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third point (up to sign). For example, look at the line 137. One should have e1*e3 = e7 (which in the picture is lk), but instead, e1*e3 = ik. This seems to be incorrect in both picture and right table. I think the fix should be to remove the ij in the middle point of the picture, leaving it as e3 = k, changing the bottom right point to e7 = ik (instead of e7 = lk), changing the top point to e5 = jk (instead of li), and changing the bottom left point to e6 = kl (instead of lj). The right table should be adjusted accordingly. Sadly, this is not an edit I can make;
1433:, the current comment either needs to be removed, changed, or a new Fano plane mnemonic needs to be added for that purpose. I guess we could replace the table and the mnemonics to all be based on the 124 triads, but that would be disruptive to the historical use of, and references to, the 123 triad representation in the article. Also, that representation putting the IJK quaternions in the upper left quadrant is also very important for understanding Cayley-Dickson constructions, so changing it would not be a good idea.
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1067:. It is how each of the algebras is derived from the previous one, so how octonions is derived from quaternions. And because of how it works each time the new algebra loses certain properties. As you move from quaternions to octonions you lose associativity, so there is no matrix representation (as matrices are associative).--
815:, whereas the table does. This is sufficient reason to change the article to a consistent format. Perhaps your browser setting result in very similar rendering, but with mine the subscript size gets really messed up with one. But I agree that this is not a reason for any choice. So really, should we go with
1745:
At the top of the page, it says of the octonions "they are alternative. They are also power associative." This is technically correct, but the language suggests that alternativity and power associativity are independent. They aren't - power associativity is a weaker form of (and thus prerequisite to)
1719:
Consistency in diagrams means forcing a selection of a single unique triad set. This would provide clarity in the visual representation, but would limit the understanding of what octonions really are. These two diagrams are likely the most commonly used among about 1/2 dozen across the literature and
1502:
I think the current diagram and the right table are incorrect. The yellow picture in
Octonion-124-137-156-235-267-346-457.svg suggests e3 = ij, while the table and text suggest the convention e4 = ij. The yellow picture also doesn't match the convention that the product of two points on a line is the
1146:
The redactrice of the Quanta
Magazine evidently knows a lot about puffery and qualification of her readership, too (see the comments section there!), but since I am neither female, nor playing an accordeon (not even a bandoneon while doing a tango), and not at all a martial artist, I won't comment on
1137:
Well, I am not scared away by pentagrams, not even with candles in their corners, so why should I shy away from a minimal projective plane. Associating selected objects from the physicists' particle zoo with distinguished elements of an algebra is free, at least for a while. Projections of predictive
632:
I assume "antiassociative" would mean (ab)c = −a(bc). This isn't always true for the octonions: it must be so since the octonions contain copies of the quaternions within them. But it will be true if {a,b} generate an algebra isomorphic to the quaternions that is orthogonal to c (essentially what the
1530:
The problem with the IJKL notation is that in trying to have the headers be consistent with the table math sequences for all 480 representations is very confusing and no longer value added. I tried to make it simple with a consistent header structure linking to e#, but included the mistake of adding
1526:
You are correct about a IJKL notation issue between the Fano diagrams' association of e# format to the header rows in the IJKL multiplication tables and your suggested fix for e3=k is the right approach. I should also remove the =lij on e7=lij=lk. These two changes remove the underlying problem with
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I don't think it's needed. The formatting matches that of the inline math above and below; changing it would make it far less consistent and so visually jarring. And a subjective reason such as 'for aesthetics purposes' is not a good reason to change formatting from one style to another; an article
1537:
IJKL is all just a holdover from extending imaginary to quaternions with only 2 possible representations where ij=k (up to sign). In extending it to octonions, the IJKL convention was largely dropped for readability and the practicality since there are now 480 possible representations (many where
1087:
I guess the question is "what do you mean by matrix representation?" Obviously it is not a subalgebra of a matrix ring because any such algebra is associative. But the elements of any finite dimensional algebra can be viewed as a linear transformation and therefore a square matrix. This is, at
1378:
To be clear, the table and both 2D and 3D Fano mnemonics are (and always have been) consistently based on the triads 123-145-176-246-257-347-365. The reference to the 124-137-156-235-267-346-457 octonion Fano plane is not present in this article (yet). FYI - my triad lists are generated
1436:
Since I was the one who generated these mnemonics from my
Mathematica codebase using any of the 480 possible triad combinations, I created a reference to the different sets (yes, I can create just the 124 Fano plane for reference w/o the triads and tables, if you wish):
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These are not the same diagram because they are not the same octonion (which is addressed in the article text). There are 480 possible and different octonion multiplication matrices. These are typically identified through a set of triads 123-145-176-246-257-347-365 or
1088:
least, the spirit of
Jacobson's structural description of general finite dimensional algebras. That being the case, I'm rewording to be more explicit about the assumption that was previously made: that a 'matrix representation' meant "subalglebra of a matrix ring".
