxesLabel->{"x","y","z"},AxesOrigin->{0,0,0},Boxed->True,PlotRange->{{-1,1},{-1,1},{-1,1}}]]

HyperPyramid Dims {2, 3, 4} as equilateral tetrahedron (orange) with 4 of 5 vertices meeting 3D circumsphere and  the 5th at the origin vs. 
vs. Simplest translated to origin (blue) with same 3D shape in all 4 dim subsets and 3 of 5 vertices meeting 3D circumradius

Show[(* Origin Point *)
Graphics3D[{PointSize@.03,Red,Point@{0,0,0}},
Axes->True,AxesLabel->{"x","y","z"},AxesOrigin->{0,0,0},Boxed->True,PlotRange->{{-1,1},{-1,1},{-1,1}}],
(* Translated Simplist **)
(**),
(* HyperPyramid points *)
,(**)
Graphics3D[{PointSize@.03,
(* Origin Point *)
Red,Point@{0,0,0},
(* Translated Simplest *)
Blue,Point/@test[[All,;;3]]/.φ->((Sqrt[5])-1)/2,
(* HyperPyramid points *)
Orange,Point/@cell5Alt[[All,2;;4]]/.φ->((Sqrt[5])-1)/2,
(* CircumCircle for Translated Simplest **)
White,Opacity@.4,Sphere[{0,0,0},Norm@test[[1,;;3]]/.φ->((Sqrt[5])-1)/2],
(* CircumCircle for HyperPyramid **)
White,Opacity@.2,Sphere[{0,0,0},Norm@cell5Alt[[2,2;;4]]/.φ->((Sqrt[5])-1)/2]
},Axes->True,AxesLabel->{"x","y","z"},AxesOrigin->{0,0,0},Boxed->True,PlotRange->{{-1,1},{-1,1},{-1,1}}]]