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Understanding Quaternions (2012)
- gfodor 13y agoAnother good reference: http://www.songho.ca/math/quaternion/quaternion.html http://www.songho.ca/math/quaternion/quaternion.html
- NAFV_P 13y agoThere's also octonions and sedenions. I prefer the blanket term onion-algebras.
- spiritplumber 13y agoThanks, my head was already hurting :P
- leoc 13y agoNow your eyes can hurt as well.
- yati 13y agoWow. Thanks for this. But I guess quaternions are much more well known because of their application in Computer Graphics/Vision (which is how I stumbled upon this article)
- NAFV_P 13y agoI think that is because quaternions form a division algebra, but matrices don't (neither do sedenions).
- wtracy 13y agoHas anyone here ever seen a good explanation of why quaternion multiplication maps to rotation concatenation?
- maxerickson 13y agoAs much as anything, it is a chosen property of the system. If you want to insist that mathematics is discovered (rather than invented), you can still say that this particular decomposition of vectors is prevalent because it has useful properties.
- ajkjk 13y agoI think of it as the opposite: there exists a Lie group of spatial rotations, so we want to use it, and we found a relatively convenient notation. There are many other kinds of 'complex' numbers (http://en.wikipedia.org/wiki/Hypercomplex_number http://en.wikipedia.org/wiki/Hypercomplex_number) - but you probably won't hear about them outside of mathematics and physics because they're less useful. The same applies to 2d - the SO(2) group exists; we map it to the complex numbers because it's convenient to do math with.
- T-A 13y agoHow about http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion_rotation_operations http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotatio...
- davidgerard 13y agoI love that plaque, and that Ireland is the sort of place that that would rate a plaque.
- thearn4 13y agoWhere the author talks about imaginary numbers being completely "made up" and suggests you shouldn't bother with trying to understand them, I think that's selling them short. Imagine, if you will, trying to explain to the ancient Greeks the idea of a number that can't be written as a division of integers (the irrational numbers). That would have seemed completely "made up" to them, but we don't really see them that way, they just "are". That concept is has since become normalized, in terms of everyday concepts (like the area of a unit circle). Similar situations arise with fractions or negative numbers to some indigenous tribes, etc. I guess what I'm saying is that complex numbers only as fictitious or imaginary as any other set of numbers that we otherwise feel like we have a good handle on.
- yati 13y agoI'm sure the author acknowledges the importance of complex numbers - the part where he says "don't bother..." is why is i^2 = -1, which makes sense, as i(or j) was indeed chosen to represent sqrt(-1) and not the other way round(no one "discovered" a magical number that, when squared, gives -1) :)
- thearn4 13y agoYeah, I actually really do like the rest of it. It's just the part: "Don’t try to actually understand this term as there is no logical reason why it exists. We just have to accept that is just something that squares to -1." that rubbed me the wrong way. But in terms of the goal of this guide, I suppose waving hands and saying "deal with it" could be chalked up to a necessary evil.
- jakobsen 13y agoBut isn't the definition of the imaginary unit axiomatic? If so it's indeed true that there is no logical reason why it exists other than it being useful.
- andrewflnr 13y agoI like this explanation. A friend said it helped him. http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ http://betterexplained.com/articles/a-visual-intuitive-guide... Basically, multiplying by i "rotates a number" 90 degrees.
- juretriglav 13y agoNot bad, but I find this to be a state of the art explanation of quaternions: http://acko.net/blog/animate-your-way-to-glory-pt2/#quaternions http://acko.net/blog/animate-your-way-to-glory-pt2/#quaterni...
- batguano 13y agoO. M. G. That is so awesome.... (Yes, so this probably counts as the kind of "+1" post that should be downvoted. But check out the link and if you don't think it's the most amazing quaternion explanation, then I will humbly accept your downvote.)
- ergoproxy 13y agoThis recently posted YouTube video by UNSW Professor Norman J. Wildberger discusses the discovery of the quaternions by Hamilton and the subsequent discovery of the octonians. It's 59 minutes, 30 seconds long, and it was published on March 5, 2014: MathHistory18: Hypercomplex numbers https://www.youtube.com/watch?v=uw6bpPldp2A https://www.youtube.com/watch?v=uw6bpPldp2A [video]