Wild Cats of Arizona

Here’s a quick followup to our discussion of the wild cats of Sumatra caught on camera by the World Wildlife Foundation. Recently there have been sightings of rare wild cats in Arizona!

• Marc Lacey, In southern Arizona, rare sightings of ocelots and jaguars send shivers, New York Times, 4 December 2011.

• Guide describes roaring, powerful jaguar, Arizona Daily Star, 23 November 2011.

For example, consider Donnie Fenn, who specializes in hunting and killing mountain lions (also known as cougars or pumas). He was taking his 10-year-old daughter out on her first hunt when his pack of hounds took off and cornered something in a tree. He then saw with the telephoto lens of his camera that it wasn’t a mountain lion—it was a jaguar, which is about twice as big!

“It’s the most amazing thing that’s ever happened to me,” said Fenn, who leads hunters to mountain lions with his dogs. “To be honest with you—I got to see it in real life, my daughter got to see it, but I hope never to encounter it again.

“I was nervous, scared, everything. It was just the aggressiveness—the power it had, the snarling. It wasn’t a snarl like a lion. It was a roar. I’ve never heard anything like it.”

Fenn was thrilled as well as scared. He had never expected to see such a large, endangered cat so early in his life, at age 32, he said. A lifelong hunter and Benson resident, he runs the mountain lion guide service as a sideline while working full time in an excavating business. He described his one-hour encounter with the jaguar as “a dream come true.”

He came away respectful of its power, speed and size.

“All my dogs took a pretty good beating. They had puncture wounds. … I got to see it in real life, and I’m glad, but I hope to never encounter it again,” he repeated.

He crept up close and took photos and a video of the jaguar:

He also notified state wildlife officials, who were later able to find hair samples left behind by the animal and a tree trunk that showed signs of being climbed by a large clawed animal. They believe he saw an adult male jaguar that weighed about 90 kilograms.

The jaguar, Panthera onca, is the third-largest cat in the world, only outranked by the lion and tiger. It’s the only surviving New World member of the genus Panthera. For example, there was once an American lion, but that went extinct 10,000 years ago, along with a lot of other large mammals, after people showed up. DNA evidence shows that the lion, tiger, leopard, jaguar, snow leopard, and clouded leopard share a common ancestor, and that this whole gang is between 6 and 10 million years old. (The so-called ‘mountain lion’, Puma concolor, is not in this group.)

Jaguars have mostly been killed off in the United States, but they survive from Mexico to Central and South America all the way down to Paraguay and northern Argentina. They are listed as ‘near threatened’ by the IUCN, or International Union for the Conservation of Nature.

The Arizona Fish and Game Department has also announced two reliable sightings of ocelots this year!

The ocelot, Leopardus pardalis, is a much smaller fellow, about the size of a domestic cat. Ocelots live in many parts of South and Central America and Mexico, and they’re listed as being of ‘least concern’ by the IUCN. Once their range extended up into the chaparral thickets of the Gulf Coast of south and eastern Texas, as well as part of Arizona, Louisiana, and Arkansas. But by now they are very hard to find in the United States. They seem to eke out an existence only in several small areas of dense thicket in South Texas… and, we now know, Arizona!

A Bet Concerning Neutrinos (Part 4)

Time for another bet!

As you may know, things are getting interesting. At first, a team of physicists claimed to see neutrinos travel from Switzerland to Italy 60 nanoseconds faster than light—but the machine made pulses of neutrinos lasting longer than that, so the whole experiment was dangerously tricky.

Now, it’s making pulses as short as 3 nanoseconds, and those physicists still see them arriving 60 nanoseconds early!

• Tomasso Dorigo, OPERA confirms: neutrinos travel faster than light!!, 17 November 2011.

This seems to have emboldened Curtis Faith, who wrote:

Care to take on another bet John?

I’d be willing to bet you:

• Two days of my time doing anything you choose within my areas of expertise.

against:

• Two days of your time doing anything I choose within your areas of expertise.

