BLOG Carl McTague rss icon
mathematician, composer, photographer, fiddler

26 Jun 2018 | categories: Mathematics, Aviation

Cayley Lane

Carl McTague on Cayley Lane (2018)

There are two things called the Cayley Plane,

  1. The projective plane over the octonions, named after the math­e­mat­ician Sir Arthur Cayley (1821–1895), and

  2. George Cayley’s Sketch of His “Boy-Carrier” (1849)
    George Cayley’s Sketch of His “Boy-Carrier” (1849)
    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.

Wydale Hall (2018)Vale of Pickering (2018)

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.”]

Carl McTague at Brompton Dale (2018)

Brompton Dale (2018)

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.


27 Nov 2017 | categories: Mathematics

A Dynamical Proof That the Ratio of Successive Fibonacci Numbers Approaches the Golden Ratio

In his 1611 essay On the Six-Cornered Snowflake, Johannes Kepler observed that the ratio of successive Fibonacci numbers $F_n/F_{n-1}$ approaches the golden ratio $\tfrac12(1+\sqrt5)$. [Recall that $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\ge2$.]

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.


17 Dec 2014 | categories: Mathematics

How to Recognize Generators for String Bordism

A key result of my paper:

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:

  1. For each integer $n>1$, there is an element $M^{4n}$ of $S$ such that for any prime $p>3$: $$\begin{align*} \mathrm{ord}_p \big( \mathrm{s}_n[M^{4n}] \big) = \begin{cases} 1 & \text{if $2n=p^i-1$ or $2n=p^i+p^j$ for some integers $0 \le i \le j$} \\ 0 & \text{otherwise} \end{cases} \end{align*}$$ where $\mathrm{s}_n[M]$ is the characteristic number corresponding to the $n$th power sum symmetric polynomial $\sum x_i^n$ in the Pontrjagin roots of the tangent bundle of $M$, and $\mathrm{ord}_p(-)$ its $p$-adic order, i.e. the heighest power of $p$ which divides it.
  2. For each prime $p>3$ and each pair of integers $0<i<j$, there is an element $N^{2(p^i+p^j)}$ of $S$ such that: $$\begin{align*} \mathrm{s}_{(p^i+p^j)/2}[N^{2(p^i+p^j)}]&=0 \\ \mathrm{s}_{(p^i+1)/2,(p^j-1)/2}[N^{2(p^i+p^j)}] &\not\equiv 0 \mod p^2 \end{align*}$$ where $\mathrm{s}_{m,n}[N]$ is the characteristic number corresponding to the symmetric polynomial $\sum_{i\ne j} x_i^mx_j^n$ in the Pontrjagin roots of the tangent bundle of $N$.

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:

  • For any integer $n>0$ and any odd prime $p$: $$\begin{align*} \mathrm{ord}_p \big( \mathrm{s}_n[M^{4n}] \big) = \begin{cases} 1 & \text{if $2n=p^i-1$ for some integer $i>0$} \\ 0 & \text{otherwise} \end{cases} \end{align*}$$

References


2 Dec 2014 | categories: Mathematics, Crime

Binomial Coefficients and Villainy

Photograph of Professor Simon Newcomb (1905–9) from the Harris & Ewing Collection

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:

HolmesMoriarty
|
CS PeirceNewcomb

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).

References

Afterword

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.

Poster for the film Greed (1924) [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.]


20 Jul 2014 | categories: Mathematics

Comments on a Paper of Karcher

Some comments on Hermann Karcher’s beautiful paper:

H. Karcher, Submersions via Projections, Geom. Dedicata 74 (1999), no. 3, pp. 249–260, MR1669359.

  • Formulas (10), (11), (12) may be written: $$\begin{align*} R(X,Y)V &= -\overbrace{R(X,Y)\mathcal{H}\cdot V}^{\text{in $HM$}} + \overbrace{R^{\mathrm{V}}(X,Y)V-\big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big]V}^{\text{in $VM$}} \\ R(X,Y)H &= \phantom{-}\underbrace{R(X,Y)\mathcal{H}\cdot H}_{\text{in $VM$}} + \underbrace{R^{\mathrm{H}}(X,Y)H-\big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big] H}_{\text{in $HM$}} \end{align*}$$
  • The last formula of (15) may be written: $$ (\nabla_Xg_\epsilon)(Y,Z) = \epsilon(\epsilon-2)\cdot g(Y,\nabla_X\mathcal{V}\cdot Z) $$
  • The equations (16) for the difference tensor $\Gamma^\epsilon(X,Y)=\nabla_XY-\nabla^\epsilon_XY$ extend to a single general formula: $$ \Gamma^\epsilon(X,Y) = \epsilon(2-\epsilon)\cdot\big(\nabla_{\mathcal{H}X}\mathcal{H}\cdot\mathcal{V}Y +\nabla_{\mathcal{H}Y}\mathcal{H}\cdot\mathcal{V}X +\nabla_{\mathcal{V}X}\mathcal{H}\cdot\mathcal{V}Y\big) $$ The first and third terms combine easily but $\Gamma^\epsilon$—being the difference of torsion free connections—is symmetric, and this three-term formula showcases that symmetry—the third term being symmetric according to (63). Some useful special cases: $$\begin{align*} \Gamma^\epsilon(X,V) &= \epsilon(2-\epsilon)\cdot\nabla_X\mathcal{H}\cdot V \\ \Gamma^\epsilon(X,H) &= \epsilon(2-\epsilon)\cdot\nabla_H\mathcal{H}\cdot\mathcal{V}X \end{align*}$$ Comparing with (9) leads to: $$\begin{align*} (\nabla^2\pi)(X,Y) &= D\pi\cdot\Gamma^1(X,Y) \end{align*}$$
  • Some formulas worth recording: $$\begin{align*} \nabla_X^\epsilon\mathcal{H}\cdot H &= \nabla_X\mathcal{H}\cdot H \\ \nabla_X^\epsilon\mathcal{H}\cdot V &= (1-\epsilon)^2\cdot\nabla_X\mathcal{H}\cdot V \\ \big[\nabla^\epsilon_X\mathcal{H},\nabla^\epsilon_Y\mathcal{H}\big] &= (1-\epsilon)^2\cdot \big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big] \end{align*}$$ Note that the second equation says that the second fundamental forms of the fibers go to zero as $\epsilon\to1$.

Next Page »