Many proofs have since been found. Here is a quick, dynamical one I thought of which I have been unable to find in the literature.
Proof: Consider the function $f(x)=(1+x)/x$. The sequence of ratios of successive Fibonacci numbers is the orbit of $F_2/F_1=1$ under $f$, i.e. $$ f(F_n/F_{n-1})=F_{n+1}/F_n. $$ On the other hand, $f$ has precisely two fixed points, the roots of $x^2-x-1$, one positive (the golden ratio), the other negative. Since $f'(x)=-1/x^2$, $f$ is a decreasing function for $x>0$. And $f(2)=3/2$ while $f(3/2)=5/3$. So $f$ maps the interval $[3/2,2]$ into itself. In fact, $f$ is a contracting mapping on $[3/2,2]$ (with Lipschitz constant $4/9$) since $$ |f(x)-f(y)|=|y-x|/|xy|\le\tfrac49|y-x| $$ for $x,y\in[3/2,2]$. By the Banach fixed point theorem, then, all orbits in $[3/2,2]$ approach the golden ratio. Although $1$ is not in $[3/2,2]$, $f(1)=2$, so the orbit of $1$ approaches the golden ratio. ∎
This proof can be extended to show that the conclusion holds even if the first two terms $F_1,F_2$ of the Fibonacci sequence are altered, provided their ratio $F_2/F_1$ is not $\tfrac12(1-\sqrt5)$ (the negative, unstable fixed point of $f$).
In a sense this explains why the limit is the golden ratio: being a fixed point of $f$ is the defining property of the golden ratio.
]]>The Cayley Plane and String Bordism, Geometry & Topology 18-4 (2014), 2045–2078.
unmentioned in the abstract or introduction—and likely of independent interest—is the following characteristic-number-theoretic criterion for a set to generate the String bordism ring (with 6 inverted).
Theorem 4. A set $S$ generates the ring $\pi_*\mathrm{MO}\langle8\rangle[1/6]$ if:
This is a consequence of Hovey’s calculation of:
The homotopy of $\mathrm{MString}$ and $\mathrm{MU}\langle6\rangle$ at large primes, Algebr. Geom. Topol. 8 (2008), 2401–2414.
which itself builds on Hopf-ring-theoretic work of Ravenel and Wilson. It should be compared with the analogous result for the oriented bordism ring (with 2 inverted):
Theorem (Novikov, cf. Stong p. 180). A sequence $\{M^{4n}\}_{n\ge1}$ generates the ring $\pi_*\mathrm{MSO}[1/2]$ if and only if:
Robert Stong, Notes on Cobordism Theory, Princeton University Press, 1968.
John Milnor, Characteristic Classes, Princeton University Press, 1974.
A New Multistage Construction Chronology for the Great Serpent Mound, USA, Edward W. Herrmann, G. William Monaghan, William F. Romain, Timothy M. Schilling, Jarrod Burks, Karen L. Leone, Matthew P. Purtill, Alan C. Tonetti, Journal of Archaeological Science, Vol. 50, pp. 117–125, Oct 2014.
Serpent Mound: A Fort Ancient Icon?, Robert V. Fletcher, Terry L. Cameron, Bradley T. Lepper, Dee Anne Wymer and William Pickard, Midcontinental Journal of Archaeology, Vol. 21, No. 1, pp. 105–143, Spring 1996.
The Serpent Mound of Ohio, F.W. Putnam, The Century Illustrated Monthly Magazine, Vol. XXXIX, No. 6, pp. 871–888, April 1890.
History Got it Wrong: Scientists Now Say Serpent Mound as Old as Aristotle by Geoffrey Sea for Indian Country Today, 7 Aug 2014.
Crazy Theories Threaten Serpent Mound, Demean Native Heritage by Mary Annette Pember for Indian Country Today, 6 June 2013.
Sherlock Holmes’s criminal archenemy Professor Moriarty was modeled on Hopkins mathematics professor Simon Newcomb (1835–1909).
“At the age of twenty-one [Moriarty] wrote a treatise upon the binomial theorem, which has had a European vogue,” and was “the celebrated author of The Dynamics of an Asteroid, a book which ascends to such rarefied heights of pure mathematics that it is said that there was no man in the scientific press capable of criticizing it” [from “The Final Problem” (1893) and The Valley of Fear (1914) by Arthur Conan Doyle].
At the age of nineteen Newcomb wrote an unpublished New Demonstration of the Binomial Theorem and was a celebrated expert on the dynamics of the solar system. He was also notoriously malicious. For example, he destroyed Charles Sanders Peirce’s career by secretly changing Hopkins president Daniel Coit Gilman’s mind about granting Peirce tenure, this after having been the favorite PhD student of Peirce’s father Benjamin Peirce at Harvard. (Newcomb was incidentally the maternal grandfather of Hassler Whitney.)
