Regular dodecahedron, vertices (exact coordinates, phi = (1+sqrt 5)/2):
   0: (1, 1, 1)   neighbours [8, 9, 10]
   1: (1, 1, -1)   neighbours [9, 11, 16]
   2: (1, -1, 1)   neighbours [10, 12, 14]
   3: (1, -1, -1)   neighbours [12, 16, 17]
   4: (-1, 1, 1)   neighbours [8, 13, 15]
   5: (-1, 1, -1)   neighbours [11, 15, 19]
   6: (-1, -1, 1)   neighbours [13, 14, 18]
   7: (-1, -1, -1)   neighbours [17, 18, 19]
   8: (0, 1/phi, phi)   neighbours [0, 4, 14]
   9: (1/phi, phi, 0)   neighbours [0, 1, 15]
  10: (phi, 0, 1/phi)   neighbours [0, 2, 16]
  11: (0, 1/phi, -phi)   neighbours [1, 5, 17]
  12: (1/phi, -phi, 0)   neighbours [2, 3, 18]
  13: (-phi, 0, 1/phi)   neighbours [4, 6, 19]
  14: (0, -1/phi, phi)   neighbours [2, 6, 8]
  15: (-1/phi, phi, 0)   neighbours [4, 5, 9]
  16: (phi, 0, -1/phi)   neighbours [1, 3, 10]
  17: (0, -1/phi, -phi)   neighbours [3, 7, 11]
  18: (-1/phi, -phi, 0)   neighbours [6, 7, 12]
  19: (-phi, 0, -1/phi)   neighbours [5, 7, 13]

Distance matrix d(i, j) of the graph of the dodecahedron (rows and columns 0..19):
   0: 0 2 2 3 2 3 3 5 1 1 1 3 3 3 2 2 2 4 4 4
   1: 2 0 3 2 3 2 5 3 3 1 2 1 3 4 4 2 1 2 4 3
   2: 2 3 0 2 3 5 2 3 2 3 1 4 1 3 1 4 2 3 2 4
   3: 3 2 2 0 5 3 3 2 4 3 2 2 1 4 3 4 1 1 2 3
   4: 2 3 3 5 0 2 2 3 1 2 3 3 4 1 2 1 4 4 3 2
   5: 3 2 5 3 2 0 3 2 3 2 4 1 4 2 4 1 3 2 3 1
   6: 3 5 2 3 2 3 0 2 2 4 3 4 2 1 1 3 4 3 1 2
   7: 5 3 3 2 3 2 2 0 4 4 4 2 2 2 3 3 3 1 1 1
   8: 1 3 2 4 1 3 2 4 0 2 2 4 3 2 1 2 3 5 3 3
   9: 1 1 3 3 2 2 4 4 2 0 2 2 4 3 3 1 2 3 5 3
  10: 1 2 1 2 3 4 3 4 2 2 0 3 2 4 2 3 1 3 3 5
  11: 3 1 4 2 3 1 4 2 4 2 3 0 3 3 5 2 2 1 3 2
  12: 3 3 1 1 4 4 2 2 3 4 2 3 0 3 2 5 2 2 1 3
  13: 3 4 3 4 1 2 1 2 2 3 4 3 3 0 2 2 5 3 2 1
  14: 2 4 1 3 2 4 1 3 1 3 2 5 2 2 0 3 3 4 2 3
  15: 2 2 4 4 1 1 3 3 2 1 3 2 5 2 3 0 3 3 4 2
  16: 2 1 2 1 4 3 4 3 3 2 1 2 2 5 3 3 0 2 3 4
  17: 4 2 3 1 4 2 3 1 5 3 3 1 2 3 4 3 2 0 2 2
  18: 4 4 2 2 3 3 1 1 3 5 3 3 1 2 2 4 3 2 0 2
  19: 4 3 4 3 2 1 2 1 3 3 5 2 3 1 3 2 4 2 2 0

