Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Added the definition of “basic triples” of octonions, and the statement that they form a torsor over .
added the actual definition to octonions
I would like to replace the Fano plane diagram for the octonion multiplication (here) with one whose labels are more well-adapted to applications: The labels , , should be on one line, and maybe best on the inner circle, so that one can readily identify them with the generators of a copy of the quaternions inside the octonions.
Checking, I see that on Wikipedia they had just the same idea for re-labeling: here.
But there they also reversed the direction of the straight inner lines. Hm, is that irrelevant up to isomorphism? Or is that a mistake?
[ Never mind. I see it’s consistent with Cayley-Dickson. ]
Okay, I have made an improved graphics of the “octonion multiplication table”, now well-adapted to inclusion of the quaternions, with the generators labeled according to their Dickson-double incarnation: here.
Now including it into the entry…
I have added pointer to the original references
Arthur Cayley, On certain results relating to quaternions, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 3 Volume 26, 1845 - Issue 171 (doi:10.1080/14786444508562684)
Leonard Dickson, On Quaternions and Their Generalization and the History of the Eight Square Theorem, Annals of Mathematics, Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 (jstor:1967865)
Then I adjusted the first sentences in the Idea-section for clarity,and then expanded the Idea-section to provide some minimum of perspective.
added pointer also to
and will add this to various related entries, too.
1 to 14 of 14