r denotes the golden ratio (1+sqrt 5)/2 = 1.61803... Type A B C D E F G H I J K L M N O P Q R S T U V W Singletons 0 0 0 0 0 0 0 4 2 4 4 4 2 4 8 10 10 4 6 6 6 6 4 Symmetries 4 4 2 2 2 2 2 3 1 2 2 1 1 3 1 2 3 2 2 1 1 1 2 Coordinates ------------------------------------------------------------------------------------- (0, 0, 0, 2) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (0, r-1, 1, r) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (0, 1-r, 1, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, 0, r-1, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 0, r-1, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r-1, 1, 0, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, 1, 0, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r-1, -1, 0, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, -1, 0, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, 0, 1-r, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 0, 1-r, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, r-1, -1, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, 1-r, -1, r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r-1, 0, r, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, 0, r, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, 1, 1, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 1, 1, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, -1, 1, 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (-1, -1, 1, 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (0, r, r-1, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, -r, r-1, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r, r-1, 0, 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (-r, r-1, 0, 1) 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (r, 1-r, 0, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 1-r, 0, 1) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, r, 1-r, 1) 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (0, -r, 1-r, 1) 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1, 1, -1, 1) 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 1, -1, 1) 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, -1, -1, 1) 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (-1, -1, -1, 1) 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r-1, 0, -r, 1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, 0, -r, 1) 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (0, 1, r, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, -1, r, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r, 0, 1, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 0, 1, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, r, 0, r-1) 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, r, 0, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, -r, 0, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, -r, 0, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r, 0, -1, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 0, -1, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, 1, -r, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, -1, -r, r-1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, 0, 2, 0) 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1, r-1, r, 0) 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 (-1, r-1, r, 0) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, 1-r, r, 0) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 1-r, r, 0) 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r-1, r, 1, 0) 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, r, 1, 0) 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 (r-1, -r, 1, 0) 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 (1-r, -r, 1, 0) 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 (r, 1, r-1, 0) 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 1, r-1, 0) 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 (r, -1, r-1, 0) 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, -1, r-1, 0) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, 2, 0, 0) 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2, 0, 0, 0) 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (-2, 0, 0, 0) 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 (0, -2, 0, 0) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r, 1, 1-r, 0) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 1, 1-r, 0) 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 (r, -1, 1-r, 0) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, -1, 1-r, 0) 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 (r-1, r, -1, 0) 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, r, -1, 0) 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 (r-1, -r, -1, 0) 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, -r, -1, 0) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, r-1, -r, 0) 0 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (-1, r-1, -r, 0) 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 (1, 1-r, -r, 0) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 1-r, -r, 0) 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 (0, 0, -2, 0) 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, 1, r, 1-r) 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, -1, r, 1-r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r, 0, 1, 1-r) 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 0, 1, 1-r) 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 (1, r, 0, 1-r) 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 (-1, r, 0, 1-r) 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 (1, -r, 0, 1-r) 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 (-1, -r, 0, 1-r) 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 (r, 0, -1, 1-r) 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 0, -1, 1-r) 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 (0, 1, -r, 1-r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 (0, -1, -r, 1-r) 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 0 1 1 (r-1, 0, r, -1) 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, 0, r, -1) 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 (1, 1, 1, -1) 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 1, 1, -1) 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 (1, -1, 1, -1) 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 0 0 (-1, -1, 1, -1) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (0, r, r-1, -1) 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 (0, -r, r-1, -1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r, r-1, 0, -1) 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, r-1, 0, -1) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (r, 1-r, 0, -1) 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (-r, 1-r, 0, -1) 1 0 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 (0, r, 1-r, -1) 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 (0, -r, 1-r, -1) 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 (1, 1, -1, -1) 1 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 (-1, 1, -1, -1) 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1, -1, -1, -1) 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 (-1, -1, -1, -1) 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 (r-1, 0, -r, -1) 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 (1-r, 0, -r, -1) 1 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 (0, r-1, 1, -r) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 (0, 1-r, 1, -r) 0 1 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 (1, 0, r-1, -r) 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 1 (-1, 0, r-1, -r) 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 (r-1, 1, 0, -r) 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, 1, 0, -r) 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 (r-1, -1, 0, -r) 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (1-r, -1, 0, -r) 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 (1, 0, 1-r, -r) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 (-1, 0, 1-r, -r) 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 1 (0, r-1, -1, -r) 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 (0, 1-r, -1, -r) 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (0, 0, 0, -2) 1 0 1 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0