The 15 half-turns about the axes through the midpoints of opposite edges
(axis = direction of the midpoint of the reversed edges; permutation i -> h(i)):
  edges (0, 8), (7, 17); 2*midpoint (1, phi, phi^2); h = [8, 6, 15, 19, 10, 12, 1, 17, 0, 14, 4, 18, 5, 16, 9, 2, 13, 7, 11, 3]
  edges (0, 9), (7, 18); 2*midpoint (phi, phi^2, 1); h = [9, 8, 5, 13, 16, 2, 17, 18, 1, 0, 15, 14, 19, 3, 11, 10, 4, 6, 7, 12]
  edges (0, 10), (7, 19); 2*midpoint (phi^2, 1, phi); h = [10, 14, 9, 4, 3, 18, 11, 19, 16, 2, 0, 6, 15, 17, 1, 12, 8, 13, 5, 7]
  edges (1, 9), (6, 18); 2*midpoint (phi, phi^2, -1); h = [11, 9, 19, 4, 3, 10, 18, 14, 17, 1, 5, 0, 13, 12, 7, 16, 15, 8, 6, 2]
  edges (1, 11), (6, 14); 2*midpoint (1, phi, -phi^2); h = [7, 11, 13, 15, 12, 16, 14, 0, 18, 17, 19, 1, 4, 2, 6, 3, 5, 9, 8, 10]
  edges (1, 16), (6, 13); 2*midpoint (phi^2, 1, -phi); h = [17, 16, 5, 9, 18, 2, 13, 8, 7, 3, 11, 10, 15, 6, 19, 12, 1, 0, 4, 14]
  edges (2, 10), (5, 19); 2*midpoint (phi^2, -1, phi); h = [12, 6, 10, 8, 17, 19, 1, 15, 3, 18, 2, 13, 0, 11, 16, 7, 14, 4, 9, 5]
  edges (2, 12), (5, 15); 2*midpoint (phi, -phi^2, 1); h = [7, 13, 12, 14, 11, 15, 16, 0, 17, 19, 18, 4, 2, 1, 3, 5, 6, 8, 10, 9]
  edges (2, 14), (5, 11); 2*midpoint (1, -phi, phi^2); h = [18, 19, 14, 4, 3, 11, 10, 9, 12, 7, 6, 5, 8, 16, 2, 17, 13, 15, 0, 1]
  edges (3, 12), (4, 15); 2*midpoint (phi, -phi^2, -1); h = [19, 6, 17, 12, 15, 8, 1, 10, 5, 13, 7, 14, 3, 9, 11, 4, 18, 2, 16, 0]
  edges (3, 16), (4, 13); 2*midpoint (phi^2, -1, -phi); h = [7, 12, 11, 16, 13, 14, 15, 0, 19, 18, 17, 2, 1, 4, 5, 6, 3, 10, 9, 8]
  edges (3, 17), (4, 8); 2*midpoint (1, -phi, -phi^2); h = [13, 18, 5, 17, 8, 2, 9, 16, 4, 6, 19, 12, 11, 0, 15, 14, 7, 3, 1, 10]
  edges (8, 14), (11, 17); 2*midpoint (0, 0, 2*phi); h = [6, 7, 4, 5, 2, 3, 0, 1, 14, 18, 13, 17, 15, 10, 8, 12, 19, 11, 9, 16]
  edges (9, 15), (12, 18); 2*midpoint (0, 2*phi, 0); h = [5, 4, 7, 6, 1, 0, 3, 2, 11, 15, 19, 8, 18, 16, 17, 9, 13, 14, 12, 10]
  edges (10, 16), (13, 19); 2*midpoint (2*phi, 0, 0); h = [3, 2, 1, 0, 7, 6, 5, 4, 17, 12, 16, 14, 9, 19, 11, 18, 10, 8, 15, 13]

Images of v = (1,1,1) (vertex 0) under the 15 half-turns, with the distance from v:
  h(v) = (0, 1/phi, phi)   distance 1
  h(v) = (1/phi, phi, 0)   distance 1
  h(v) = (phi, 0, 1/phi)   distance 1
  h(v) = (-1, -1, 1)   distance 3
  h(v) = (-1, 1, -1)   distance 3
  h(v) = (-phi, 0, 1/phi)   distance 3
  h(v) = (0, 1/phi, -phi)   distance 3
  h(v) = (1, -1, -1)   distance 3
  h(v) = (1/phi, -phi, 0)   distance 3
  h(v) = (-1/phi, -phi, 0)   distance 4
  h(v) = (-phi, 0, -1/phi)   distance 4
  h(v) = (0, -1/phi, -phi)   distance 4
  h(v) = (-1, -1, -1)   distance 5
  h(v) = (-1, -1, -1)   distance 5
  h(v) = (-1, -1, -1)   distance 5
  -> identical to the table in the proof of Lemma 7.2 of the note
  vertices at distance 2 from v: (1, 1, -1), (1, -1, 1), (-1, 1, 1), (0, -1/phi, phi), (-1/phi, phi, 0), (phi, 0, -1/phi)

For every vertex v: the multiset of the 15 distances d(v, h(v)) is [((1, 3), (3, 6), (4, 3), (5, 3))]
d(v, h(v)) is never 0 and never 2, for all 20 vertices and all 15 half-turns: True
ALL CHECKS PASSED
