Talk:Octonion/Archive 1

Archive 1

External links

The external link to the fractal site 404s mavhc — Preceding unsigned comment added by Mavhc (talk • contribs) 19:34, 25 January 2005 (UTC)

Untitled

Could you describe what are they used for ? --Taw

---

I have a couple of problems with this:

Octonions form the largest normed division algebra. These are algebras in which it is possible to define division and the norm of a number, in the algebra of complex numbers the norm of a number is equal to that number multiplied by its complex conjugate.The norm plays a vital role in the study of abstract algebras.

However, a norm cannot be defined for the elements of any algebra larger than the octonions, a notable example is the case of the sedenions these 16 dimensional numbers have little or no algebraic structure and have very little value.

The second property mentioned above is that of division being defined in an algebra, the octonions form the largest algebra in which division is defined, again take the case of the sedenions, it is possible to choose two non-zero sedenions, S and T such that S*T=0.

This is not possible in any smaller algebra, such as the real numbers, complex numbers, quaternions or octonions.

These facts have brought octonions to the forefront of research in modern physics at present, in superstring theory the existence of norms and division in algebras of dimensions 1, 2, 4 and 8 endows spaces of dimensions 3, 4, 6 and 10 with exceptional properties, the 10 dimensional case being very important to string theory.

First, our article on normed division algebras claims that the quaternions are the largest. This is probably due to the fact that our division algebras have to be associative.

Second, there are smaller algebras where division is not defined; take for instance the dual numbers.

These things need to be cleaned up. We should probably start with cleanly defining the norm of an octonion. AxelBoldt 05:03 Jan 4, 2003 (UTC)

Also the claim that there are five exceptional Lie groups, as opposed to five types or classes of exceptional Lie group. There are five exceptional Dynkin diagrams, but that's not the same thing. (I took out five; that's almost certainly not the best solution. Let who can improve it.) Septentrionalis 03:26, 26 September 2005 (UTC)

If a real number represents the mass of an object, i, j, and k represent the position, and l represents time, than you'd end up with one number for the mass, position, velocity, and time of an object. If you square the position or velocity, you'd always get a negetive real number. Using that, than if mass is positive, than energy (mass times velocy squared) is negitive, and vice versa. — Daniel 22:15, 20 November 2005 (UTC)

Question

How to build that multiply table? --80.178.62.64 12:58, 19 August 2007 (UTC)

• Hello - One way is through the Cayley-Dickson construction. Another would be to begin with quaternions to basis $\{1,i,j,k\}$ and propose another basis l with $l^{2}=-1$ and demanding that the resulting algebra is a vector space over the reals equipped with a multiplicative norm. Jens Koeplinger 12:45, 20 August 2007 (UTC)

It looks backward to me. For example, $i(kl)=ik\times l=-ki\times l=-jl$. What am I doing wrong? Phoenix1304 (talk) 14:58, 6 April 2008 (UTC)

You need to take non-associativity into account: If you look closely at the brackets, you see that $i(kl)=jl$, however, $i(kl)=-(ik)l=-(-j)l=jl$. So, in your equation, the step from the first expression to the second expression misses a minus sign.

But - in case you're interested - not every 3-tuple is non-associative. In general, any three octonion basis elements $\{a,b,c\}$ are associative if they satisfy one (ore more) of these properties:

1. One (or more) of the $\{a,b,c\}$ is 1.
2. Two of the $\{a,b,c\}$ multiply to $\pm 1$.
3. The set $\{a,b,c\}$ is multiplicatively closed (with a factor $\pm 1$), i.e. $ab=\pm c$.

If any of these three conditions are satisfied, then a triplet of octonion basis elements is associative: $(ab)c=a(bc)$. If none are satisfied, as it was in your example, then the triplet is anti-associative, i.e. $(ab)c=-a(bc)$.

Does this answer your question? Thanks, Koeplinger (talk) 16:23, 6 April 2008 (UTC)

I guess at the root of my problem is I don't fully understand where the non-associativity originates. I mean, with quaternions the non-commutivity came about (at least in my education) as an attempt to eliminate the bcij and bcji terms when normalizing. Phoenix1304 (talk) 16:51, 6 April 2008 (UTC)

It goes to the requirement of adding an imaginary basis element to your existing set of basis elements, while maintaining a multiplicative norm, i.e. $||a||||b||=||ab||$ for every element of your number systems. Together with the requirement that every non-zero number has a non-zero norm (to arrive at a "normed division algebra"), you must lose commutativity for quaternions, and you must lose associativity for octonions. Let's see:

For quaternions, beginning with the complex basis $\{1,i\}$, you want to add another imaginary basis $j$ with $j^{2}=-1$ such that j is different from i, any non-zero element of your space has non-zero norm, and the norm is multiplicative. If you take two numbers $A:=i+j$ and $B:=i-j$, then their product is $AB=-1-ij+ji+1$. As you already said, if ij were to be ji, then the norm of the product would be zero, which violates our requirements. With $ji:=-ij$, however, we are not violating our requirements. After checking all the other basis elements, one can see that the quaternions satisfy are OK.

