On Quaternions and Octonions, by John Conway and Derek Smith
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On Quaternions and Octonions, by John Conway and Derek Smith
 
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Has anyone read this?
http://math.ucr.edu/home/baez/octonions/conway_smith/ I've not read enough Conway and I'm not sure where to start! -- Owen ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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Owen . . .
Hmmm . . . several potential issues here . . . If your goal is to "get to know Conway" better, then you really ought to start with ONAG (On Numbers and Games), which is a "classic" of sorts -- a non-standard way of developing number systems (but, if you are thinking about going there, you'd probably have more fun starting with Knuth's "Surreal Numbers" -- yes, that Knuth). If this kind of stuff gets you going, then the Winning Ways (with Berlekamp and Guy) volume(s) will give you more than you could ever want (and still yet again more :-) . . . On the other hand, my guess is that more interesting is likely to be quaternions (aka hamiltonians), and their applications, in which case Conway is probably not the best starting place. Specifically, this would largely boil down to, are you more interested in "division algebras", or in a "unified" framework for (-1, 1, 1, 1) signature metrics for "minkowski space" where you can do special (and eventually general) relativity in a "non-kludgey" way? :-) If you want to see quaternions in action (and they are quite fun :-), a reasonable place to look is here: http://world.std.com/~sweetser/quaternions/ps/book.pdf More generally, it might be worth noting that although "Conway's Game of Life" gets lots of airplay, in Conway's intellectual life it is almost certainly just about what it was . . . an evening's amusement on a cocktail napkin! :-) Remember that cellular automata basically got their start with von Neuman's efforts to "automate" exploration of the universe :-) Here's the problem: If you assume the universe is isotropic (the "same" in all directions), then the "search space" grows quadratically (at least) as you go out radially from the solar system, and given (e.g., the Challenger example, which says we can barely put minimal mass in low earth orbit) that there's no way we could "carry with us" the supplies, or launch enough probes, to survey the universe, the only "solution" is to send out a self replicating probe (it lands on some planet, makes copies of itself, which then land on other planets, make copies, etc.). Around here though (actually, around 1950), we got the so-called "Fermi paradox": "Where are they?" In other words, if the universe is generally (locally) temporally (as well as spatially) isotropic, so that "we" are just sort of average, then there must have been other planets in other solar systems where intelligent life evolved long (say millions or billions of years) ahead of us. In which case, since "curiosity" is clearly part of "intelligence", at least one of those species would have sent out a self replicating probe . . . and, since exponential is bigger than quadratic (or cubic), the universe should be "full" of copies of that original probe, so we should have seen at least one by now! Von Neumann asked the question, "Is there some theoretical reason there can't be self replicating machines?" (although one might argue that "life itself" is such a self replicating machine, so perhaps we ourselves are just the current stage in the development of an earlier "probe" that landed on earth long ago . . . :-). Anyway, von Neumann set himself the task of designing a self replicating machine (and in the process more or less invented cellular automata --where appropriate credit should also go to Ulam). He did "solve the problem" in the sense of a mathematical "existence proof". Von Neumann's "machine" lived in a 2-d space, used orthogonal (4 neighbors . . .) neighborhoods, and each cell had 29 possible states, and quite complicated "transition rules". You can look here: http://en.wikipedia.org/wiki/Von_Neumann_cellular_automaton Conway, at some point, said, "von Neumann's machine is way too messy. Is there a simpler version?" The answer he found was "yes" -- 2-d space, 2-state cells (although 8 neighbors rather than 4 . . .), and very simple "transition rule". Cute little simplification of von Neumann's original, but not particularly "deep" . . . Of course, Martin Gardner's Mathematical Games Column in Scientific American deserves most of the credit for the popularity of Conway's CA . . . Oh, well . . . tom p.s. Some disclosure . . . my dissertation was on the (localized) homotopy of the classical Lie groups (orthogonal, unitary, and symplectic). The symplectic group is the group of (isometric) rotations in n-dimensionsional (or, eventually, infinite dimensional) quaternionic space . . . One of the nice tools I used was a representation of quaternions as skew-symmetric 2x2 complex matrices, and a representation of complex numbers as skew-symmetric 2x2 real matrices, which induce mappings . . . --> Sp(n) --> U(2n) --> O(4n) -- > Sp(4n) --> . . . hence, my longstanding enjoyment of quaternions (and, of course, category theory, etc. . . .) Also, it turns out I'm somewhat of a bigot -- I like my algebraic structures to be associative, so I really don't like the octonians -- too weird for me!!! (the quaternions are non-commutative, and that's bad enough! :-) tom On Oct 10, 2009, at 8:26 PM, Owen Densmore wrote: > Has anyone read this? > http://math.ucr.edu/home/baez/octonions/conway_smith/ > I've not read enough Conway and I'm not sure where to start! > > -- Owen > > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by Owen Densmore
No, and I cannot help you pick which Conway to read, either.