These and further similarities between Moriarty and Newcomb were first noted by Ronald Schorn in 1978, and the case for direct inspiration was convincingly made by Bradley Schaefer in 1993 (by arguing that Doyle would have known of Newcomb through close friend Alfred Drayson):
Sherlock Holmes and some astronomical connections, Journal of the British Astronomical Association, Vol. 103, No. 1 (1993), pp. 30–34.
The possibility of completing the square:
Holmes | — | Moriarty |
⋮ | | | |
CS Peirce | — | Newcomb |
is suggested by the existence of the book:
Umberto Eco & Thomas A. Sebeok (eds.), The Sign of Three: Dupin, Holmes, Peirce, Indiana University Press (1984).
Joseph Brent, Charles Sanders Peirce, A Life, Indiana University Press, Bloomington, 1998.
Prof Stephen Hawking: make me a Bond villain, The Telegraph, London, 1 Dec 2014. [via Jack Morava]
I got onto this after lecturing on Bernstein polynomials and Bézier curves. The (constructive) proof of the Stone-Weierstrass approximation theorem in that case requires some finesse with binomial coefficients. To lighten the mood I made an off-the-cuff remark about Moriarty’s young interest in binomial coefficients, and the idea that a genuine rival for Holmes could have no background but mathematics. After lecture I started digging through some references and was led to Newcomb.
I later pointed out that, although Holmes scoffs when in their first meeting Dr Watson makes the comparison, Holmes was himself modeled on C. Auguste Dupin, who first appeared in “The Murders in the Rue Morgue” (1841) by Edgar Allan Poe (†1849 in Baltimore). [And Watson himself, ironically, on Poe’s anonymous narrator.]
Having begun the course with an analysis of the word “analysis”:
From the Greek ana·lysis=“up”·“to loosen”=“to loosen up”. To be compared with “dissolve”→“solve”. But not to be confused with anal·ysis—“anal” coming from “anus”, the Latin for “ring”. Which is relevant since it leads to the mathematical word “annulus”. And which illustrates that etymology is nonassociative: (ana)·(lysis)≠(anal)·(ysis). Don’t you think it’s a little strange, though, that the Greek and Latin are not entirely unrelated?
I ended with the opening of “Rue Morgue”:
The mental features discoursed of as the analytical, are, in themselves, but little susceptible of analysis. We appreciate them only in their effects. We know of them, among other things, that they are always to their possessor, when inordinately possessed, a source of the liveliest enjoyment. As the strong man exults in his physical ability, delighting in such exercises as call his muscles into action, so glories the analyst in that moral activity which disentangles. He derives pleasure from even the most trivial occupations bringing his talent into play. He is fond of enigmas, of conundrums, of hieroglyphics; exhibiting in his solutions of each a degree of acumen which appears to the ordinary apprehension præternatural. His results, brought about by the very soul and essence of method, have, in truth, the whole air of intuition.
The faculty of re-solution is possibly much invigorated by mathematical study, and especially by that highest branch of it which, unjustly, and merely on account of its retrograde operations, has been called, as if par excellence, analysis. Yet to calculate is not in itself to analyse. A chess-player, for example, does the one without effort at the other. It follows that the game of chess, in its effects upon mental character, is greatly misunderstood. I am not now writing a treatise, but simply prefacing a somewhat peculiar narrative by observations very much at random…
[As an aside, I visited Thomas Browne’s monument in Norwich this summer. His Hydriotaphia, Urn Burial, or, a Discourse of the Sepulchral Urns lately found in Norfolk (1658)—from which Poe’s epigraph is taken—is otherworldly.]
Observe that the opening paragraph of Poe’s manuscript did not survive to the printed edition:
]]>“It is not improbable that a few further steps in phrenological science will lead to a belief in the existence, if not to the actual discovery and location of an organ of analysis.”
My solution:
So to see the archive of past talks one scrolls up! I’ve not seen this approach before but think it’s quite natural. Additionally:
All this is achieved automatically with a few lines of jQuery code at the end of an easily maintained static webpage. The code parses the very same dates which humans read.
The page also automatically announces upcoming talks by email: A cron job (“seminarbot”) on the department server uses PhantomJS to fetch and execute an email-generating function reminderEmail() embedded in the webpage and drops the output into sendmail.
H. Karcher, Submersions via Projections, Geom. Dedicata 74 (1999), no. 3, pp. 249–260, MR1669359.