Wikipedia:Reference desk/Archives/Mathematics/2024 March 1

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March 1

Planarise quaternion graph

Cayley Q8 graph showing the six cycles of multiplication by i, j and k – in the SVG file, cycles can be highlighted by hovering over or clicking on them

I drew this diagram of the quaternion multiplication table some time ago, and am wondering if it's possible to make it (more) planar so it's easier to understand. Is that possible? Thanks, cmɢʟee⎆τaʟκ 06:57, 1 March 2024 (UTC)

It might not be the most elegant solution, but you could have -1 form an entire outer ring, with the three semicircular circles becoming straight lines passing through, sort of like a point at infinity. Given that -1 is "antipodal" to 1 on the three relevant cycles, I think it would make sense intuitively, but again there might be other much better ways to go about it. GalacticShoe (talk) 08:13, 1 March 2024 (UTC)

Thanks. It's more or less already like that: the dotted semicircles all passing through -1 like a Metro map. cmɢʟee⎆τaʟκ 09:42, 1 March 2024 (UTC)

What would be clearer—to me, at least—would be to just list out the six cycles without trying to force them into the same graph. (As it stands, if you're trying to show the patterns in the cycles, the graph honestly obfuscates that more than elucidates it.) 2603:8001:4542:28FB:D83C:41C6:8B71:EABA (talk) 17:22, 1 March 2024 (UTC) (Send talk messages here)

Ok, thanks. cmɢʟee⎆τaʟκ 07:08, 2 March 2024 (UTC)

Not planar, but cycles are clearer

cmɢʟee⎆τaʟκ 10:13, 2 March 2024 (UTC)

The symmetries are also more perspicuous. The two-step paths $(\times \mathbf {i} )^{2},(\times \mathbf {j} )^{2},(\times \mathbf {k} )^{2}$ represented by pairs of smooth solid arcs passing through the $\pm 1$ vertices, could be mistaken for one-step arcs passing under the marbles representing the $\pm 1$ vertices, bypassing these vertices. I expect that letting the "hidden part" of the smooth arcs go more closely to the centres of these marbles will make it clearer that they are meant to proceed through these vertices. Also, in this version the inner and outer arcs hardly interfere; IMO there is little point in expressing a distinction with a solid and a dashed style.

For the reader whose colour perception is challenged, it may help to make the arcs a bit fatter, and also to use three colours whose brightness is more distinct, e.g.  #ff8888 ,  #009900 ,  #0000cc .  --Lambiam 09:58, 3 March 2024 (UTC)

Good recommendations, @Lambiam: I've updated the diagram using line style as an alternative to colour for the colour-blind. Your red is a bit too intense, so I've retained #cc0000. With the line styles, colour differences are now less important. Cheers, cmɢʟee⎆τaʟκ 13:30, 3 March 2024 (UTC)

P.S. Would it be better to rotate the diagram 90° clockwise so that the ±1 and ±i orbs are in their respective directions on the complex plane? cmɢʟee⎆τaʟκ 16:28, 3 March 2024 (UTC)

There is no connection between specifically the basis element $\mathbf {i}$ of the quaternions and the imaginary unit $i.$ The roles of $\mathbf {i} ,$ $\mathbf {j}$ and $\mathbf {k}$ are perfectly interchangeable: we have $\mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}={-}1.$ In fact, any valid algebraic identity remains valid under a cyclic permutation. Take, for example, $(a{+}\mathbf {i} )(b{+}\mathbf {i} )=ab{-}1+(a{+}b)\mathbf {i} .$ So then it is also the case that $(a{+}\mathbf {j} )(b{+}\mathbf {j} )=ab{-}1+(a{+}b)\mathbf {j} .$ Therefore there is no point in paying special attention to the orientation of $\mathbf {i} .$ You could introduce this basis-element symmetry in the image by rotating the dotted green $1\to \mathbf {j} \to {-}1\to {-}\mathbf {j} \to 1$ cycle 60° clockwise and the dashed blue $1\to \mathbf {k} \to {-}1\to {-}\mathbf {k} \to 1$ cycle 120° clockwise, with obvious changes for the outer three cycles. Then the vertices $\mathbf {i} ,$ $\mathbf {j}$ and $\mathbf {k}$ will form the vertices of an equilateral triangle.  --Lambiam 17:18, 3 March 2024 (UTC)