For octonions, we begin at the quaternion basis $\{1,i,j,k\}$ and require an additional imaginary basis, l, with $l^{2}=-1$. You can show, just as for the quaternions, that l must also be anti-commutative with all your quaternion basis elements. For associativity, we now need three numbers, for example $A:=i+j$, $B:=i-l$, and $C:=j-l$ and their product is:

$(AB)C=((i+j)(i-l))(j-l)$

$=(-1-il+ji-jl)(j-l)$

$=-j-l-(il)j+(il)l+(ji)j-(ji)l-(jl)j+(jl)l$

Assuming that associativity would hold, we would have:

$ABC=-j-l+kl-i+i+kl-l-j=-2j-2l+2kl$

The norm of this would be ${\sqrt {12}}$, which is wrong. Instead, the two terms $-(il)j$ and $-(ji)l$ are anti-associative and cancel each other out; the correct result is:

$(AB)C=-j-l+kl-i+i-kl-l-j=-2j-2l$

It has norm ${\sqrt {8}}$ as it should be. Hope this helps! (And I hope there are no more errors in the calculations!) Thanks, Jens Koeplinger (talk) 21:00, 6 April 2008 (UTC)

(removed the "PS" that was here, because it was wrong; the above example is correct ... unless you can prove me differently? I would have loved to have an example where the product evaluates to zero under the assumption of associativity) -- sorry for that. Thanks, Jens Koeplinger (talk) 04:12, 7 April 2008 (UTC)

Thanks a lot! That really helps me get my head around the article. Phoenix1304 (talk) 14:34, 7 April 2008 (UTC)

Koeplinger's argument can be considerably simplified. Assume we have a real associative algebra with elements $i,j,k=ij,l$ which all anticommute. We make no assumptions as to the squares. Then

$kl=ijl=-ilj=+lij=lk=-kl$

so $kl=0$. This algebra has zero divisors and so the norm cannot be multiplicative. Worse, if $k^{2}=-1$ then $l=0$, or if $l^{2}=-1$ then $k=0$.

In other words, if we assume that $l$ anticommutes with both $i$ and $j$ then it must commute with the product $ij$. One then runs into the same problems as before. The only way to make $l$ anticommute with $k$ and still get a nonzero product is to drop the assumption of associativity. -- Fropuff (talk) 19:29, 7 April 2008 (UTC)

Thanks for that example! It doesn't require a particular definition of norm, either, which is desirable as well. Koeplinger (talk) 22:20, 7 April 2008 (UTC)

There needs to be a section (more than a link?) on Ferdinand Degen - the Dane who found the 8-square norm identity needed for the octonions. He pre-dates others in his discovery - yet remains almost unknown.

66.130.86.141 (talk) 03:51, 18 November 2009 (UTC)

Format changes

The recent format changes I'm afraid render the table less easy to read, not more, at least for me as the font used is a particularly ugly one on my machine. The manual of style is very clear on such changes: at WP:MOSCOLOR it says it is "almost never a good idea" to change the font, with reasons. It is not clear from the edit summaries why the changes were made, and there is nothing wrong with the default font, so if they can't be re-done in a way in line with the style guidelines they should be reverted.--JohnBlackburne^words_deeds 23:31, 20 April 2010 (UTC)

The font change aimed at was simply from l to math font $\ l\$ used in the text. Unfortunately, the table formatting doesn't seem to recognize the math script so I used Cambria: l, which looks pretty much the same as math $\ l\$ used in the text, both in Firefox and IE on my PC. Otherwise, the simple font l looks a lot like the number ‘1’ or a capital ‘I’ Brews ohare (talk) 00:37, 21 April 2010 (UTC)

That does not help users (like me) who do not have that font: it's for precisely that reason specifying individual fonts is bad idea, as you can't guarantee users has a particular font, or if they do it will look the same on all machines. Again, the manual of style is very clear on this.--JohnBlackburne^words_deeds 08:31, 21 April 2010 (UTC)

When I expressed a similar problem with your use of a symbol for the wedge in the wedge product, you were most unaccommodating, and simply told me to change my browser settings. which I did (to Arial Unicode MS). I also could change to Cambria math. Maybe you can find a setting that reads Cambria fonts? Brews ohare (talk) 14:58, 21 April 2010 (UTC)

The wedge symbol is a standard unicode symbol. WP is unicode based, has been for years, and few OSes don't support Unicode (I think the most recent is Windows ME). You don't even need to know how to type or or copy and paste it, as it's in the edittools, in the Math and logic set, about in the middle of them. Cambria is a commercial font. To view it I, and many others would have to pay for it, for $35. Are you suggesting users should have to pay to view the page properly?--JohnBlackburne^words_deeds 17:32, 21 April 2010 (UTC)

I'd suggest you explore a bit and come up with a font that makes an l that looks like l. Brews ohare (talk) 22:44, 21 April 2010 (UTC)

Here's Corbel l; Georgia l; Bookman Old Style l; Calisto MT l.