For simplicity, we could assume the same criteria for determining the winner as your existing bet.

I replied:

Hi, Curtis! In principle I’m willing to take that bet… but I’m just curious, what sort of things might you want me to do? What sort of things might I want you to do?

I don’t want you to say I need to dig a ditch in your backyard. I don’t mind digging ditches too much, but I don’t really want to fly over to your place to do it, especially since if I lose my bet to Frederik I may be allowed just one round-trip flight for a year.

Seriously, I’m actually curious about what I can do, that you would value, that I’m not already doing.

Curtis replied:

John,

It would be really mean of me to take your only flight. Besides, it seems to me that your expertise is probably not ditch digging.

I was thinking that I’d probably come over to Singapore (or Riverside) with my wife and spend the two days learning from you how to take some ideas that I’ve been working on forward. Getting advice for what sort of math I’d need to develop the ideas, where to learn more, prior work that might be relevant that I don’t know about that you might, etc. I’m guessing it would be the sort of discussions you have with graduate students but with someone who doesn’t have the same level of math skills (yet).

Since this is a very mild penalty, I said okay. We just need to write up an official contract here.

Okay: it’s your move, Curtis.

Babylon and the Square Root of 2

joint with Richard Elwes

Sometimes you can learn a lot from an old piece of clay. This is a Babylonian clay tablet from around 1700 BC. It’s known as “YBC7289”, since it’s one of many in the Yale Babylonian Collection.

It’s a diagram of a square with one side marked as having length 1/2. They took this length, multiplied it by the square root of 2, and got the length of the diagonal. And our question is: what did they really know about the square root of 2?

Questions like this are tricky. It’s even hard to be sure the square’s side has length 1/2. Since the Babylonians used base 60, they thought of 1/2 as 30/60. But since they hadn’t invented anything like a “decimal point”, they wrote it as 30. More precisely, they wrote it as this:

Take a look.

So maybe the square’s side has length 1/2… but maybe it has length 30. How can we tell? We can’t. But this tablet was probably written by a beginner, since the writing is large. And for a beginner, or indeed any mathematician, it makes a lot of sense to take 1/2 and multiply it by to get .

Once you start worrying about these things, there’s no end to it. How do we know the Babylonians wrote 1/2 as 30? One reason is that they really liked reciprocals. According to Jöran Friberg’s book A Remarkable Collection of Babylonian Mathematical Texts, there are tablets where a teacher has set some unfortunate student the task of inverting some truly gigantic numbers such as 3²⁵ · 5. They even checked their answers the obvious way: by taking the reciprocal of the reciprocal! They put together tables of reciprocals and used these to tackle more general division problems. To calculate they would break up into factors, look up the reciprocal of each, and take the product of these together with . This is cool, because modern algebra also sees reciprocals as logically preceding division, even if most non-mathematicians disagree!

So, we know from tables of reciprocals that Babylonians wrote 1/2 as 30. But let’s get back to our original question: what did they know about ?

On this tablet, they used the value

This is an impressively good approximation to

But how did they get this approximation? Did they know it was just an approximation? And did they know is irrational?

There seems to be no evidence that they knew about irrational numbers. One of the great experts on Babylonian mathematics, Otto Neugebauer, wrote:

… even if it were only due to our incomplete knowledge of the sources that we assume that the Babylonians did not know that had no solution in integer numbers and , even then the fact remains that the consequences of this result were never realized.

But there is evidence that the Babylonians knew their figure was just an approximation. In his book The Crest of the Peacock, George Gheverghese Joseph points out that a number very much like this shows up at the fourth stage of a fairly obvious recursive algorithm for approximating square roots! The first three approximations are

and

The fourth is

but if you work it out to 3 places in base 60, as the Babylonians seem to have done, you’ll get the number on this tablet!

The number 577/408 also shows up as an approximation to in the Shulba Sutras, a collection of Indian texts compiled between 800 and 200 BC. So, Indian mathematicians may have known the same algorithm.

