Great geophysicists #12: Gauss

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Carl Friedrich Gauss was born on 30 April 1777 in Braunschweig (Brunswick), and died at the age of 77 on 23 February 1855 in Göttingen. He was a mathematician, you've probably heard of him; he even has his own Linnean handle: Princeps mathematicorum, or Prince of mathematicians (I assume it's the royal kind, not the Purple Rain kind — ba dum tss).

Gauss's parents were poor, working class folk. I wonder what they made of their child prodigy, who allegedly once stunned his teachers by summing the integers up to 100 in seconds? At about 16, he was quite a clever-clogs, rediscovering Bode's law, the binomial theorem, and the prime number theorem. Ridiculous.

His only imperfection was that he was too much of a perfectionist. His motto was pauca sed matura, meaning "few, but ripe". It's understandable how someone so bright might not feel much need to share his work, but historian Eric Temple Bell reckoned that if Gauss had published his work regularly, he would have advanced mathematics by fifty years.

He was only 6 when Euler died, but surely knew his work. Euler is the only other person who made comparably broad contributions to what we now call the exploration geophysics toolbox, and applied physics in general. Here are a few: 

  • He proved the fundamental theorems of algebra and arithmetic. No big deal.
  • He formulated the Gaussian function — which of course crops up everywhere, especially in geostatistics. The Ricker wavelet is a pulse with frequencies distributed in a Gaussian.
  • The gauss is the cgs unit of magnetic flux density, thanks to his work on the flux theorem, one of Maxwell's equations.
  • He discovered the Cauchy integral theorem for contour integrals but did not publish it.
  • The 'second' or 'total' curvature — a coordinate-system-independent measure of spatial curvedness — is named after him.
  • He made discoveries in non-Euclidean geometry, but did not publish them.

Excitingly, Gauss is the first great geophysicist we've covered in this series to have been photographed (right). Unfortunately, he was already dead. But what an amazing thing, to peer back through time almost 160 years.

Next time: Augustin-Jean Fresnel, a pioneer of wave theory.

At home with Leonardo

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Well, OK, Leonardo da Vinci wasn't actually there, having been dead 495 years, but on Tuesday morning I visited the house at which he spent the last three years of his life. I say house, it's more of a mansion — the Château du Clos Lucé is a large 15th century manoir near the centre of the small market town of Amboise in the Loire valley of northern France. The town was once the royal seat of France, and the medieval grandeur still shows. 

Leonardo was invited to France by King Francis I in 1516. Da Vinci had already served the French governor of Milan, and was feeling squeezed from Rome by upstarts Rafael and Michelangelo. It's nice to imagine that Frank appreciated Leo's intellect and creativity — he sort of collected artists and writers — but let's face it, it was probably the Italian's remarkable capacity for dreaming up war machines, a skill he had honed in the service of mercenary and cardinal Cesare Borgia. Leonardo especially seemed to like guns; here are models of a machine gun and a tank, alongside more peaceful concoctions:

Inspired by José Carcione's assertion that Leonardo was a geophysicst, and plenty of references to fossils (even Palaeodictyon) in his notebooks, I scoured the place for signs of Leonardo dablling in geology or geophysics, but to no avail. The partly-restored Renaissance floor tiles did have some inspiring textures and lots of crinoid fossils... I wonder if he noticed them as he shuffled around?

If you are ever in the area, I strongly recommend a visit. Even my kids (10, 6, and 4) enjoyed it, and it's close to some other worthy spots., specifically Chenonceau (for anyone) and Cheverny (for Tintin fans like me). The house, the numerous models, and the garden (below — complete with tasteful reproductions from Leonardo's works) were all terrific.

Check out José Carcione's two chapters about Leonardo and
his work in 52 Things You Should Know About Geophysics.
Download the chapter for free! [PDF, 3.8MB]

Great geophysicists #11: Thomas Young

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Painting of Young by Sir Thomas LawrenceThomas Young was a British scientist, one of the great polymaths of the early 19th century, and one of the greatest scientists. One author has called him 'the last man who knew everything'¹. He was born in Somerset, England, on 13 June 1773, and died in London on 10 May 1829, at the age of only 55. 