Surely one will be readable by you? Brews ohare (talk) 22:59, 21 April 2010 (UTC)

They are all readable: my OS does a good job of substituting other fonts. But I would think they look nothing like they do on your machine, and will look different again for other users with different installed fonts. I.e. there is no alternative font that will work better for all users. I assure you the table looks worse for me than it did, and as the manual of style is very clear on this the fonts should be removed.--JohnBlackburne^words_deeds 23:25, 21 April 2010 (UTC)

Well, these fonts all look very similar on my PC, which is probably the same as 3/4 of the PC's out there, and very similar to $\ l\$, and you say your machine does a "fine job" with other fonts, so possible we are just talking about aesthetics, eh? Let's see what others have to say on this matter. Brews ohare (talk) 00:19, 22 April 2010 (UTC)

What "others have to say" is the style guidelines. As you have not fixed it yourself, and the style guidelines are very clear on this, I've just restored it to the state it was before you started changing the formatting. --JohnBlackburne^words_deeds 09:33, 22 April 2010 (UTC)

Thanks a bunch, John. I reintroduced the table with my changes and replaced the l with ‘l’ according to What "others have to say" is the style guidelines. You could have done that too, instead of reverting everything, making more trouble. Brews ohare (talk) 19:49, 22 April 2010 (UTC)

Here's a Unicode example, but users might not yet have a font installed that supports the code points in all the Math ranges. (Also when editing, cursor placement using the mouse can be a little tricky in Firefox). I could upload it as an image, or convert it to a real html table, if that would help with the WP:MATH wiki mark–up, (I'm not familiar with it).

Another thing that might work is doing it like so:

<span style='font-family: "Bookman Old Style", Times, "Times New Roman", serif; font-size:110%; font-style:italic;'>l</span>

It should default correctly that way if users don't have "Bookman Old Style". Unfortunately, the Unicode solution will probably just be blank for unsupported characters, Or there's option 4, I can just go away—no worries! ̼⌂̺͛ᴖ̲̿ᴥ̲̿ᴖ̺͛⌂̼

𝑒₀	𝑖, 𝑒₁	𝑗, 𝑒₂	𝑘, 𝑒₃	𝑙, 𝑒₄	𝑖𝑙, 𝑒₅	𝑗𝑙, 𝑒₆	𝑘𝑙, 𝑒₇
𝑖, 𝑒₁	−𝟷	𝑘	−𝑗	𝑖𝑙	−𝑙	−𝑘𝑙	𝑗𝑙
𝑗, 𝑒₂	−𝑘	−𝟷	𝑖	𝑗𝑙	𝑘𝑙	−𝑙	−𝑖𝑙
𝑘, 𝑒₃	𝑗	−𝑖	−𝟷	𝑘𝑙	−𝑗𝑙	𝑖𝑙	−𝑙
𝑙, 𝑒₄	−𝑖𝑙	−𝑗𝑙	−𝑘𝑙	−𝟷	𝑖	𝑗	𝑘
𝑖𝑙, 𝑒₅	𝑙	−𝑘𝑙	𝑗𝑙	−𝑖	−𝟷	−𝑘	𝑗
𝑗𝑙, 𝑒₆	𝑘𝑙	𝑙	−𝑖𝑙	−𝑗	𝑘	−𝟷	−𝑖
𝑘𝑙, 𝑒₇	−𝑗𝑙	𝑖𝑙	𝑙	−𝑘	−𝑗	𝑖	−𝟷

Octonion multiplication table Note antisymmetry about diagonal.

Octonion multiplication table

Note antisymmetry about diagonal.

Regards—Machine Elf 1735 (talk) 15:00, 23 April 2010 (UTC)

That displays as lots of little rectangles for me: both when views normally and when editing it I see letter shaped rectangles. Odd as I normally don't have problems with Unicode, but it seems those are too obscure for the browser. I only see them if I copy and paste them into a text editor. Then they just look like italic letters, so just using i, j, or i, j, would be far better. Your table also uses non-standard subscripts which are not recommended (see MOS:MATH#Superscripts and subscripts), and the fonts, though better than what was added before as you've listed alternatives, is still against the recommendations at WP:MOSCOLOR.

Interestingly it now does work for me. I installed the font Code2001, from here, which includes all the math Unicode symbols and seems to have cured the problems with your table. But that's not something we can expect readers to do, and it hardly seems necessary as they are just italic serif letters, and not the best looking ones at that.--JohnBlackburne^words_deeds 18:35, 23 April 2010 (UTC)

I've read them but the alpha characters were going to be a show stopper anyway. I thought it was a little prettier this way, (just a Unicode 'font' demo, and now a picture).