But what is this algorithm, exactly? Joseph describes it, but Sridhar Ramesh told us about an easier way to think about it. Suppose you’re trying to compute the square root of 2 and you have a guess, say . If your guess is exactly right then

so

But if your guess isn’t right, won’t be quite equal to . So it makes sense to take the average of and , and use that as a new guess. If your original guess wasn’t too bad, and you keep using this procedure, you’ll get a sequence of guesses that converges to . In fact it converges very rapidly: at each step, the number of correct digits in your guess will approximately double!

Let’s see how it goes. We start with an obvious dumb guess, namely 1. Now 1 sure isn’t equal to 2/1, but we can average them and get a better guess:

Next, let’s average 3/2 and 2/(3/2):

We’re doing the calculation in painstaking detail for two reasons. First, we want to prove that we’re just as good at arithmetic as the ancient Babylonians: we don’t need a calculator for this stuff! Second, a cute pattern will show up if you pay attention.

Let’s do the next step. Now we’ll average 17/12 and 2/(17/12):

Do you remember what 17 times 17 is? No? That’s bad. It’s 289. Do you remember what 12 times 24 is? Well, maybe you remember that 12 times 12 is 144. So, double that and get 288. Hmm. So, moving right along, we get

which is what the Babylonians seem to have used!

Do you see the cute pattern? No? Yes? Even if you do, it’s good to try another round of this game, to see if this pattern persists. Besides, it’ll be fun to beat the Babylonians at their own game and get a better approximation to .

So, let’s average 577/408 and 2/(577/408):

Do you remember what 577 times 577 is? Heh, neither do we. In fact, right now a calculator is starting to look really good. Okay: it says the answer is 332,929. And what about 816 times 408? That’s 332,928. Just one less! And that’s the pattern we were hinting at: it’s been working like that every time. Continuing, we get

So that’s our new approximation of , which is even better than the best known in 1700 BC! Let’s see how good it is:

So, it’s good to 11 decimals!

What about that pattern we saw? As you can see, we keep getting a square number that’s one more than twice some other square:

and so on… at least if the pattern continues. So, while we can’t find integers and with

because is irrational, it seems we can find infinitely many solutions to

and these give fractions that are really good approximations to . But can you prove this is really what’s going on?

We’ll leave this as a puzzle in case you’re ever stuck on a desert island, or stuck in the deserts of Iraq. And if you want even more fun, try simplifying these fractions:

and so on. Some will give you the fractions we’ve seen already, but others won’t. How far out do you need to go to get 577/408? Can you figure the pattern and see when 665,857/470,832 will show up?

If you get stuck, it may help to read about Pell numbers. We could say more, but we’re beginning to babble on.

References

You can read about YBC7289 and see more photos of it here:

• Duncan J. Melville, YBC7289.

• Bill Casselman, YBC7289.

Both photos in this article are by Bill Casselman.

If you want to check that the tablet really says what the experts claim it does, ponder these pictures:

The number “1 24 51 10” is base 60 for

and the number “42 25 35” is presumably base 60 for what you get when you multiply this by 1/2 (we were too lazy to check). But can you read the clay tablet well enough to actually see these numbers? It’s not easy.

For a quick intro to what Babylonian mathematicians might have known about the Pythagorean theorem, and how this is related to YBC7289, try:

• J. J. O’Connor and E. F. Robertson, Pythagoras’s
theorem in Babylonian mathematics.

We got our table of Babylonian numerals from here:

• J. J. O’Connor and E. F. Robertson, Babylonian numerals.

For more details, try:

• D. H. Fowler and E. R. Robson, Square root approximations in Old Babylonian mathematics: YBC 7289 in context, Historia Mathematica 25 (1998), 366–378.

We also recommend this book, an easily readable introduction to the history of non-European mathematics that discusses YBC7289:

• George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Princeton U. Press, Princeton, 2000.

To dig deeper, try these:

• Otto Neugebauer, The Exact Sciences in Antiquity, Dover Books, New York, 1969.

