#  MAPLE. June 1, ,2006.
# 
# Example 29 in HCD table of spherical designs by Munemasa.
#
# This is the 24-cell.
d := 4;

v :=   [1 ,  8 , 6 , 8 , 1 ];
verts := 24;

Q := matrix([
[1 ,   4 ,   9 ,   8 ,   2],
[1 ,   2 ,   0 ,  -2 ,  -1],
[1 ,   0 ,  -3 ,   0 ,   2],
[1 ,  -2 ,   0 ,   2 ,  -1],
[1 ,  -4 ,   9 ,  -8 ,   2]
]);

#  This tridiagonal matrix is L1-star.
L := matrix([
[0  ,  4   , 0  ,  0  ,  0],
[1  ,  0   , 3  ,  0  ,  0],
[0  , 4/3  , 0  , 8/3 ,  0],
[0  ,  0   , 3  ,  0  ,  1],
[0  ,  0   , 0  ,  4  ,  0]
]);


         [0    8    0    0    0]
         [                     ]
         [1    3    3    1    0]
         [                     ]
 L_1 =   [0    4    0    4    0]
         [                     ]
         [0    1    3    3    1]
         [                     ]
         [0    0    0    8    0]

         [0    0    6    0    0]
         [                     ]
         [0    3    0    3    0]
         [                     ]
 L_2 =   [1    0    4    0    1]
         [                     ]
         [0    3    0    3    0]
         [                     ]
         [0    0    6    0    0]

         [0    0    0    8    0]
         [                     ]
         [0    1    3    3    1]
         [                     ]
 L_3 =   [0    4    0    4    0]
         [                     ]
         [1    3    3    1    0]
         [                     ]
         [0    8    0    0    0]

         [0    0    0    0    1]
         [                     ]
         [0    0    0    1    0]
         [                     ]
  L_4 =  [0    0    1    0    0]
         [                     ]
         [0    1    0    0    0]
         [                     ]
         [1    0    0    0    0]


    [1     8     6     8     1]
    [                         ]
    [1     4     0    -4    -1]
    [                         ]
P = [1     0    -2     0     1]
    [                         ]
    [1    -2     0     2    -1]
    [                         ]
    [1    -4     6    -4     1]

   [1     4     9     8     2]
   [                         ]
   [1     2     0    -2    -1]
   [                         ]
Q= [1     0    -3     0     2]
   [                         ]
   [1    -2     0     2    -1]
   [                         ]
   [1    -4     9    -8     2]


       [0     4     0     0     0]
       [                         ]
       [1     0     3     0     0]
       [                         ]
 Ls1 = [0    4/3    0    8/3    0]
       [                         ]
       [0     0     3     0     1]
       [                         ]
       [0     0     0     4     0]

         [0    0    9    0    0]
         [                     ]
         [0    3    0    6    0]
         [                     ]
  Ls2 =  [1    0    6    0    2]
         [                     ]
         [0    3    0    6    0]
         [                     ]
         [0    0    9    0    0]

      [0     0     0     8      0]
      [                          ]
      [0     0     6     0      2]
      [                          ]
Ls3 = [0    8/3    0    16/3    0]
      [                          ]
      [1     0     6     0      1]
      [                          ]
      [0     4     0     4      0]

         [0    0    0    0    2]
         [                     ]
         [0    0    0    2    0]
         [                     ]
  Ls4 =  [0    0    2    0    0]
         [                     ]
         [0    1    0    1    0]
         [                     ]
         [1    0    0    0    1]