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So this incorrectly implies there is a difference in the octonion triads used in the table and the mnemonics. While I understand the motivation to clarify and explain how to perform the interesting 7-cycle math based on the triangular rotation of the Fano plane
862:
Thanks, I missed the anchor. Anyhow, I'm not trying to dispute the principle that style should not be changed without reason, I only thought it was a link that needed fixing. That still leaves us with my observation that this article mixes HTML with and without
1154:
The linked remarks from 2011 by J. Baez at al., meanwhile ripened during a period of seven years and however prophetic, may also be applicable, but is there any reliable source (not talking about secondary), that is beyond speculative expectations, since
1375:, deleted the tag by adding text that inferred the multiplication table and the mnemonic were different octonions: "...by using the triangular multiplication diagram, or Fano plane, below (and instead of the multiplication table, above)."
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is all there is that I am aware of, for the time being. I am, however, really scared away by the alleged multiple tensor(!) product of groups, or is it rather vector spaces, or division algebrae, being myself really unsure about a valid
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I now see you moved the text (vs. deleting it), but this was not the crux of my point - which was just trying to outline the complex history of the changes related to the confusing set of points being made in the
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It's likely these are identical under relabeling, but it would be a lot easier if a consistent definition was used. I'm just learning this for the first time and got a little confused!
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703:), since the semantics are not of variables. Controlling column width (and making them a little wider) in the table would be good. Ordinarily, I would not agree with removing the
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they are neither commutative nor anticommutative. I don't even know what antiassociative means; there's not a simple negative form of associativity, as there is of commutativity.--
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alternativity. This should probably be rephrased to something like "they are alternative, and thus power associative", but I can't find the best way to phrase that sentence.
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Just wanted to add that I regret having written an uncharitable summary for that edit, I don't think there was malicious intent and I should have assumed good faith anyway.
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but if a quaternion can be represented by a matrix, and octonions can be defined as 4 quaternions then it could be represented uniquely by 4x4x4 (3D) matrix.. No?
1649:
I see what you mean. The diagram should be replaced by separate, smaller diagrams, but I don't have the facilities for doing that. Hopefully someone else will.—
869:: the table uses it, but the only place outside the table where it is used is in the sentence immediately after the table. We should change the article to use
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don't match. For instance, e3 is the center of one, but the other has e4. Also the inner circle is counterclockwise in one while the other is clockwise.
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her authority in estimating the scientific weight of her starring protegee, nor on that of the latter's article. The article in Quanta
Magazine has
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and moved before the citation, which does not involve the term "associative" at all. This makes me suspect it wasn't meant very seriously.
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their differences are interesting (IMO). For the complete set of 480 diagrams and matrices (which includes split octonions by triad), see
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first it was stated: Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions
802:
I see the redirect at the link MOS:STABILITY that you provided does not go anywhere but WP:MOS – any idea which section it should go to?
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template, but here the subscripts end up too large on my browser. An alternative is removing the subscript from the font modification:
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but
Hamilton was already using the term "associative character" to talk about quaternions in his paper presented November 13, 1843:
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Having said this, maybe dropping the all of the IJKL references in the Fano diagrams and deleting the IJKL tables would be better.
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and next time you wrote: Because of their non-associativity, octonions do not have matrix representations, unlike quaternions.
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Snark aside, I fail to see why this isn't serious contemporary research worthy of note on
Knowledge. Flesh it out for me.
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I removed a remark: "Hamilton invented the word associative so that he could say that octonions were not associative."
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Please, feel free to edit according to your estimation what is best for the encyclopedicity of this article.
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Hopefully this is sufficient explanation to remove the remark rather than sticking a citation request on it.
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What do you guys think about slightly increasing the width of each column in the table? What about replacing
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Sorry, my mastery of the
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One of the graphs stick out past the Tools bar. I don't know how to fix this, could someone else do it?
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goes to the third para of the lead: "Style and formatting..." with links to relevant arbitration cases.