• Jöran Fridberg, A Remarkable Collection of Babylonian Mathematical Texts, Springer, Berlin, 2007.

---

Here a sad story must indeed be told. While the field work has been perfected to a very high standard during the last half century, the second part, the publication, has been neglected to such a degree that many excavations of Mesopotamian sites resulted only in a scientifically executed destruction of what was left still undestroyed after a few thousand years. – Otto Neugebauer.

Quantum Theory Talks in Asia and Australia

Next Wednesday the Centre for Quantum Technologies is putting on a show:

• The Famous, the Bit and the Quantum, CQT Annual Symposium 2011, Centre for Quantum Technologies, Singapore, 7 December 2011.

There will be three talks. Since I’m not famous, I must either the ‘bit’ or the ‘quantum’. (Seriously, I have no idea what the title of this workshop means.)

• 3 pm. Immanuel Bloch (Max-Planck-Institut für Quantenoptik): Controlling and exploring quantum gases at the single atom level.

Abstract: Over the past years, ultracold quantum gases in optical lattices have offered remarkable opportunities to investigate static and dynamic properties of strongly correlated bosonic or fermionic quantum many-body systems. In this talk I will show how it has recently not only become possible to image such quantum gases with single atom sensitivity and single site resolution, but also how it is now possible to coherently control single atoms on individual lattice sites, how one can measure hidden order parameters and how one can follow the propagation of entangled quasiparticles in a many-body setting. In addition I will present recent results on the generation of strong effective magnetic fields for ultracold atoms in optical lattices, which has opened a new avenue for realizing fractional quantum Hall like states with atomic gases.

• 4.30 pm. Harry Buhrman (Centrum Wiskunde & Informatica & University of Amsterdam): Position-based cryptography.

Abstract: Position-based cryptography uses the geographic position of a party as its sole credential. Normally digital keys or biometric features are used. A central building block in position-based cryptography is that of position-verification. The goal is to prove to a set of verifier that one is at a certain geographical location. Protocols typically assume that messages can not travel faster than the speed of light. By responding to a verier in a timely manner one can guarantee that one is within a certain distance of that verifier. Quite recently it was shown that position-verification protocols only based on this relativistic principle can be broken by two attackers who simulate being at a the claimed position while physically residing elsewhere in space. Because of the no-cloning property of quantum information (qubits) it was believed that with the use of quantum messages one could devise protocols that were resistant to such collaborative attacks. Several schemes were proposed that later turned out to be insecure. Finally it was shown that also in the quantum case no unconditionally secure scheme is possible. We will review the field of position-based quantum cryptography and highlight some of the research currently going on in order to develop, using reasonable assumptions on the capabilities of the attackers, protocols that are secure in practice.

• 6 pm. John Baez (U.C. Riverside & CQT): Probabilities versus amplitudes.

Abstract: Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of “chemical reaction networks”, which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas•but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as fermions or bosons?

People who have been following my network theory course will know this stuff already. I’ll give a more detailed mini-course on network theory here:

• Expository Quantum Lecture Series 5 (EQuaLS5), Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, Malaysia, 9-13 January 2012.

This looks like fun because a number of people will be giving such mini-courses:

• Do Ngoc Diep (Inst of Math, Hanoi): A procedure for quantization of fields.

• Maurice de Gosson (Univ. of Vienna): The symplectic camel and quantum mechanics.

• Fredrik Stroemberg (Technical Univ. of Darmstadt): Arithmetic quantum chaos.

• S. Twareque Ali (Concordia University, Montreal): Coherent states: theory and applications.

The ‘symplectic camel’, in case you’re wondering, is an allusion to Mikhail Gromov’s result on classical mechanics limiting our ability to squeeze a region of phase space into a long and skinny shape. It’s like trying to squeeze a camel through the eye of a needle!

Later, I’ll give a version of my talk ‘Probabilities versus amplitudes’ at this workshop:

• Coogee ’12: Sydney Quantum Information Theory Workshop, Coogee Bay Hotel, Australia, 30 January – 2 February 2012.

