\phi for the golden ratio?

\phi for the golden ratio?

[Oswald Wyler]

In seventh or eighth grade -- a long time ago -- , I learned the name
"goldener Schnitt" (golden ratio, ratio aurea) for the positive solution
of the equation x^2 = x + 1.  Recently, I read an article, I forgot
where, discussing this number and using \phi as the "accepted symbol"
for it.  The old name was never mentioned.

So far, I have only met three real or complex numbers with universally
accepted one-letter symbols:  \pi, e, i.  Have I missed something?

Oswald Wyler

Re: \phi for the golden ratio?

[Robert Seely]

[note from moderator: several replies were received on this item - which
is admittedly off-topic; they are digested below.]

There are a number of books on the subject  - or on the related Fibonacci
numbers - which call it "the golden ratio", "the golden number", "the divine
proportion", as well as phi.  (Check out Amazon for example.)  I think the
name isn't as "fixed" as pi, e, or i, but it's pretty standard, within a
small orbit.  (Maple doesn't know it - unlike Pi!)

Actually, in a sense, i isn't as standard as pi or e - at least in
engineering circles, some prefer J (or j).  (I've never seen a
*mathematical* writer use J though - anyone?)

Of course there are lots of non-real, non-complex, numbers with universally
accepted one-letter symbols.  (omega ... do you allow subscripts?)  But real
or complex?  (And presumably not constants from physics ...)  No others
spring to mind, though I'm willing to bet that 5 minutes after I hit "send"
one will do so!  Close?  There are a few, like the Euler constant gamma
(lim_{n->\infty} (sum_{k=1}^n 1/k - ln n), but I suspect most mathematicians
would have to look such cases up, so they hardly qualify.  (I only know
about gamma because it came up in conversation this past semester at work!)

-= rags =-

On Sat, 29 May 2004, Oswald Wyler wrote:

> In seventh or eighth grade -- a long time ago -- , I learned the name
> "goldener Schnitt" (golden ratio, ratio aurea) for the positive solution
> of the equation x^2 = x + 1.  Recently, I read an article, I forgot
> where, discussing this number and using \phi as the "accepted symbol"
> for it.  The old name was never mentioned.
>
> So far, I have only met three real or complex numbers with universally
> accepted one-letter symbols:  \pi, e, i.  Have I missed something?
>
> Oswald Wyler
>

--
<rags@math.mcgill.ca>
<www.math.mcgill.ca/rags>

------------------------------------------------------------------------------

From: Robert Knighten <Robert@Knighten.org>
Date: Sun, 30 May 2004 16:46:52 -0700

I have no idea if you've missed something, but at least in the English
speaking world \phi as the name for the golden ratio has indeed become the
"accepted symbol".  Asking Google about 'phi number "golden ratio"' produced
8120 pages containing all of those.  This includes, for example,

A biography of the number phi:
The Golden Ratio: The Story of Phi: The World's Most Astonishing Number
Mario Livio
Broadway Books, 320 pp, $24.95

I don't know if the use of \phi for the golden ratio is common in other
languages, but I suspect the pervasive effect of the English usage has had its
effect.

-- Bob

--
Robert L. Knighten
Robert@Knighten.org

-----------------------------------------------------------------------

Date: Sun, 30 May 2004 19:05:29 -0400
From: Steve Stevenson <steve@cs.clemson.edu>

This month's "Discover" magazine had an article on the golden ratio.

steve

----------------------------------------------------------------------

Date: Mon, 31 May 2004 10:21:41 -0300
From: "Robert J. MacG. Dawson" <Robert.Dawson@smu.ca>

    As a generalization: "Golden ratio" is used by geometers being
informal, autodidacts, historians, popular math writers, and crackpots.

    $\phi$ is used by popular math writers, autodidacts, and the better
class of crackpots.

    $\tau$ is used by most mathematicians.

    A not-entirely-correct analogy:

    golden ratio:$\phi$:\tau$ :: blue vitriol : copper sulphate: copper(II)
sulphate pentahydrate

    -Robert Dawson

------------------------------------------------------------------------------------

Date: Mon, 31 May 2004 10:29:36 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
Subject: "phi and gamma"

Oswald Wyler writes:

  I have only met three real or complex numbers with universally
  accepted one-letter symbols: \pi, e, i.  Have I missed something?