But, if you really want to know about Quaternions, there are several digitized editions of Sir William Rowan Hamilton's Elements of Quaternions available, both the original (1866) single volume prepared by his son and the two volumes edited by Joly (1899). Just search "hamilton quaternions" at archive.org
-- rec -- On Sat, Oct 10, 2009 at 9:26 PM, Owen Densmore <[hidden email]> wrote: Has anyone read this? ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by Owen Densmore
Thus spake Owen Densmore circa 09-10-10 08:26 PM:
> Has anyone read this? > http://math.ucr.edu/home/baez/octonions/conway_smith/ > I've not read enough Conway and I'm not sure where to start! So, is it fair to say that octonions are a geometric algebra, even though they aren't associative? I think I remember reading somewhere that they were considered a geometric algebra... perhaps in Hestenes book or in Penrose's Road to Reality. But wikipedia claims that a geometric norm _must_ be associative and that a geometric algebra must be over a vector space with an associative norm. WTF? Is wikipedia oversimplifying? Or are octonions really not considered a geometric algebra despite their deep relevance to geometry? And, more importantly, why do my searches for Clifford fail in Adobe Reader, but succeed in Evince, while reading the following file: http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf ??? -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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Look octonions (denizens of Octonia, which borders Philistia?) up in
'This Weeks Finds'....John Baez wrote on 'em a bit awhile back... glen e. p. ropella wrote: > Thus spake Owen Densmore circa 09-10-10 08:26 PM: > >> Has anyone read this? >> http://math.ucr.edu/home/baez/octonions/conway_smith/ >> I've not read enough Conway and I'm not sure where to start! >> > > So, is it fair to say that octonions are a geometric algebra, even > though they aren't associative? I think I remember reading somewhere > that they were considered a geometric algebra... perhaps in Hestenes > book or in Penrose's Road to Reality. > > But wikipedia claims that a geometric norm _must_ be associative and > that a geometric algebra must be over a vector space with an associative > norm. WTF? Is wikipedia oversimplifying? Or are octonions really not > considered a geometric algebra despite their deep relevance to geometry? > > And, more importantly, why do my searches for Clifford fail in Adobe > Reader, but succeed in Evince, while reading the following file: > > http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf > > ??? > > ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by glen e. p. ropella-2
On Wed, Oct 14, 2009 at 6:06 PM, glen e. p. ropella <[hidden email]> wrote:
Thus spake Owen Densmore circa 09-10-10 08:26 PM: I've don't recall ever seeing the octonions turn up in Hestenes, and looking at their multiplication table makes my head hurt. -- rec -- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by Carl Tollander
More specifically, http://math.ucr.edu/home/baez/octonions/
Carl Tollander wrote: > Look octonions (denizens of Octonia, which borders Philistia?) up in > 'This Weeks Finds'....John Baez wrote on 'em a bit awhile back... > > glen e. p. ropella wrote: >> Thus spake Owen Densmore circa 09-10-10 08:26 PM: >>> Has anyone read this? >>> http://math.ucr.edu/home/baez/octonions/conway_smith/ >>> I've not read enough Conway and I'm not sure where to start! >> >> So, is it fair to say that octonions are a geometric algebra, even >> though they aren't associative? I think I remember reading somewhere >> that they were considered a geometric algebra... perhaps in Hestenes >> book or in Penrose's Road to Reality. >> >> But wikipedia claims that a geometric norm _must_ be associative and >> that a geometric algebra must be over a vector space with an associative >> norm. WTF? Is wikipedia oversimplifying? Or are octonions really not >> considered a geometric algebra despite their deep relevance to geometry? >> >> And, more importantly, why do my searches for Clifford fail in Adobe >> Reader, but succeed in Evince, while reading the following file: >> >> http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf >> >> >> ??? >> > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by glen e. p. ropella-2
Glen -
Ligature . . . ff is (sometimes) a "single glyph" . . . tom On Oct 14, 2009, at 5:06 PM, glen e. p. ropella wrote: And, more importantly, why do my searches for Clifford fail in Adobe ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by glen e. p. ropella-2
Glen -