Except for the size, the style setting should default to serif on your box and look about the same as the example you sent, {{math|<big>i, j</big>}}, unless you have "Times" or "Times New Roman" installed. How do they look? Here's the html table:

e0	i, e1	j, e2	k, e3	l, e4	il, e5	jl, e6	kl, e7
i, e1	−1	k	−j	il	−l	−kl	jl
j, e2	−k	−1	i	jl	kl	−l	−il
k, e3	j	−i	−1	kl	−jl	il	−l
l, e4	−il	−jl	−kl	−1	i	j	k
il, e5	l	−kl	jl	−i	−1	−k	j
jl, e6	kl	l	−il	−j	k	−1	−i
kl, e7	−jl	il	l	−k	−j	i	−1

Octonion multiplication table Note antisymmetry about diagonal.

—Machine Elf 1735 (talk) 19:12, 23 April 2010 (UTC)

The last one looks better, but you don't need to specify the font to get italic letters so should not do so: better to leave the fonts as the default ones, which look fine, for consistency and so the user can override them with a style sheet or by choosing a different skin.

Not sure if my 2c are relevant, but in general I would suggest that if beauty or typesetting is getting relevant, to rather create an SVG image and embed it. But when it doesn't have to be typeset perfectly, then I vote for MediaWiki markup and simplicity almost all of the time. This gives portability across many different uses, including handheld devices, print to PDF, usability helpers, and the various skins and rendering choices people prefer. In the case of the above discussion, my vote is to use <math>...</math> or {{math|...}}. I would generally ask to avoid specifying fonts, sizes, or forcing PNG rendering by injecting "\ " into the TeX instruction (again, if MediaWiki markup proves to be insufficient in demonstrating the content, then this could be moved into an SVG image). I'm using Firefox on openSuSE Linux, and Opera on Windows CE, and what I'm seeing rendered mostly does not match the description of a particular suggested style change above. Thanks, Jens Koeplinger (talk) 13:11, 26 April 2010 (UTC)

Serif & italic

Started a new section, the above was getting long. I think User Talk:Brews ohare would like a serif font as well as italics.—Machine Elf 1735 (talk) 19:41, 23 April 2010 (UTC)

Because its font family, serif will be what you've set your browser's serif font to be.—Machine Elf 1735 (talk) 19:59, 23 April 2010 (UTC)

I'm not sure as he's redone it in a sans-serif style: he could just prefer the Cambria font, which was unacceptable as (as with any font) it's one not all users have. As for the default font it's set by WPs style sheets, as if I change my browser font to something else it makes no difference here. Of course you can override using style sheets in your own user space.--JohnBlackburne^words_deeds 20:09, 23 April 2010 (UTC)

Because lowercase L and the number 1 are so similar, he suggested various serif fonts like like the WP:MATH text: $/l/$ & / l /. Its reasonable to change the table from sans-serif to serif in order to clear that up.

For avoidance of doubt, I'll change the table so that serif is the only font family specified. I'll also change the style from italic to normal on the diagonal negative ones.

I think that achieves maximum respect for the reader's preference and adequately addresses the visual ambiguity.—Machine Elf 1735 (talk) 22:15, 23 April 2010 (UTC)

As noted above it should not be done with font styles or families, as it will break the formatting for many people reading it. The manual of style is very clear on this, and it's completely unnecessary as it's easy to do without specifying the style or family.--JohnBlackburne^words_deeds 22:22, 23 April 2010 (UTC)

I see you've now redone it using fonts again, despite the manual of style being clear on this. Here's one reason why, i.e. one example of it not working. Or this which for me replaces my default, readable, font with a less readable one in the table. If you did it without specifying the font family it would work optimally in all skins and with whatever other preferences the user sets.--JohnBlackburne^words_deeds 22:51, 23 April 2010 (UTC)

Please reflect and consider that you may be overstating the matter in absolute terms. It reads "…except in special cases." It would typically not be a good idea for the majority of users because css is complicated. There's a very good reason here.—Machine Elf 1735 (talk) 22:57, 23 April 2010 (UTC)

Both of those presentations look entirely acceptable to me. Can you please take screen shots of what you're seeing? As I said, the default "serif" font is set in the browser.—Machine Elf 1735 (talk) 23:05, 23 April 2010 (UTC)

I'm sorry, I don't understand what you mean to say exactly with "redone it using fonts again". Thank you.—Machine Elf 1735 (talk) 23:08, 23 April 2010 (UTC)

I mean again after User:Brews ohare did it - the version I reverted, after pointing out the problems to him and suggesting he do it another way.