The workshop will focus on quantum computation with spin lattices, quantum memory, and quantum error correction. The speakers include:

• Sean Barrett (Imperial College London, UK)
• Hector Bombin (Perimeter Institute, Canada)
• Andrew Doherty (Sydney, Australia)
• Guillaume Duclos-Cianci (Sherbrooke, Canada)
• Steve Flammia (Caltech/Washington, USA)
• Jeongwan Haah (Caltech, USA)
• Robert Koenig (IBM, USA)
• Tobias Osborne (Leibniz Universitat Hannover, Germany)
• David Poulin (Sherbrooke, Canada)
• Jiannis Pachos (Leeds, UK)
• Norbert Schuch (Caltech, USA)
• Frank Verstraete (Vienna, Austria)
• Guifre Vidal (Perimeter Institute, Canada)

I’d better stop blogging and get my talk ready!

Liquid Light

Elisabeth Giacobino works at the Ecole Normale Supérieure in Paris. Last week she gave a talk at the Centre for Quantum Technologies. It was about ‘polariton condensates’. You can see a video of her talk here.

What’s a polariton? It’s a strange particle: a blend of matter and light. Polaritons are mostly made of light… with just enough matter mixed in so they can form a liquid! This liquid can form eddies just like water. Giacobino and her team of scientists have actually gotten pictures:

Physicists call this liquid a ‘polariton condensate’, but normal people may better appreciate how wonderful it is if we call it liquid light. That’s not 100% accurate, but it’s close—you’ll see what I mean in a minute.

Here’s a picture of Elisabeth Giacobino (at right) and her coworkers in 2010—not exactly the same team who is working on liquid light, but the best I can find:

How to make liquid light

How do you make liquid light?

First, take a thin film of some semiconductor like gallium arsenide. It’s full of electrons roaming around, so imagine a sea of electrons, like water. If you knock out an electron with enough energy, you’ll get a ‘hole’ which can move around like a particle of its own. Yes, the absence of a thing can act like a thing. Imagine an air bubble in the sea.

All this so far is standard stuff. But now for something more tricky: if you knock an electron just a little, it won’t go far from the hole it left behind. They’ll be attracted to each other, so they’ll orbit each other!

What you’ve got now is like a hydrogen atom—but instead of an electron and a proton, it’s made from an electron and a hole! It’s called an exciton. In Giacobino’s experiments, the excitons are 200 times as big as hydrogen atoms.

Excitons are exciting, but not exciting enough for us. So next, put a mirror on each side of your thin film. Now light can bounce back and forth. The light will interact with the excitons. If you do it right, this lets a particle of light—called a photon—blend with an exciton and form a new particle called polariton.

How does a photon ‘blend’ with an exciton? Umm, err… this involves quantum mechanics. In quantum mechanics you can take two possible situations and add them and get a new one, a kind of ‘blend’ called a ‘superposition’. ‘Schrödinger’s cat’ is what you get when you blend a live cat and a dead cat. People like to argue about why we don’t see half-live, half-dead cats. But never mind: we can see a blend of a photon and an exciton! Giacobino and her coworkers have done just that.

The polaritons they create are mostly light, with just a teeny bit of exciton blended in. Photons have no mass at all. So, perhaps it’s not surprising that their polaritons have a very small mass: about 10^-5 times as heavy as an electron!

They don’t last very long: just about 4-10 picoseconds. A picosecond is a trillionth of a second, or 10^-12 seconds. After that they fall apart. However, this is long enough for polaritons to do lots of interesting things.

For starters, polaritons interact with each other enough to form a liquid. But it’s not just any ordinary liquid: it’s often a superfluid, like very cold liquid helium. This means among other things, that it has almost no viscosity.

So: it’s even better than liquid light: it’s superfluid light!

The flow of liquid light

What can you do with liquid light?