Like his contemporary Joseph Fourier, Young was an early Egyptologist. With Jean-François Champollion he is credited with deciphering the Rosetta Stone, a famous lump of granodiorite. This is not very surprising considering that at the age of 14, Young knew Greek, Latin, French, Italian, Hebrew, Chaldean, Syriac, Samaritan, Arabic, Persian, Turkish and Amharic. And English, presumably. 

But we don't include Young in our list because of hieroglyphics. Nor  because he proved, by demonstrating diffraction and interference, that light is a wave — and a transverse wave at that. Nor because he wasn't a demented sociopath like Newton. No, he's here because of his modulus. 

Elasticity is the most fundamental principle of material science. First explored by Hooke, but largely ignored by the mathematically inclined French theorists of the day, Young took the next important steps in this more practical domain. Using an empirical approach, he discovered that when a body is put under pressure, the amount of deformation it experiences is proportional to a constant for that particular material — what we now call Young's modulus, or E:

This well-known quantity is one of the stars of the new geophysical pursuit of predicting brittleness from seismic data, and a renewed interested in geomechanics in general. We know that Young's modulus on its own is not enough information, because the mechanics of failure (as opposed to deformation) are highly nonlinear, but Young's disciplined approach to scientific understanding is the best model for figuring it out. 

Sources and bibliography

Footnote

¹ Thomas Young wrote a lot of entries in the 1818 edition of Encyclopædia Britannica, including pieces on bridges, colour, double refraction, Egypt, friction, hieroglyphics, hydraulics, languages, ships, sound, tides, and waves. Considering that lots of Wikipedia is from the out-of-copyright Encyclopædia Britannica 11th ed. (1911), I wonder if some of Wikipedia was written by the great polymath? I hope so.

Great geophysicists #10: Joseph Fourier

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Joseph Fourier, the great mathematician, was born on 21 March 1768 in Auxerre, France, and died in Paris on 16 May 1830, aged 62. He's the reason I didn't get to study geophysics as an undergraduate: Fourier analysis was the first thing that I ever struggled with in mathematics.

Fourier was one of 12 children of a tailor, and had lost both parents by the age of 9. After studying under Lagrange at the École Normale Supérieure, Fourier taught at the École Polytechnique. At the age of 30, he was an invited scientist on Napoleon's Egyptian campaign, along with 55,000 other men, mostly soldiers:

Citizen, the executive directory having in the present circumstances a particular need of your talents and of your zeal has just disposed of you for the sake of public service. You should prepare yourself and be ready to depart at the first order.
Herivel, J (1975). Joseph Fourier: The Man and the Physicist, Oxford Univ. Press.

He stayed in Egypt for two years, helping found the modern era of Egyptology. He must have liked the weather because his next major work, and the one that made him famous, was Théorie analytique de la chaleur (1822), on the physics of heat. The topic was incidental though, because it was really his analytical methods that changed the world. His approach of decomposing arbitrary functions into trignometric series was novel and profoundly useful, and not just for solving the heat equation. 

Fourier as a geophysicist

Late last year, Evan wrote about the reason Fourier's work is so important in geophysical signal processing in Hooray for Fourier! He showed how we can decompose time-based signals like seismic traces into their frequency components. And I touched the topic in K is for Wavenumber (decomposing space) and The spectrum of the spectrum (decomposing frequency itself, which is even weirder than it sounds). But this GIF (below) is almost all you need to see both the simplicity and the utility of the Fourier transform. 

In this example, we start with something approaching a square wave (red), and let's assume it's in the time domain. This wave can be approximated by summing the series of sine waves shown in blue. The amplitudes of the sine waves required are the Fourier 'coefficients'. Notice that we needed lots of time samples to represent this signal smoothly, but require only 6 Fourier coefficients to carry the same information. Mathematicians call this a 'sparse' representation. Sparsity is a handy property because we can do clever things with sparse signals. For example, we can compress them (the basis of the JPEG scheme), or interpolate them (as in CGG's REVIVE processing). Hooray for Fourier indeed.