As for 'except in special cases' this is not one, especially as serif fonts can be easily done without recourse to them, e.g. with the math template as I illustrated above. See MOS:MATH#Font formatting for more details.

As for they look "acceptable to me", the point is that by specifying fonts you are overriding someone's CSS. That might be in a different skin, or it might be their custom CSS, or it might be a special purpose one made for e.g. a mobile device or for accessibility. I've already shown you two examples, one where it doesn't work and one where it looks worse as it ignores the user's preferred font (assuming you've specified one in your browser). And that's just seen by me without changing browser or browser settings.--JohnBlackburne^words_deeds 23:25, 23 April 2010 (UTC)

The one below uses <math></math>. Does that work for you? Looks OK to me.

If you don't want to send me screenshots, I'll take your word for it that the 2 presentations were not acceptable to you. If its so easy to render serifs & italics in a more compatible way, please demonstrate.—Machine Elf 1735 (talk) 01:06, 24 April 2010 (UTC)

$e$0	$i,e$1	$j,e$2	$k,e$3	$l,e$4	$il,e$5	$jl,e$6	$kl,e$7
$i,e$1	$-1$	$k$	$-j$	$il$	$-l$	$-kl$	$jl$
$j,e$2	$-k$	$-1$	$i$	$jl$	$kl$	$-l$	$-il$
$k,e$3	$j$	$-i$	$-1$	$kl$	$-jl$	$il$	$-l$
$l,e$4	$-il$	$-jl$	$-kl$	$-1$	$i$	$j$	$k$
$il,e$5	$l$	$-kl$	$jl$	$-i$	$-1$	$-k$	$j$
$jl,e$6	$kl$	$l$	$-il$	$-j$	$k$	$-1$	$-i$
$kl,e$7	$-jl$	$il$	$l$	$-k$	$-j$	$i$	$-1$

The easiest way is using the {{math}} template as mentioned above. Easiest way to show you is do it myself, so please have a look.--JohnBlackburne^words_deeds 09:35, 24 April 2010 (UTC)

Looks good. Thanks for taking care of it.—Machine Elf 1735 (talk) 10:41, 24 April 2010 (UTC)

Not so good folks; on my PC (probably typical of most) the subscripts are jammed up against the e′s, and the whole thing is a mess. Brews ohare (talk) 01:10, 25 April 2010 (UTC) I've changed the subscript treatment from e₇ to e₇, which both look nearly the same here, but in the table the second form avoids the subscript crowding. On my PC this looks better. However, the changes made here have not made the ‘l’ display differently from plain vanilla form before all this started. If Blackburne insists upon MOS:MATH#Superscripts_and_subscripts, I'd prefer to go back to the earlier version repeated below, which looks better and meets all Blackburne's objections.:

e0	i, e1	j, e2	k, e3	l, e4	il, e5	jl, e6	kl, e7
i, e1	-1	k	−j	il	−l	−kl	jl
j, e2	−k	-1	i	jl	kl	−l	−il
k, e3	j	−i	-1	kl	−jl	il	−l
l, e4	−il	−jl	−kl	-1	i	j	k
il, e5	l	−kl	jl	−i	-1	−k	j
jl, e6	kl	l	−il	−j	k	-1	−i
kl, e7	−jl	il	l	−k	−j	i	-1

Brews ohare (talk) 06:44, 25 April 2010 (UTC)

I've just undone the last change as it again conflicted with the manual of style which says

"To start with, we generally use italic text for variables (but never for numbers or symbols)"

and

"Subscripts and superscripts should be wrapped in <sub> and <sup> tags, respectively, with no other formatting info. Font sizes and such should be entrusted to be handled with stylesheets"

The changes (adding <small> tags and italicising numbers) are clearly not allowed by the manual of style. But the other reason for quoting the above is that if you have problems on your machine then the way to fix is using style sheets, not to break it for everyone else. If you really thing the manual of style is wrong then bring it up at WT:MSM to see if there's consensus for a change.--JohnBlackburne^words_deeds 10:12, 25 April 2010 (UTC)

Version using {{math|

in Firefox with any Unicode font: e.g. Lucida Sans Unicode]]

Same table but with Arial Unicode MS as Sans-serif font and Times New Roman as serif font. These options are a result of the help given on this Talk page subsequent to the above version.

Same table but with Safari, Mac, Vector skin

OK John. I have reinstalled a previous version with no unicode violations and no special subscript treatment, including no italics. It displays just fine on my PC, which is probably true for any PC out of the box. Unfortunately, it doesn't have a script-style display of ‘l’, which is what the original machinations here were aimed to achieve. I can live with that, inasmuch as the version using {{math| does not display well (see figure). It jams all the subscripts too tightly to the letters,and doesn't lower the subscripts sufficiently. And ‘l’ looks the same as before we began.

