# Tricapped Triangular Bipyramid.
# A beautiful set of 11 pts in 3-D which is not a 3-design.
# (in fact it is only a spherical 1-design);

n:=3:
N:=11:
r:=1/sqrt(5);
t:=r;
s:=sqrt(3)*r;
d:=r;
b:=-2*d;
a[1]:=[0,0,1];
a[2]:=[0,0,-1];
a[3]:=[1,0,0];
a[4]:=[-1/2,sqrt(3)/2,0];
a[5]:=[-1/2,-sqrt(3)/2,0];
a[6]:=[r,s,t];
a[7]:=[r,-s,t];
a[8]:=[r,s,-t];
a[9]:=[r,-s,-t];
a[10]:=[b,0,d];
a[11]:=[b,0,-d];

e1:=r^2+s^2+t^2-1;
with(linalg):
dist:=proc(x,y) expand(dotprod(a[x]-a[y],a[x]-a[y])); end;
> for i from 1 to 11 do print(evalf(a[i])); od;

                                   [0, 0, 1.]

                                  [0, 0, -1.]

                                   [1., 0, 0]

                   [-.5000000000000000, .8660254037844385, 0]

                  [-.5000000000000000, -.8660254037844385, 0]

           [.4472135954999580, .7745966692414834, .4472135954999580]

           [.4472135954999580, -.7745966692414834, .4472135954999580]

           [.4472135954999580, .7745966692414834, -.4472135954999580]

          [.4472135954999580, -.7745966692414834, -.4472135954999580]

                   [-.8944271909999160, 0, .4472135954999580]

                  [-.8944271909999160, 0, -.4472135954999580]