Going by today's final arbiter of the universal, Google switches into
calculator mode when queried for "e", "phi" and "pi" (delivering for
each its numerical value to 8 decimal places -- it also names "phi" as
"the golden ratio"). Google does not switch into calculator mode for
"gamma" or "i" but it does for "1*gamma" (naming it as "1 * Euler's
constant") and "1*i" (with no name). If you're willing to count
dimensioned numbers then it also recognizes (and names) "c", "h", "k",
and "u" if, in each case, you preface it with "1*".

If you go beyond single letters then who knows? A few I've found are
"1*au", "1*googol" and "1*mark twain". (Alas, Google has not yet
learned to recognize "1*millihelen".)

------------------------------------------------------------------------

Date: Mon, 31 May 2004 10:12:38 -0700 (PDT)
From: "John Baez" <baez@math.ucr.edu>


I don't know what this has to do with categories, but it gives me
the chance to recycle some trivia I recently learned.  This number
1.618... is widely called phi, but also Phi - and also tau.   It was
named Phi after the Greek sculptor Phidias, who helped design the Parthenon.
But it was named this only in 1914, in a book called The Curves of Life,
by the artist Theodore Cook.  And it was Cook who first started calling
1.618... the golden ratio!  Before him, 0.618... was called the golden
ratio.  Cook dubbed this number "phi", the lower-case baby brother of Phi.

In fact, the whole "golden" terminology can only be traced back to 1826,
when it showed up in a footnote to a book by one Martin Ohm, brother
of Georg Ohm, the guy with the law about resistors.  Before then, a lot
of people called 1/G the "Divine Proportion".  And the guy who started
that was Luca Pacioli, a pal of Leonardo da Vinci who translated Euclid's
Elements.  In 1509, Pacioli published a 3-volume text entitled Divina
Proportione, advertising the virtues of this number.

> So far, I have only met three real or complex numbers with universally
> accepted one-letter symbols:  \pi, e, i.  Have I missed something?

Sure - the Euler-Mascheroni constant,

gamma = 0.5772156649015...

      = lim_{n->infinity} (integral_1^n dx/x) - (1/1 + 1/2 + ... + 1/n)

See:

http://mathworld.wolfram.com/Euler-MascheroniConstant.html

By the way, engineers often call i "j".

Best,
jb

Re: \phi for the golden ratio?

[Paul B Levy]

[note from moderator: this will be the last of this thread...]

> So far, I have only met three real or complex numbers with universally
> accepted one-letter symbols:  \pi, e, i.  Have I missed something?

What about "r" (pronounced "rev"), denoting 2*pi, in the archaic but still
widely understood phrase "33 rpm record"?

A shame the Greeks chose pi rather than rev, for I think the latter is
more fundamental.

Paul

Re: \phi for the golden ratio?

[Nils Andersen]

[and one more...]

In reply to the question from Oswald Wyler <owyler@suscom-maine.net>
let me quote from Donald E. Knuth, The Art of Computer Programming
(1968, 1973), Section 1.2.8:

The number $\phi$ itself has a very interesting history.  Euclid
called it the "extreme and mean ratio"; the ratio of A to B
is the ratio of (A+B) to A, if the ratio of A to B is $\phi$.
Renaissance writers called it the "divine proportion"; and in the
last century it has commonly been called the "golden ratio".  In
the art world, the ratio of $\phi$ to 1 is said to be the most
pleasing proportion aesthetically, and this opinion is confirmed
from the standpoint of computer programming aesthetics as well.
For the story of $\phi$, see the excellent article "The Golden
Section, Phyllotaxis, and Whythoff's Game", by H.S.M. Coxeter,
Scripta Math. 19 (1953), 135-143, and see also Chapter 8 of
The 2nd Scientific American Book of Mathematical Puzzles and Diversions,
by Martin Gardner (New York:  Simon and Schuster, 1961).

-- Nils Andersen

>In seventh or eighth grade -- a long time ago -- , I learned the name
>"goldener Schnitt" (golden ratio, ratio aurea) for the positive solution
>of the equation x^2 = x + 1.  Recently, I read an article, I forgot
>where, discussing this number and using \phi as the "accepted symbol"
>for it.  The old name was never mentioned.
>
>So far, I have only met three real or complex numbers with universally
>accepted one-letter symbols:  \pi, e, i.  Have I missed something?
>
>Oswald Wyler