I see no reason to expect anyone to run into “problems on your machine then the way to fix is using style sheets” in order to read this table, when the version installed here is perfectly readable without special attention. Brews ohare (talk) 13:14, 25 April 2010 (UTC)

I've done a screenshot of what the previous version looks like on my machine with my settings for comparison. I would say the rendering on yours is off: the subscripts look too high up, i.e. they've not moved down enough. The size looks fine: full height numbers are reduced to about the height of a lower case letter as a subscript. That they are not low enough also explains the crowding, though another reason might be your letters seem too slanted, like applying italics to a font that is already an italic font. This also makes your table much wider: not a problem but it looks a bit odd.

I preferred the serif version, but I'm also happy with the san-serif default font: for simple math such as this in e.g. the body of articles it's what I tend to use myself, though I may start using {{math}} more. I was previously concerned that italic text, as produced by <math> tags and {{math}}, generate too small text but yesterday got the css for the vector skin changed to fix this: MediaWiki talk:Vector.css#texhtml size increase. This is another way to fix rendering problems you are seeing, if they affect more than you and are particular to WP.--JohnBlackburne^words_deeds 13:50, 25 April 2010 (UTC)

John: Your rendering looks fine on your machine. I am not a sophisticated computer person, and my rendering is on my PC in Firefox using either of the two fonts listed there that read unicode: Cambria math or Arial unicode MS. I changed to them so I could read the wedge-product symbol, which did not show up in the original default setting. However, I have no clue what to make of your suggestions for fixing problems. It appears you are suggesting that I do some reprogramming of templates, or engage in some conversation on that subject with others able to do that. Is that the case?

You also mention Safari, Mac, Vector skin, which appears to refer to some machine options for a Mac, not applicable to a PC. Can you explain?

So I am left with this idea: at present the WP formatting macros are not going to lead to the rendering on a PC that you see on the Mac, and the real solution to these problems is to improve the WP macros or templates. Until those improvements are made, WP is better off with the simpler version of the table presently installed, because that version is better rendered on a wider variety of machines. Comment? Brews ohare (talk) 14:15, 25 April 2010 (UTC)

Safari is the Safari browser and Mac is Mac OS X. But that just means it's a standards compliant browser on a modern OS, so should render the same as other browsers (and in my experience does). Vector is the Vector skin, the one I switched to from the Monobook one. This does affect rendering dramatically: some skins use serif fonts for body text, different colours and they all use different layouts. You can change and preview them through the appearance tab of your preferences, and preview any page with a bit of URL editing. For example this [1] is this article with the "Cologne blue" skin.

I doubt it's a Mac vs. PC problem. My experience of using different browsers on different machines is the differences now are minimal, or are largely explained by user settings and options, especially on sites like WP which follows standards well so usually does nothing that will look different on different browsers. And I certainly can't imagine it's more broken on the PC than the Mac, as the majority of users are on PCs and any problems that affected a large number of PCs would be quickly fixed or worked around.

You might try other fonts. Code2000 + Code2001 give very good Unicode coverage and are free. Apart from that the things we've discussed already, such as changing settings, browser and WP skin, can all make a difference.--JohnBlackburne^words_deeds 14:41, 25 April 2010 (UTC)

@Brews ohare please don't roll this back to a version that is not serif and italic. JohnBlackburne's version was better.—Machine Elf 1735 (talk) 16:47, 25 April 2010 (UTC)

To clarify an error in my 18:47, 25 April 2010 WP:EDSUM, "Revert to version by User:JohnBlackburne at 10:01, 25 April 2010 and 09:17, 24 April 2010 per talk. Maybe the rendering indirection of {{math|i_1}} would help?", I see only <math>i_1</math> produces $i_{1}$, while {{math|i_1}} renders as i_1. The proximity of the subscripts is merely cosmetic but the ambiguity of Il1 is an accessibility issue.—Machine Elf 1735 (talk) 19:05, 25 April 2010 (UTC)

I agree the serif version looks better, though I'm not that concerned about whether it's serif or not. the "i_1" syntax is LaTeX syntax, so for use inside <math> taqs. The {{math}} template just changes the font so it looks the same as LaTeX rendered math, or is as close as possible, and you still need to use wiki markup and HTML inside it.--JohnBlackburne^words_deeds 19:23, 25 April 2010 (UTC)

I have provided a screen shot of the table as it appears as of now on my PC. It stinks. Blackburne has provided a screen shot of the rendering on his Mac: it looks fine.

Here is the question: What exactly do you think is going to happen to the bulk of readers looking at this table? My guess is that they will have to have sufficient sophistication to monkey with their machine to get a proper rendering, because out-of-the-box settings won't work. I don't know how to do that. I don't think that most will bother. And I don't know what other ramifications such changes may have, not only on WP but elsewhere.

What is the answer? Brews ohare (talk) 04:52, 26 April 2010 (UTC)

Your screenshot looks similar to what I see.

Using Template:Math.

