May 2008 - mathfun@mailman.xmission.com

[math-fun] phi, pho, phoo (was: golden)
by Steve Witham 28 May '08

28 May '08

>From: James Propp <jpropp(a)cs.uml.edu> > >I believe that historically both (1+sqrt(5))/2 and its reciprocal >(-1+sqrt(5))/2 have been called the golden ratio (maybe one is the >golden ratio and the other is the golden section?), so it'd be handy >if one of the two numbers were denoted phi and the other were denoted >by tau. >From: "Fred lunnon" <fred.lunnon(a)gmail.com> > >You're right about the ambiguity. I'd prefer to ignore (sqrt5 - 1)/2 though, >on the grounds that the conjugate of tau = (1 + sqrt5)/2 is actually >taubar = (1 - sqrt5)/2 = -1/tau, the negative of the smaller ratio. WFL Wikipedia & Mathworld both say (1 - sqrt5)/2 is called the "golden ratio conjugate" & Mathworld adds "silver ratio". Wikipedia rationalizes (heh) that it's the absolute value of the actual conjugate. Both use the convention phi = (1+sqrt5)/2 and Phi = (1-sqrt5)/2, -+- / (|) or (/) capital Phi -+- / but Mathworld sez that Phi for the larger and phi the smaller "are sometimes also used (Knott), although this usage is not necessarily recommended." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/ I have personally been fiddling with the number phi^-2 =~ .381966. Might as well call them phi, 1, pho, phoo. Which naturally extends to ...phooo, phoo, pho, 1, phi, phii, phiii... Although I don't know all the Greek letters for those. I do think we should redefine 1 as phi^0 and use the less confusing notation "ph". --Steve

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[math-fun] Knuth circle product [was ... Zeckendorf representation]
by Fred lunnon 27 May '08

27 May '08

CONJECTURE: x o y = 3 x y - x [(y+1)/tau^2] - y [(x+1)/tau^2] . PROOF: Presently skimped. WANTED: x o y o z = symmetric function of x,y,z. NOTES: x o y == x (*) y previously; tau == (1 + sqrt(5))/2; [...] == floor, integer part. WFL

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Re: [math-fun] Salutations on COORDINATE Day
by N. J. A. Sloane 27 May '08

27 May '08

Bill said: > I'll add them to www.tweedledum.com/rwg/idents.htm soon. There are some pretty amazing identities there already! Wow! Neil

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Re: [math-fun] golden
by Dan Asimov 27 May '08

27 May '08

<< I believe that historically both (1+sqrt(5))/2 and its reciprocal (-1+sqrt(5))/2 have been called the golden ratio (maybe one is the golden ratio and the other is the golden section?), so it'd be handy if one of the two numbers were denoted phi and the other were denoted by tau. (Maybe someone more historically and/or notationally savvy than I could chime in here.) >> One (unrealistic) way to avoid ambiguity might be to redefine the golden ratio as the element [(1+sqrt5)/2] of R/Z. (Where [x] here means the equivalence class of x mod Z, i.e., the coset x + Z.) Idea: since square bracket [x] to mean the greatest integer <= x has largely been replaced by the computer science notation |_x_|, perhaps we can standardize the notation at least in math-fun by using _x_ to mean floor(x) from now on. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] golden
by Dan Asimov 27 May '08

27 May '08

Neil wrote: << there are millions of places where "phi" is used and where "tau" is used for (1+sqrt(5))/2 ! it is too late to try to unify the notation, sorry! >> I don't think it's ever too late to *try* to unify notation. But, the current state where two symbols are used for the same thing is not especially bad. On the other hand, to try to redefine one of these to mean its reciprocal (as a previous poster suggested) would be only making any current ambiguity worse. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] golden
by James Propp 27 May '08

27 May '08

I believe that historically both (1+sqrt(5))/2 and its reciprocal (-1+sqrt(5))/2 have been called the golden ratio (maybe one is the golden ratio and the other is the golden section?), so it'd be handy if one of the two numbers were denoted phi and the other were denoted by tau. (Maybe someone more historically and/or notationally savvy than I could chime in here.) Jim

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Re: [math-fun] Knuth circle product [was ... Zeckendorf representation]
by Dan Asimov 27 May '08

27 May '08

Fred wrote: [various neat formulas for x o y, x @ y, and x o y o z] Alas, I forget the definition of x @ y. Fred, could you please remind me? Thanks, Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Salutations on COORDINATE Day
by rwg＠sdf.lonestar.org 27 May '08

27 May '08

CAROTENOID Day? CORONADITE Day? You'll figure it out. I just found a dvi of some slides of weird identities I gave at MIT. E.g. inf z ==== -n 1 + - \ 2 c > (sqrt(z) - c) (sqrt(sqrt(z)) - c) ...(z - c) = -----, / 1 + c ==== n = 0 |1-c| < 1. I'll add them to www.tweedledum.com/rwg/idents.htm soon. --rwg

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[math-fun] iterated integer part
by Fred lunnon 26 May '08

26 May '08

On 5/24/08, Fred lunnon <fred.lunnon(a)gmail.com> wrote: > [There must be something about this expression --- the version below > is/was wrong in OEIS too!] Apologies to OEIS --- the Fibonacci formula _is_ correct there --- I just need a new pair of spectacles --- or maybe a new pair of eyes --- new brain --- witter, witter ... Speaking of which --- coincidentally with RCS revealing unsuspected depths in the intger part function --- I've turned up a number of apparently elementary identities for which I can discover no obvious proof technique. The following example [occurring in the investigation of the "shifted down 2" variation of the Knuth circle product] is typical: for integer x, x - [[x/tau^2 + 1/tau^2]*tau^2 + 1/tau] = ( - ([(x+3)*tau^2] - 2*[(x+2)*tau^2] + [(x+1)*tau^2]) )mod 3 - 1 The sequence concerned, starting from x = 0, is 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 0, 1, ... The second expression is much easier to deal with than the first, since the iterated integer part has been eliminated. But why are the two equal? Fred Lunnon

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[math-fun] Number Theory on YouTube
by Christian Boyer 26 May '08

26 May '08

Journal of Number Theory on YouTube: http://www.youtube.com/user/JournalNumberTheory Christian.

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