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Re: OT: Quaternions and the United Nations

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  • doriscott20002000
    Sharani, you made me chuckle! I am glad I am not the only one who understood almost nothing. I expected Dharmaja the mathematician to respond. But I am amazed
    Message 1 of 4 , Oct 26, 2006
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      Sharani, you made me chuckle! I am glad I am not the only one who
      understood almost nothing. I expected Dharmaja the mathematician to
      respond. But I am amazed of what thoughts can flow through ones
      mind.

      I become more and more aware of the stupidity of some of my thoughts
      that are flowing freely without asking me whether I agree or not. By
      God's grace they can't put an "y" before an "x" (German saying).

      Doris



      --- In Sri_Chinmoy_Inspiration@yahoogroups.com, sharani_sharani
      <no_reply@...> wrote:
      >
      > Hi Shane,
      > Even though I understood little or nothing about the imaginary
      numbers
      > and whatnot that you described, I enjoyed reading this post
      immensely.
      > You offer a synthesis of running, mathematics, physics and the
      United
      > Nations that makes them seem like they are all related - and I'm
      sure
      > on some cosmic level indeed they are :-) This might just get my
      vote
      > as my favorite of your writings. How that is possible when I do not
      > understand science very well has me admittedly intrigued...
      >
      > Sharani
      >
      > --- In Sri_Chinmoy_Inspiration@yahoogroups.com, shane_dublincentre
      > <no_reply@> wrote:
      > >
      > >
      > > Last week, I was doing my final long run before the upcoming
      Dublin
      > > marathon. On long runs, I generally like to take my stopwatch
      and run
      > > for a certain length of time, rather than for a certain number of
      > > miles. So today I aimed for three hours.
      > >
      > > My brother Aidan came out with me to do a shorter run, and the
      > > chatting and laughing made the first hour pass very quickly. We
      did a
      > > loop so that he could return home easily (he's just come up from
      home
      > > to live in town and the possiblilities for getting lost are
      many). Now
      > > alone, I took a journey around the woodland tracks of the
      Phoenix Park
      > > (Europe's largest city park), before joining the Royal Canal at
      > > Ashtown and heading towards the sea.
      > >
      > > Whilst running it occured to me that I should soon be running
      under
      > > Brougham Bridge, a quaint stone bridge that I'd heard quite a lot
      > > about in my undergraduate days studying theoretical physics at
      > > Trinity, in connection with a certain William Rowan Hamilton.
      > >
      > > Hamilton was born in 1805 and quickly marked himself out as a
      child
      > > prodigy; at the age of 13 he had mastered Hebrew, Arabic,
      Sanskrit and
      > > Hindi (as well all the classical and modern European languages
      > > expected of any child prodigy worth his salt). His mathematics
      > > education was largely self-taught, and Trinity recognised his
      > > achievements by appointing him to a professorship at the age of
      22,
      > > when he had not even completed his university studies.
      > >
      > > Hamilton's most famed achievement is the discovery of
      quaternions.
      > > Those of you who didn't drop out of maths at the earlies legally
      > > allowable stage will probably have come across imaginary numbers
      at
      > > some stage. If you did drop out, then grab a calculator, type
      in -1
      > > and press the square root sign, whereupon your calculator should
      start
      > > vehemently protesting. Why? Because the square root of -1 cant be
      > > written as a number. So what?, you might ask. Well that also
      means
      > > equations like
      > >
      > > x^2 + 1 = 0
      > >
      > > dont have any solution. (by x^2, i mean x squared) Unless we
      make up a
      > > whole batch of new numbers, that is. And that's precisely what
      was
      > > done: We let the square root of -1 be equal to **i**, a so-called
      > > imaginary number. These imaginary numbers tend to crop up at an
      > > alarming rate when doing calculations in physics and engineering;
      > > however, they always manage to get out of the way just before any
      > > answers pertaining to the real world are obtained.
      > >
      > > Back to our friend Hamilton. From primary school you'll remember
      we
      > > could put all numbers on a numberline (Im assuming none of you
      dropped
      > > out then!). However, imaginary numbers don't fit on this line
      and need
      > > one of their own. If you put the imaginary line at a 90 degree
      angle
      > > to the real one you get something that looks a bit like a 2D
      graph
      > > (called an Argand diagram) upon which you can plot any
      combination of
      > > real and imaginary numbers.
      > >
      > > This set Hamilton thinking as to whether he could extrapolate
      this to
      > > three dimensions; this would mean creating another number, j,
      with its
      > > own numberline 90 degrees to the first two. This actually proved
      to be
      > > a bit of a problem.
      > >
      > > I remember writing an article about how on many occasions great
      > > scientists have striven to the end of their tether for an answer
      to
      > > the problem they were working on and how the answer just came to
      them
      > > whilst they were doing something comparatively innocuous. This
      was
      > > especially true in this case. Hamilton kept notoriously long
      hours
      > > working on his problems; however one day , he was walking along
      the