I'd say "stinks" is an exaggeration. In answer to your question, my response is: Nothing whatsoever is going to happen and I don't think it bothers people that much. WP:DEADHORSE—Machine Elf 1735 (talk) 05:49, 26 April 2010 (UTC)

Machine Elf:

Version using {{math| in Firefox with any Unicode font: e.g. Lucida Sans Unicode

My rendering is this, not much like yours. The ‘l’'s are not serif and the subscripts are too large, too high, and jammed into the letters. Your response WP:DEADHORSE is neither polite nor helpful. You haven't responded to the issues raised regarding the unsophisticated viewer. . Brews ohare (talk) 12:29, 26 April 2010 (UTC)

To add to this: I have gone to the skin page as suggested by Blackburne and changed to "vector". However, that doesn't help. The table looks fine if I switch to Times Roman as a font, but then I cannot see the ∧ symbol. To see ∧, I have to pick Cambria math or Arial unicode MS or Lucida Sans Unicode and all of these display as indicated. The appearance shown is in Firefox; IE subscripts look better. Brews ohare (talk) 13:20, 26 April 2010 (UTC)

Vector and monobook are very close: they render text identically I think, with the main differences being in the headings and sidebar, i.e. the stuff common to every page. Other skins might have more of an effect as they also change the fonts. The HTML for one of the cells with subscript is:

<i>i</i>, <i>e</i>₁

i.e. it's just a subscript, in bold as it's a table heading cell, so you should be seeing this problem far more. The font makes a difference but every time <math> tags generate HTML instead of an image, or {{math}} is used, the same font is used, so it should look the same in probably thousands of maths articles, not just this one. So you either need to get used to it, find a way to fix it on your machine (there are other browsers than those two for example), or work out what the problem is and get it fixed in the style sheets, as I did for the {{math}} rendering size. But changing just this page makes no sense, especially if it involves fonts or other changes certain to cause problems for someone else.--JohnBlackburne^words_deeds 13:23, 26 April 2010 (UTC)

John: Thanks for your patience and help. Inasmuch as IE seems to handle the subscripts well and Firefox doesn't, I'd guess it's a Firefox issue with how it handles unicode. This problem doesn't show up with formulas like e₁ or e₁ or $\mathbf {e_{1}}$ or ${\boldsymbol {e_{1}}}$, so the Table formatting in Wiki is handled differently by Firefox than IE.

Font rendering issues

However, the font size in <math></math> rendering is smaller and in inline text is elevated above the other letters. That is the case with both Firefox and IE. The problem with math in inline text is of recent orgin: it always was sized a bit smaller, but the failure to line up with other text is a new development. Misalignment doesn't happen when no subscripts are present as in $\ell$ or ${\mathit {l}}$. However,<math>\ell</math> appears slightly smaller than surrounding text, while <math>\mathit l </math> appears much larger than regular text. Brews ohare (talk) 14:06, 26 April 2010 (UTC)

I wonder if this

/* Reduce line-height for <sup> and <sub> */
sup, sub {
    line-height: 1em;
}

might be your problem. It's in MediaWiki:Common.css, and is intended to stop lines with superscripts and subscripts from being too wide: important not just on maths pages but on any with references. It's applied to all sub and sup tags, regardless of skin or font. I wonder if Firefox is moving the subscript up because of it, probably contrary to web standards. You could put the following

sup, sub {
    line-height: normal;
}

In your own monobook.css or vector.css. Go to e.g. Special:MyPage/monobook.css and save the file with just the above in it - see my contribution history for the change before this one for an example. It will break line spacing but if it fixes your subscripts it at least shows the source of the problem. --JohnBlackburne^words_deeds 15:04, 26 April 2010 (UTC)

Hmm, the things I am learning about WP under the hood! I did this - - no change. But thank you for taking an interest. Brews ohare (talk) 15:24, 26 April 2010 (UTC)

You might need to force a reload of a page, or empty your browser cache, to tell it that something's changed.--JohnBlackburne^words_deeds 15:31, 26 April 2010 (UTC)

Hi John: I used the instructions on Special:MyPage/monobook.css (press Ctrl-Shift-R) to do that, and also used the Tools/Clear recent history tab on Firefox. I am not sure that does the whole refresh, maybe you have some additional ideas? Brews ohare (talk) 15:55, 26 April 2010 (UTC)

Clearing the cache is different. In Safari it's in a menu ("Empty Cache...") but in other browsers it's hidden away in the settings, e.g. in the same place you set the browser cache size.--JohnBlackburne^words_deeds 16:26, 26 April 2010 (UTC)

Font selection in Firefox for good Unicode display and good subscript display

Thanks John: I discovered the method is described here. Possibly this link should go on the Special:MyPage/monobook.css instructions. Nonetheless, no improvement. However, going back to some earlier suggestions about the choice of Serif font, I discovered the Tools/Options/Content/Advanced menu that allows separate identification of the Serif and Sans-serif fonts and specification of a Default Character Encoding. It turns out that the treatment of subscripts in the table is a strong function of the font choice, with most fonts leaving the subscripts too high. However, Times New Roman is better than most. I've updated the figures with the choices shown in the menu.

