Possibly the oldest henge in the British Isles – the Standing Stones of Stenness in Orkney, Scotland. The Stenness Watch Stone is visible in the distance. The Ness of Brogdar, Ring of Brogdar, Maeshowe, and Skara Brae are nearby but not visible.
Compare my photo with the following still from Powell & Pressburger’s first collaboration, The Spy in Black (1939).
In it Conrad Veidt is sneaking with a motorcycle between moonlit standing stones in Orkney, on his way from his U-boat near The Old Man of Hoy to a clandestine rendezvous in a house overlooking the British Grand Fleet in Scappa Flow.
I incidentally once dressed up for Halloween as one of Veidt’s earliest screen roles – the somnambulist in The Cabinet of Dr. Caligari (1920).
Besides Powell & Pressburger – and of course Kubrick – the sunset at Stenness also made me think of “The Mathematician’s Nightmare” from Bertrand Russell’s Nightmares of Eminent Persons and Other Stories (1954).
There are two things called the Cayley Plane,
The projective plane over the octonions, named after the mathematician Sir Arthur Cayley (1821–1895), and
The groundbreaking aeroplane designed by his distant cousin, the aviation pioneer Sir George Cayley (1774–1857).
I could write a lot about the first. In fact, I’ve written a paper about it,
The Cayley Plane and String Bordism, Geometry & Topology 18-4 (2014), 2045–2078.
But last week I travelled to Brompton [outside Scarborough, UK] to pay homage to the second.
I visited Sir George’s workshop at Brompton Hall, where he designed his plane. [Cayley Lane, pictured above, runs beside it.] And I spent the night at Sir George’s annex Wydale Hall, now a retreat centre run by the Diocese of York.
After breakfast a kind 83-year-old pastor walked me to Brompton Dale, where in 1853 Sir George’s terrified coachman John Appleby flew the plane [afterwards saying “Please, Sir George, I wish to give notice. I was hired to drive, and not to fly.”]
There is, as far as I know, no connection with the Brompton folding bicycle – my primary mode of transport, named by its inventor Andrew Ritchie after the Brompton Oratory in London, where Alfred Hitchcock got hitched, and which Ritchie could see from his workshop window while building prototypes in 1976 – outside the fact that Sir George invented the wire wheel for his plane.
Speaking of London, a few days earlier I visited Lincoln’s Inn, where in the 1840’s Sir Arthur would meet to discuss invariant theory with his friend JJ Sylvester – not to be confused with Sylvester II.
]]>Bierstadt Lake is perched on a moraine near the Continental Divide in Rocky Mountain National Park. After spotting the painter’s surname on a map, Vitaly Lorman and I climbed 600 feet to reach the moraine, then made our way through a dense pine forest to arrive at the lake, which instantly called to my mind Albert Bierstadt’s 1876 painting Mount Corcoran. [The lake is incidentally 7 miles from the Stanley Hotel, where in 1974 Stephen King wrote The Shining.]
]]>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.”
[Moriarty, incidentally, had an associate named McTague in the 1899 play Sherlock Holmes, cowritten by Doyle and William Gillette. In fact, “Lightfoot” McTague appears in the 1916 silent film adaptation of the play, considered among the holy grails of lost films until a nitrate dupe negative was discovered mislabeled in the vaults of the Cinémathèque Française in 2014. For a far more cinematic appearance of a McTague in silent film, however, see Erich von Stroheim’s 1924 masterpiece film adaptation Greed of Frank Norris’s 1899 novel McTeague. [Is it a coincidence that McTeague and the Holmes play both date to 1899?] Seeing Greed at Eastman House inspired me to recreate one of its famous title cards—compare with the original and notice that the recreation even shakes like a projected film.]
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