Hi Peter,
If you reverse all of the arrows in a Fano Plane representation of an octonion multiplication rule set, you get another octonion rule set of the opposite handedness, but still an octonion.
There are 8 "Right Octonion" rules and 8 "Left Octonion" rules for a given set of 7 permutation triplets. For every right(left) there is a left(right) representation that is the negation of all 7 permutations == switch order in the permutation == change arrow direction in the Fano Plane representation.
Joel and LC,
The statement that there are 480 "different" octonion multiplication tables has to be qualified by a definition of "different" and a blind eye towards what is actually also different but not coming up with the desired number of 480. Proponents of 480 will not dispute that the multiplication rules for unlike octonion basis elements are described by a set of 7 permutation triplet rules where each basis element shows up in 3 of the 7. Once you have a set of 7 permutations, your only degree of freedom for change is negation of individual permutation rules, i.e. (e1 e2 e3) -> (e3 e2 e1). This would permit a maximum of 2^7=128 possible multiplication tables for a given set of 7 permutations. But alas, only 16 of these are alternative composition algebras, a requirement to be called octonion. These are the 16 (8 left and 8 right) I have mentioned.
So to get past 16, you have to say there is something significant with a specific subset of aliasing on the enumeration of the set of 7 vector basis elements. There are after all, 7! ways to alias, a number way bigger than 480. So once you open the door for aliasing, there are actually 7!*16 different tables. The abstractionist often states certain differences are unimportant, yet rarely carries the burden of proof that the assumption is still correct into further extensions of their logic.
When I am asked how many octonion tables there are, I provisionally state 2, since all left are isomorphic algebras and all right are isomorphic algebras. By this I mean they are structurally identical, one can be had from another by a one-to-one and onto re-enumeration, which is NOT possible between left and right. There is no algebraic significance to how we order or re-order the basis elements, we do this to keep them separate in our minds. So I see no fundamental claim of difference out of aliasing.
However, I have exploited the differences between all 16 for a non-aliased representation, so there is certainly physical significance to these differences. These operations I call extra-algebraic since one can only formulate mathematical expressions born out of a single pre-established rule for multiplication. My Sieve Process does break this rule, but only to demonstrate algebraic variance and invariance to sort out just where each product term fits in.
Jens,
Looking forward to your paper on non-associativity. How is it coming? I need to get smarter on your application of octonions and the issues you see from this application, not to mention non-associativity in general.
As you know from our previous dialog, I do not fret much over octonion non-associativity, since it is a central feature of the algebra I love, and I do not try to get outside the confines of the algebra with applications where associativity is assumed or at least required. I see octonion product chains as having both associative and non-associative components. To me this is a good thing, for if we see things in reality that are both, we should have an algebra that likewise has both. The vector triple product is not unknown in physics, so one can't say all of physical reality is associative. The possible lack of associativity is one of nature's way of telling us that product order has significance. This product history is central to the determination of octonion algebraic variance and invariance, and as I have shown, physical reality seems to be built on algebraic invariant forms.
Drs Dray and Monague, I really would like to know your impressions about what I and others have discussed here. Could you be so kind as to drop a line or two?
Thanks,
RL