It's probably worth remembering that collections of spatio- temporally located mathematicians will choose to use the "definitions" that give them the amount of "traction" they want. They'll use definitions that are sufficiently general as to cover the cases they're most interested in, but specific enough to make theorem statements and proofs appropriately concise and straightforward. Also, in general, a "mathematical definition" is ordinarily just a brief name for a collection of "axioms"; and, in various cases, there is a "new name" when you add an additional "axiom" . . . so, for example, a *semigroup* is a set with an associative binary operation. A *monoid* is a semigroup with an identity element. A *group* is a monoid with inverses. The classic example of the process of "choosing the definition that you like best" is the "definition" of a *ring* . . . Some mathematicians require a *ring* to have a unit, others don't. So you get slightly odd locutions like *rng* (called a "ring without unit" by others), and "ring with unit" used by those who like the "more general" version without the requirement of a unit. (see, e.g., http://en.wikipedia.org/wiki/Pseudo-ring ) These days, most mathematicians are so comfortable with associativity that they'll go ahead and include that as part of "the definition" (of, e.g., a geometric algebra) . . . and then also they won't have a bunch of theorems that start out, "Let A be an associative geometric algebra . . ." rather than "Let A be a geometric algebra . . ." (for example . . .) It's possible that as, perhaps, String Theory gains traction (and with it interest in exceptional Lie groups like, e.g., E8), the insistence on including associativity in "the definition" will lessen, but . . . (see, e.g., http://en.wikipedia.org/wiki/E8_(mathematics) ) tom On Oct 14, 2009, at 5:06 PM, glen e. p. ropella wrote: > Thus spake Owen Densmore circa 09-10-10 08:26 PM: >> Has anyone read this? >> http://math.ucr.edu/home/baez/octonions/conway_smith/ >> I've not read enough Conway and I'm not sure where to start! > > So, is it fair to say that octonions are a geometric algebra, even > though they aren't associative? I think I remember reading somewhere > that they were considered a geometric algebra... perhaps in Hestenes > book or in Penrose's Road to Reality. > > But wikipedia claims that a geometric norm _must_ be associative and > that a geometric algebra must be over a vector space with an > associative > norm. WTF? Is wikipedia oversimplifying? Or are octonions really > not > considered a geometric algebra despite their deep relevance to > geometry? > > And, more importantly, why do my searches for Clifford fail in Adobe > Reader, but succeed in Evince, while reading the following file: > > http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf > > ??? > > -- > glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com > > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by Tom Carter
Thus spake Tom Carter circa 10/14/2009 10:49 PM:
> Ligature . . . ff is (sometimes) a "single glyph" . . . Yeah, I figured it was something like that. "Clifford" appears in the abstract; so I cut-n-pasted that into the Adobe Reader search box. It showed one of those weird little ctrl-character boxes. But the search still didn't work. Leave it to Adobe to fsck up something as simple as a string search. Thanks. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: On Quaternions and Octonions, by John Conway and Derek Smith
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In reply to this post by Tom Carter
Thus spake Tom Carter circa 10/14/2009 11:30 PM:
> These days, most mathematicians are so comfortable with associativity > that they'll go ahead and include that as part of "the definition" (of, > e.g., a geometric algebra) . . . and then also they won't have a bunch > of theorems that start out, "Let A be an associative geometric algebra . > . ." rather than "Let A be a geometric algebra . . ." (for example . . .) After searching last night, I can't find the origins of my conflation between the two (division and geometric algebras). Perhaps in my earlier, sloppy, efforts, I just wasn't well enough informed to see the distinction. But they are definitely different, though some of them are both division and geometric, obviously. Your thought above may be right. Perhaps I stumbled across a lower quality paper, wherein they conflated the two. It's not in amongst the geometric algebra papers I archived on my hard disk, which would indicate that it wasn't very useful to me at the time (IF that's what happened ;-). Thanks for the help. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
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