Good point, thanks. I'll keep the current orientation then. I think keeping i, j, k together is clearer than going i, −k, j, −i, k and −j. cmɢʟee⎆τaʟκ 08:08, 5 March 2024 (UTC)

Hello sirs

Why does 1+1=2? -Alex Salamander 63.151.121.148 (talk) 23:36, 1 March 2024 (UTC)

Because it is a theorem in Principia Mathematica. Bubba73 ^{You talkin' to me?} 23:57, 1 March 2024 (UTC)

You might be interested in this discussion from May 17 last year. GalacticShoe (talk) 23:58, 1 March 2024 (UTC)

Because that is the definition of 2. —Tamfang (talk) 00:21, 2 March 2024 (UTC)

Curiously, this questioner and that of last year geolocate to locations less than 9 miles apart. Perhaps there is something in the air in Minneapolis–Saint Paul that evokes this question.  --Lambiam 01:01, 2 March 2024 (UTC)

"Why does Minneapolis + Saint Paul = Twin Cities?" GalacticShoe (talk) 01:07, 2 March 2024 (UTC)

It depends on the context. One cloud plus one cloud makes one cloud. One rabbit plus one rabbit makes three or more rabbits. One gunmam plus one gunmand quite possibly makes no gunmen. In the context of schools 1+1 is 2 is the correct answer and marked with a tick - what more do you need? NadVolum (talk) 11:23, 2 March 2024 (UTC)

Perhaps one might wish to see an explanation of why, in the context of schools, it is "the" correct answer. Was it decided by a conference of school boards? Or legislated after the powerful pressure of a lobby for publishers of books on arithmetic? Or is it because the teachers check the answers on their pocket calculators, most of which show as the result of calculating 1+1=?  --Lambiam 14:10, 2 March 2024 (UTC)

My first suspicion in cases like this is normally the reptilians from Tibet - but I think in this case it's because of a powerful cabal of mathematicians pushing the Peano axioms. S(0) is called 1 and S(1) is called 2 and 1+1 = 1+S(0) = S(1+0) = S(1) = 2 and no one has yet found a secret backdooor in those axioms allowing other values to be assigned to 1+1. That would be equivalent to finding an x so S(x) = 0. NadVolum (talk) 17:17, 2 March 2024 (UTC)

But how do we know these axioms are correct? Maybe there is a largest number – admittedly veeery large – where you hit a limit with this S. You could call it a fixed point. At least Boolean algebra (see below) keeps it simple. I hear that many mathematicians believe it has some truth value.  --Lambiam 21:29, 2 March 2024 (UTC)

Axioms are not "correct" or "incorrect", they are basic assumptions upon which further logic is built. {The poster formerly known as 87.81.230.195} 176.24.44.161 (talk) 07:09, 6 March 2024 (UTC)

If reality uses Saturation arithmetic then the Peano axioms don't model it! That sounds like a reasonable way for some supercomputer to simulate u all but I think it's more likely it wouldn't detect overflow and just produce a huge negative number instead 😁 NadVolum (talk) 19:17, 3 March 2024 (UTC)

In Boolean algebra, 1+1 = 1. CodeTalker (talk) 19:26, 2 March 2024 (UTC)

Boole seems to have got in a bit of a mess with + and -. He'd have been much beter off using + for exclusive or and forgetting about -. With that 1+1=0, not x is 1+x, and the usual x or y is x + y + xy. NadVolum (talk) 22:10, 2 March 2024 (UTC)