It's been a struggle. Thanks for the guidance. Brews ohare (talk) 17:17, 26 April 2010 (UTC)

BTW, the choice of fonts helps with numerical subscripts but letter subscripts are still too large; the following all are versions of subscripts ‘one’: e₁ $e_{1}$ and of subscripts ‘ℓ’: e_l $e_{l}$ e_$\ell$ e_$l$. Brews ohare (talk) 19:27, 26 April 2010 (UTC)

Please take this discussion to Template talk:Math. Thank you.—Machine Elf 1735 (talk) 19:31, 26 April 2010 (UTC)

Directional arrow in Fano plane diagram

Hello, I don't see the directional arrow on the line through points 2,4,6. I assume it would be directed from midpoint 2 toward vertex 6, as are the other two medians? Fizzbowen (talk) 21:51, 26 September 2010 (UTC)

Yes, you're right. There's an older complete and correct version of it which I've restored for now. The editor that created the newer version is on a break at the moment so I don't know when it will be fixed, unless someone else wants to do so.--JohnBlackburne^words_deeds 22:12, 26 September 2010 (UTC)

What 'data loss'?

Here's a summary of the changes I made, expanded from the edit summaries.

In [2] I merged two short sections, the lead and history, then copy-edited, mostly making some redirects into direct links, but also moving a reference that had ended up in the text to the refs section, then editing that for sentence length and removing a second link to quaternion. Lastly I moved the TOC and inserted a clear to eliminate the white space to the right of the TOC. In most articles that space is filled by an image or part of an infobox but this article has few of the former and none of the latter so moving the TOC made sense.

In [3] I copyedited the definition section, fixing a significant number of problems with the math formatting, including some inappropriate colouring as per MOS:COLOR. I removed a table which served no purpose, as a key for different indexing schemes for the basis elements that appear nowhere else in the article. And I removed a quote that was an unnecessary copyright violation. I rewrote some of the text for clarity and added some text in place of the removed copyvio.

These were both reverted with the reason 'too much data loss', but I cannot see why. Little data was removed: all the mathematical content was still there and hopefully clearer: certainly the formatting was much better and more consistent. The only actual content excised was the quote, but it's very odd to use sources by copying wholesale from them: it's much more normal to summarise them as I did. Quotes should only be used if relevant, i.e. if they reader gains something that cannot be gained from good paraphrasing.--JohnBlackburne^words_deeds 02:36, 5 October 2011 (UTC)

The quote was informative, on a number of levels. -- cheers, Michael C. Price ^talk 07:30, 5 October 2011 (UTC)

The quote is from a copyright source. WP:FUC (a "Wikimedia policy with legal considerations") is very clear on this: "Non-free content is used only where no free equivalent is available, or could be created, that would serve the same encyclopedic purpose." – if possible it should be paraphrased, not copied. If the quote itself were e.g. being commented on and were needed for that commentary, then fair use would apply. But that is not the case here. As that is covered by policy, and you do not seem to object to any other of my changes, I am restoring them.--JohnBlackburne^words_deeds 13:39, 5 October 2011 (UTC)

Extension

Shouldn't this link to the extension of Octonions, the 16-nions (don't know the proper name/spelling) just as the Quaternions linked to the Octonions?

I do not know much algebra, as I have always been more of an analysis person, but I think the 16-nions have a lot of special properties. For instance, they do not yield a cross product in 15-space, as the quaternions (resp octonions) do in 3 (resp 7) space. I think this fact was proved by Frobenius, although I have not found a proof of it online. --Jpawloski 14:50, 16 February 2006 (UTC)

It does link to the sedenions (in the see-also section). --Zundark 15:45, 16 February 2006 (UTC)

Missed that. Thanks. --Jpawloski 16:02, 16 February 2006 (UTC)

Yep, I agree. There should be a mention to sedenions. IntegralPython (talk) 01:39, 1 June 2018 (UTC)

Is there octonion analysis?

Do there exist octonion analysis? I try to find but it I don't find any mentions in google. I think here should be some explanations — Preceding unsigned comment added by 178.122.147.92 (talk) 16:40, 1 September 2012 (UTC)

I don't think it makes sense to construct one as octonions are non-associative, and to do analysis you need an associative product. Without that even the simplest constructions break down. I don't think you'd be left with any useful mathematics. Besides if you want to do analysis in eight dimensions you can use geometric calculus, which generalises vector and quaternion calculus to any dimension.--JohnBlackburne^words_deeds 19:18, 1 September 2012 (UTC)