      > > Royal Canal with his wife on his way to Dunsink observatory,
      where he
      > > was based. As he passed under the bridge, a flash of
      inspiration came
      > > to him. He wasn't able to extrapolate to three dimensions (later
      > > mathematicians showed this to be impossible) but he could
      extrapolate
      > > to *four*. According to legend, Hamilton took out his
      pocketknife and
      > > scratched out the formulae which numbers on the four numberlines
      (i,
      > > j, k, 1) would have to obey:
      > >
      > > i^2 = j^2 = k^2 = ijk = -1
      > >
      > >
      > > The formula is not to be seen on the bridge (indeed it would be
      hard
      > > to make it out under all the graffiti), but the obligatory
      plaque has
      > > been put up. The bridge has become kind of a place of
      pilgrimage, and
      > > in my PhD days it wasnt unusual to overhear a visiting
      mathematician
      > > asking directions to get there.
      > >
      > > A major conceptual breakthrough was the fact that none of these
      > > numbers commuted when you multiplied them. Normally, if we have
      two
      > > numbers a and b then we take it for granted that ab=ba. However
      in the
      > > case of quaternions we get
      > >
      > > ij= -ji
      > >
      > > kj=-jk
      > >
      > > ik= -ki
      > >
      > >
      > > This was to prove an important conceptual stepping stone in the
      path
      > > to *quantum mechanics*, where data such as the position and
      energy of
      > > a particle are found by applying things called matrices which
      > > sometimes don't commute either. This non-commuting quantity is
      > > responsible for the famous Heisenberg uncertainty principle; you
      can't
      > > find out the exact position and energy of very small objects at
      the
      > > same time because their matrices don't commute. Indeed, there are
      > > enough of Hamilton's writings to suggest that had he been around
      in
      > > the 1920's when quantum mechanics were being developed, he could
      very
      > > well have played a significant role in its development. As for
      > > quaternions, they are still being used today in such diverse
      fields as
      > > computer game design and spacecraft control.
      > >
      > > Out of interest, you can also extrapolate to eight dimensions
      > > (*octonions*) and sixteen dimensions (*sedenions*), indeed to any
      > > power of two, but it all gets very messy very quickly.
      > >
      > > My run took me past the bridge at the 1:50 mark and again at the
      2:20
      > > mark. I've become very fond of the Royal Canal as a running route
      > > (although I usually go the other way, inland towards Clonsilla
      and
      > > Leixlip). I returned home via Stoneybatter, one of the last
      remaining
      > > parts of 'old' Dublin, where you can still smell turf fires
      burning.
      > > Irish and United Nations flags towered above a memorial to the
      Irish
      > > soilders who lost their lives serving in the Lebanon. The
      solitary
      > > figure mowing the lawn, is not quite so elderly as to make me
      wonder
      > > if perhaps he served in the Lebanon himself. The words 'United
      > > Nations' still evoke some higher ideal within, some vision of how
      > > things could be, which years of exposure to the crassness of
      > > international politics has not yet dimmed. I think of Uruguyan
      and
      > > Bangladeshi peacekeepers in the rain forests of the Congo,
      Ghanaians
      > > and Indians in the Lebanon, and I cannot think that they are
      there
      > > solely out o their governments' self-interest. My meditation
      teacher,
      > > Sri Chinmoy, has written many times of the United Nations. Here
      are
      > > some extracts from his book `A Real Member of the United Nations
      > > <http://www.srichinmoylibrary.com/member-united-
      nations/toc.html>,
      > > written in 1989:
      > >
      > > *A real member of the United Nations is he who claims the
      United
      > > Nations to be his own, very own. Unless and until he claims the
      United
      > > Nations to be his own, he will not be richly inspired to change
      its
      > > face and fate lovingly and surprisingly for the better.*
      > >
      > > *A real member of the United Nations carries in his heart-
      pocket a
      > > valid visa to humanity's oneness-heart.*
      > >
      > > *A real member of the United Nations does not expect the
      complete
      > > cure of world maladies by a 44-year-old United Nations. Countless
      > > problems were born long before it was born. Slow and steady wins
      the
      > > race. Let us hope for the best. Let us fervently hope that the
      world
      > > of peace the United Nations envisages will without fail be
      manifested
      > > on earth.*
      > >
      > > *A real member of the United Nations forgives the United
      Nations
      > > mistakes, for he knows that the United Nations has been in
      existence
      > > for only 44 years. He hopes that in the forthcoming endless days
      the
      > > United Nations will not only rectify all its mistakes but also
      become
      > > the embodiment of truth-perfection.*
      > >
      > > --------------Related Links:----------------
      > >
      > > - Hamilton's original paper on quaternions
      > > <http://www.emis.de/classics/Hamilton/OnQuat.pdf>
      > >
      > > - The Inner Role of the United Nations:
      > > <http://www.srichinmoylibrary.com/inner-role-united-
      nations/20.html>
      > > A talk by Sri Chinmoy.
      > >
      > > ---------------------------------------------
      > >
      > > This article, complete with pictures, is to be found on my blog
      > > http://www.srichinmoycentre.org/Members/shane_magee/blog
      > >
      >
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