The watercolour caricature of Fourier is by Julien-Leopold Boilly from his work Album de 73 Portraits-Charge Aquarelle’s des Membres de I’Institute (1820); it is in the public domain.

Read more about Fourier on his Wikipedia page — and listen to this excellent mini-biography by Marcus de Sautoy. And check out Mostafa Naghizadeh's chapter in 52 Things You Should Know About Geophysics. Download the chapter for free!

Great geophysicists #9: Ernst Chladni

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Ernst Chladni was born in Wittenberg, eastern Germany, on 30 November 1756, and died 3 April 1827, at the age of 70, in the Prussian city of Breslau (now Wrocław, Poland). Several of his ancestors were learned theologians, but his father was a lawyer and his mother and stepmother from lawyerly families. So young Ernst did well to break away into a sound profession, ho ho, making substantial advances in acoustic physics. 

Chladni, 'the father of acoustics', conducted a large number of experiments with sound, measuring the speed of sound in various solids, and — more adventurously — in several gases too, including oxygen, nitrogen, and carbon dioxode. Interestingly, though I can find only one reference to it, he found that the speed of sound in Pinus sylvestris was 25% faster along the grain, compared to across it — is this the first observation of acoustic anisotropy? 

The experiments Chladni is known for, however, are the plates. He effectively extended the 1D explorations of Euler and Bernoulli in rods, and d'Alembert in strings, to the 2D realm. You won't find a better introduction to Chladni patterns than this wonderful blog post by Greg Gbur. Do read it — he segués nicely into quantum mechanics and optics, firmly linking Chladni with the modern era. To see the patterns forming for yourself, here's a terrific demonstration (very loud!)...

The drawings from Chladni's book Die Akustik are almost as mesmerizing as the video. Indeed, Chladni toured most of mainland Europe, demonstrating the figures live to curious Enlightenment audiences. When I look at them, I can't help wondering if there is some application for exploration geophysics — perhaps we are missing something important in the wavefield when we sample with regular acquisition grids?

References

Chladni, E, Die Akustik, Breitkopf und Härtel, Leipzig, 1830. Amazingly, this publishing company still exists.

Read more about Chladni in Wikipedia and in monoskop.org — an amazing repository of information on the arts and sciences. 

This post is part of a not-very-regular series of posts on important contributors to geophysics. It's going rather slowly — we're still in the eighteenth century. See all of them, and do make suggestions if we're missing some!

July linkfest

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It's another linkfest! All the good stuff from our newsfeed over the last few weeks.

We mentioned the $99 supercomputer in April. The Adapteva Parellella is a bit like a Raspberry Pi, but with the added benefit of a 16- or 64-core coprocessor. The machines are now shipping, and a version is available for pre-order.

In April we also mentioned the University of Queensland's long-running pitch drop experiment. But on 18 July a drop fell from another similar experiment, but which has even slower drops...

A gem from history:

In the British Islands alone, twice as much oil as the navy used last year could be produced from shale. — Winston Churchill, July 1913.

This surprising quote was doing the rounds last week (I saw it on oilit.com), but of course Churchill was not fortelling hydraulic fracturing and the shale gas boom; he was talking about shale oil. But it's still Quite Interesting.

Chris Liner's blog is more than quite interesting — and the last two posts have been especially excellent. The first is a great tutorial video describing a semi-automatic rock volume estimation workflow. You can get grain size and shape data from the same tool (tip: FIJI is the same but slightly awesomer). And the most recent post is about a field school in the Pyrenees, a place I love, and contains some awesome annotated field photos from an iPhone app called Theodolite.

Regular readers will already know about the geophysics hackathon we're organizing in Houston in September, timed perfectly as a pre-SEG brain workout. You don't need to be a coder to get involved — if you're excited by the idea of creating new apps for nerds like you, then you're in! Sign up at hackathon.io.

If you crave freshness, then check my Twitter feed or my pinboard. And if you have stuff to share, use the comments or get in touch — or jump on Twitter yourself!