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Both of these changes would only be for aesthetic purposes. It would match the corresponding table for
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Graves wrote to
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on
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Could you be specific, please? It would be easier to address your issue if you made it clear.—
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The lead says that they are noncommutative and nonassociative. They are actually
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correctly added a tag that some text related to a 7-cycle construction based on
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A problematic reference to the multiplication table and mnemonics used based on
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The formatting is not consistent: the inline math does not in general use
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It's the IJKL graph, I'm on a Chromebook so that might be the reason.
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No: ℝ and ℂ are both subspaces which are commutative, while any two
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the math elements in the e3 and e7 Fano plane nodes. My apologies.
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Hello. I do not think this is resolved. Actually, it is true that
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trying to link IJKL headers to the e# on all 480 diagrams.
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Since this ia a Talk page, references stay in sections. —
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They look fine to me. Which graph are you referring to?—
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379:{\displaystyle e_{i}e_{j}=-e_{j}e_{i}\neq e_{j}e_{i}\,}
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power of these associations are nevertheless missing.
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should not be changed just for such reasons. See e.g.
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https://www.math.columbia.edu/~woit/wordpress/?p=3665
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Much easier to simply reduce the image size - fixed.
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765:. Do you think we should go ahead with the change?
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459:) It corrected the affirmation in the article.--
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1364:deleted some of that text, but not the tag.
1534:So I have corrected those WP Fano diagrams.
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749:. There isn't much difference between
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452:{\displaystyle e_{0}e_{i}=e_{i}e_{0}}
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488:Thank you, that helps clear it up.
95:This article is within the scope of
38:It is of interest to the following
923:this was credited to Cayley, then
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1775:Low-priority mathematics articles
1206:Wolchover, Natalie (2018-07-20).
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1426:{\displaystyle e_{1}e_{2}=e_{4}}
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1371:, with some clarification by
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399:It is not true if i or j =0 (
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227:and antiassociative, right?
109:and see a list of open tasks.
1770:C-Class mathematics articles
1716:124-137-156-235-267-346-457.
1563:English next time please???
1119:Still not even wrong or no?
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1022:21:16, 11 October 2017 (UTC)
1517:13:10, 6 October 2021 (UTC)
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1065:Cayley–Dickson construction
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910:Invention of "associative"
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879:uniformly, or remove the
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1629:UnqaidIntern
1600:UnqaidIntern
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1215:. Retrieved
1211:
1201:
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1148:
1139:
1115:Than again:
1114:
1112:
1109:
1042:— Preceding
1039:
1036:
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982:
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635:Double sharp
631:
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256:
251:
247:
242:
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222:
203:
177:
169:
156:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
767:BradBentz44
677:BradBentz44
274:quaternions
112:Mathematics
103:mathematics
59:Mathematics
1764:Categories
1695:this image
1691:this image
1594:UI problem
1565:Carlimited
1489:Pianostar9
1483:I believe
1217:2018-07-27
1194:References
1727:Jgmoxness
1665:Jgmoxness
1651:Anita5192
1615:Anita5192
1580:Anita5192
1548:Jgmoxness
1544:Thoughts?
1470:Jgmoxness
1178:kencf0618
1140:Promising
1121:kencf0618
977:: 424–434
952:: 338–341
673:sedenions
461:Cbigorgne
1748:Cyversch
1523:article.
1143:outcome.
1090:Rschwieb
1056:contribs
1048:Xakepp35
1044:unsigned
1000:Agashlin
985:Agashlin
965:(1844),
963:Hamilton
940:(1848),
938:Hamilton
885:usage. —
882:{{math}}
872:{{math}}
866:{{math}}
840:MOS:MATH
705:{{math}}
484:Resolved
204:365 days
170:Archives
1485:my edit
1014:Rgdboer
888:Quondum
829:Quondum
719:Quondum
701:''...''
490:Bubba73
392:Bubba73
229:Bubba73
139:on the
30:C-class
1702:Pstetz
1559:Huh???
36:scale.
1160:Purgy
1155:then?
1075:deeds
854:deeds
792:deeds
661:with
284:deeds
271:: -->
1752:talk
1731:talk
1706:talk
1693:and
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1655:talk
1633:talk
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1604:talk
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1474:talk
1453:123
1440:124
1182:talk
1164:talk
1125:talk
1094:talk
1052:talk
1018:talk
1004:talk
989:talk
875:and
820:math
810:math
771:talk
758:and
681:talk
639:talk
624:talk
592:and
465:talk
255:for
246:and
917:In
826:? —
784:.--
692:var
560:if
131:Low
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259:≠
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42::
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