A Tree for Computing the Cayley-Dickson Twist

The Eight Cayley–Dickson Doubling Products

“…Applying rule 9.2 on the previous page for n = 1, 2, 3 gives the quaternion triplets (1, 2, 3), (1,4,5) and (1,6,7). The remaining four can be obtained from (1, 2, 3) and rule 9.4 on the preceding page: (2, 6, 4), (2, 5, 7), (6,3,5) and (4,7,3).…”

“…Reverse these to obtain the triplets for P ⊤ 0 through P ⊤ 3 . P ⊤ 0 : (1, 3, 2), (1,5,4), (1,7,6), (2, 4, 6), (7, 5, 2), (5, 3, 6), (3, 7, 4) P ⊤ 1 : (1, 3, 2), (1,5,4), (1,7,6), (2,4,6), (5, 7, 2), (7, 3, 4), (3, 5, 6) P ⊤ 2 : (1, 3, 2), (1,5,4), (1,7,6), (2, 6, 4), (7, 5, 2), (5, 3, 6), (3, 7, 4) P ⊤ 3 : (1, 3, 2), (1,5,4), (1,7,6), (2,6,4), (5, 7, 2), (7,3,4), (3,5,6) For P 0 , P 3 there are permutations of integers 1 through 7 which take any triple (p, q, r) through all the triples for that product as well as its transpose. For example: P 0 : (1263457)|(1, 2, 3), (2, 6, 4), (6, 3, 5), (3, 4, 7), (4, 5, 1), (5, 7, 2), (7, 1, 6) P 3 : (1243675)|(1, 2, 3), (2, 4, 6), (4, 3, 7), (3, 6, 5), (6, 7, 1), (7, 5, 2), (5,1,4)…”

“…For each of the eight doubling products, the sense of the three sides is the same-either clockwise (←) around the triangle, or counter-clockwise (→). If clockwise, then the three sides of the triangle represent (5, 2, 7), (7,4,3) and (3,6,5). If counter-clockwise then the three sides represent (7, 2, 5), (5,6,3) and (3,4,7).…”

“…The three altitudes may all be in an 'up' sense from base to vertex (↑) or may all be in a 'down' sense from vertex to base (↓). So the altitudes must either be (1, 2, 3), (1,4,5), and (1,6,7) or (1,3,2), (1,5,4), and (1,7,6). All altitudes must have the same sense.…”

“…In [3] it was demonstrated how to use a tree diagram to determine ω 3 (p, q) for any two non-negative integers p and q. Figure 16.1 is the twist tree for ω 2 .…”