Great geophysicists #8: d'Alembert

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Jean-Baptiste le Rond d'Alembert was a French mathematician, born on 16 or 17 November 1717 in Paris, and died on 29 October 1783, also in Paris. His father was an artillery officer, but his mother was much more interesting. Having been a nun, she sought papal dispensation in 1714 for a new career as a fun-loving socialite, benefiting from the new government banknote printing scheme of John Law. She left her illegitimate child on the steps of Église St Jean Le Rond de Paris, whence he was taken to an orphanage. When his father returned from duty, he arranged for the boy's care.

Perhaps d'Alembert's greatest contribution to the world was helping Denis Diderot 'change the way people think' by editing the great Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des métiers of 1751. There were many contributors, but d'Alembert was listed as co-editor on the title page (left). This book was an essential ingredient in spreading the Enlightenment across Europe, and d'Alembert was closely involved in the project for at least a decade. 

But that's not why he's in our list of great geophysicists. As I mentioned when I wrote about Euler, d'Alembert substantially progressed the understanding of waves, making his biggest breakthrough in 1747 in his work on vibrating strings. His paper was the first time the wave equation or its solution had appeared in print:

Though Euler and d'Alembert corresponded on waves and other matters, and strongly influenced each other, they eventually fell out. For example, Euler wrote to Lagrange in 1759:

d'Alembert has tried to undermine [my solution to the vibrating strings problem] by various cavils, and that for the sole reason that he did not get it himself... He thinks he can deceive the semi-learned by his eloquence. I doubt whether he is serious, unless perhaps he is thoroughly blinded by self-love. [See Morris Kline, 1972]

D'Alembert did little mathematics after 1760, as he became more involved in other academic matters. Later, ill health gradually took over. He lamented to Lagrange (evidently an Enlightenment agony aunt) in 1777, six years before his death:

What annoys me the most is the fact that geometry, which is the only occupation that truly interests me, is the one thing that I cannot do. [See Thomas Hankins, 1970]

I imagine he died feeling a little hollow about his work on waves, unaware of the future impact it would have—not just in applied geophysics, but in communication, medicine, engineering, and so on. For solving the wave equation, d'Alembert, we salute you.

References

Read more on Wikipedia and The MacTutor History of Mathematics.

D'Alembert, J-B (1747). Recherches sur la courbe que forme une corde tenduë mise en vibration. (Researches on the curve that a tense cord forms [when] set into vibration.) Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 214–219. Read on Google Books, with its sister paper, 'Further researches...'.

Portrait is a pastel by Maurice Quentin de La Tour, 1704–88.

Great geophysicists #7: Leonhard Euler

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Leonhard Euler (pronounced 'oiler') was born on 15 April 1707 in Basel, Switzerland, but spent most of his life in Berlin and St Petersburg, where he died on 18 September 1783. Has was blind from the age of 50, but took this handicap stoically—when he lost sight in his right eye at 28 he said, "Now I will have less distraction".

It's hard to list Euler's contributions to the toolbox we call seismic geophysics—he worked on so many problems in maths and physics. For example, much of the notation we use today was invented or at least popularized by him: f (x), e, i, π. He reconciled Newton's and Liebnitz's versions of calculus, making huge advances in solving difficult real-world equations. But he made some particularly relevant advances that resonate still:

  • Leonardo and Galileo both worked on mechanical stress distribution in beams, but didn't have the luxuries of calculus or Hooke's law. Daniel Bernoulli and Euler developed an isotropic elastic beam theory, and eventually convinced people you could actually build things using their insights. 
  • Euler's equations of fluid dynamics pre-date the more complicated (i.e. realistic) Navier–Stokes equations. Nonetheless, this work continued into vibrating strings, getting Euler (and Bernoulli) close to a general solution of the wave equation. They missed the mark, however, leaving it to Jean-Baptiste le Rond d'Alembert. 
  • optics (also wave behaviour). Though many of Euler's ideas about dispersion and lenses turned out to be incorrect (e.g. Pedersen 2008, DOI 10.1162/posc.2008.16.4.392), Euler did at least progress the idea that light is a wave, helping scientists move away from Newton's corpuscular theory.

