(* Mathematica Graphics3D program for A047751 examples.
This program produces two examples (1 and 5 cells) of the tetrahedral clusters with maximal symmetry (type K). The output can be manipulated with the cursor to rotate the figures. *)
newp[a_, b_, c_, d_ ] := 2 (a + b + c)/3 - d;
p1 = {0, 0, 0}; p2 = {0, 1, 1}; p3 = {1, 0, 1}; p4 = {1, 1, 0};
p5 = p1 + {-2, 2, 0}; p6 = p2 + {-2, 2, 0}; p7 = p3 + {-2, 2, 0}; p8 = p4 + {-2, 2, 0};
p9 = newp[p5, p6, p7, p8];
p10 = newp[p6, p7, p8, p5];
p11 = newp[p7, p8, p5, p6];
p12 = newp[p8, p5, p6, p7];
Show[Graphics3D[{{RGBColor[0, 1, 0], 
    Sphere[{p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}, 
     0.05]}, {RGBColor[1, 0.6, 0.6], 
    Cylinder[{{p1, p2}, {p1, p3}, {p1, p4}, {p2, p3}, {p2, p4}, {p3, 
       p4}, {p5, p6}, {p5, p7}, {p5, p8}, {p6, p7}, {p6, p8}, {p7, 
       p8}}, 0.025]}, {RGBColor[0.6, 0.6, 1], 
    Cylinder[{{p5, p9}, {p6, p9}, {p7, p9}, {p6, p10}, {p7, p10}, {p8,
        p10}, {p7, p11}, {p8, p11}, {p5, p11}, {p6, p12}, {p8, 
       p12}, {p5, p12}}, 0.025]}}], Boxed -> False, 
 ViewPoint -> {3, 4, 5}]