The moment of Euler's death was described by the Marquis de Condorcet in a eulogy:

He had full possession of his faculties and apparently all of his strength... after having enjoyed some calculations on his blackboard concerning the laws of ascending motion for aerostatic machines... [he] spoke of Herschel's planet and the mathematics concerning its orbit and a little while later he had his grandson come and play with him and took a few cups of tea, when all of a sudden the pipe that he was smoking slipped from his hand and he ceased to calculate and live.

"He ceased to calculate," I love that.

Great geophysicists #6: Robert Hooke

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Robert Hooke was born near Freshwater on the Isle of Wight, UK, on 28 July 1635, and died on 13 March 1703 in London. At 18, he was awarded a chorister scholarship at Oxford, where he studied physics under Robert Boyle, 8 years his senior. 

Hooke's famous law tells us how things deform and, along with Newton, Hooke is thus a parent of the wave equation. The derivation starts by equating the force due to acceleration (of a vibrating particle, say), and the force due to elastic deformation:

where m is mass, x is displacement, the two dots denote the second derivative with respect to time (a.k.a. acceleration), and k is the spring constant. This powerful insight, which allows us to compute a particle's motion at a given time, was first made by d'Alembert in about 1742. It is the founding principle of seismic rock physics.

Hooke the geologist

Like most scientists of the 17th century, Hooke was no specialist. One of his best known works was Micrographia, first published in 1665. The microscope was invented in the late 1500s, but Hooke was one of the first people to meticulously document and beautifully draw his observations. His book was a smash hit by all accounts, inspiring wonder in everyone who read it (Samuel Pepys, for example). Among other things, Hooke described samples of petrified wood, forams, ammonites, and crystals of quartz in a flint nodule (left). Hooke also wrote about the chalk formations in the cliffs near his home town.

Hooke went on to help Wren rebuild London after the great fire of 1666, and achieved great respect for this work too. So esteemed is he that Newton was apparently rather jealous of him, and one historian has referred to him as 'England's Leonardo'. He never married, and lived in his Oxford college all his adult life, and is buried in Bishopsgate, London. As one of the fathers of geophysics, we salute him.

The painting of Hooke, by Rita Greer, is licensed under a Free Art License. It's a interpretation based on descriptions of him ("his chin sharp, and forehead large"); amazingly, there are no known contemporary images of him. Hear more about this.

You can read more about the relationship between Hooke's law and seismic waves in Bill Goodway's and Evan's chapters in 52 Things You Should Know About Geophysics. Download their chapters for free!

Great geophysicists #5: Huygens

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Christiaan Huygens was a Dutch physicist. He was born in The Hague on 14 April 1629, and died there on 8 July 1695. It's fun to imagine these times: he was a little older than Newton (born 1643), a little younger than Fermat (1601), and about the same age as Hooke (1635). He lived in England and France and must have met these men.

It's also fun to imagine the intellectual wonder life must have held for a wealthy, educated person in these protolithic Enlightenment years. Everyone, it seems, was a polymath: Huygens made substantial contributions to probability, mechanics, astronomy, optics, and horology. He was the first to describe Saturn's rings. He invented the pendulum clock. 

Then again, he also tried to build a combustion engine that ran on gunpowder. 

Geophysicists (and most other physicists) know him for his work on wave theory, which prevailed over Newton's corpuscles—at least until quantum theory. In his Treatise on Light, Huygens described a model for light waves that predicted the effects of reflection and refraction. Interference has to wait 38 years till Fresnel. He even explained birefringence, the anisotropy that gives rise to the double-refraction in calcite.

The model that we call the Huygens–Fresnel principle consists of spherical waves emanating from every point in a light source, such as a candle's flame. The sum of these manifold wavefronts predicts the distribution of the wave everywhere and at all times in the future. It's a sort of infinitesimal calculus for waves. I bet Newton secretly wished he'd thought of it.