For starters, you can watch it flow around obstacles. Semiconductors have ‘defects’—little flaws in the crystal structure. These act as obstacles to the flow of polaritons. And Giacobimo and her team have seen the flow of polaritons around defects in the semiconductor:

The two pictures at left are two views of the polariton condensate flowing smoothly around a defect. In these pictures the condensate is a superfluid.

The two pictures in the middle show a different situation. Here the polariton condensate is viscous enough so that it forms a trail of eddies as it flows past the defect. Yes, eddies of light!

And the two pictures at right show yet another situation. In every fluid, we can have waves of pressure. This is called… ‘sound’. Yes, this is how ordinary sound works in air, or under water. But we can also have sound in a polariton condensate!

That’s pretty cool: sound in liquid light! But wait. We haven’t gotten to the really cool part yet. Whenever you have a fluid moving past an obstacle faster than the speed of sound, you get a ‘shock wave’: the obstacle leaves an expanding trail of sound in its wake, behind it, because the sound can’t catch up. That’s why jets flying faster than sound leave a sonic boom behind them.

And that’s what you’re seeing in the pictures at right. The polariton condensate is flowing past the defect faster than the speed of sound, which happens to be around 850,000 meters per second in this experiment. We’re seeing the shock wave it makes. So, we’re seeing a sonic boom in liquid light!

It’s possible we’ll be able to use polariton condensates for interesting new technologies. Giacobimo and her team are also considering using them to study Hawking radiation: the feeble glow that black holes emit according to Hawking’s predictions. There aren’t black holes in polariton condensates, but it may be possible to create a similar kind of radiation. That would be really cool!

But to me, just being able to make a liquid consisting mostly of light, and study its properties, is already a triumph: just for the beauty of it.

Scary technical details

All the pictures of polariton condensates flowing around a defect came from here:

• A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre, E. Giacobino, C. Ciuti, and A. Bramati, Hydrodynamic solitons in polariton superfluids.

and this is the paper to read for more details.

I tried to be comprehensible to ordinary folks, but there are a few more things I can’t resist saying.

First, there are actually many different kinds of polaritons. In general, polaritons are quasiparticles formed by the interaction of photons and matter. For example, in some crystals sound acts like it’s made of particles, and these quasiparticles are called ‘phonons’. But sometimes phonons can interact with light to form quasiparticles—and these are called ‘phonon-polaritons’. I’ve only been talking about ‘exciton-polaritons’.

If you know a bit about superfluids, you may be interested to hear that the wavy patterns show the phase of the order parameter ψ in the Landau-Ginzburg theory of superfluids:

If you know about quantum field theory, you may be interested to know that the Hamiltonian describing photon-exciton interactions involves terms roughly like

where is the annihilation operator for photons, is the annihilation operator for excitons, the Greek letters are various constants, and the third term describes the interaction of photons and excitons. We can simplify this Hamiltonian by defining new particles that are linear combinations of photons and excitons. It’s just like diagonalizing a matrix; we get something like

where and are certain linear combinations of and . These act as annihilation operators for our new particles… and one of these new particles is the very light ‘polariton’ I’ve been talking about!

New Climate Sensitivity Estimate

Devoted readers will remember my interview of Nathan Urban in week302—week305 of This Week’s Finds. We talked about how he estimated the probability that global warming will cause the biggest current in the North Atlantic to collapse.

Now he and a bunch of coauthors have a new paper using paleoclimate data and some of the same mathematical techniques to estimate of how much the Earth will warm if we double the amount of CO₂ in the atmosphere:

• A. Schmittner, N.M. Urban, J.D. Shakun, N.D. Mahowald, P.U. Clark, P.J. Bartlein, A.C. Mix and A. Rosell-Melé, Climate sensitivity estimated from temperature reconstructions of the last glacial maximum, Science, 2011.

The average global temperature rise when we double the amount of CO₂ in the atmosphere is called the climate sensitivity.

The paper claims that the “likely” (66% probability) climate sensitivity is between 1.7 and 2.6 °C. They say it’s “very likely” (90% probability) that the climate sensitivity is between 1.4 and 2.8 °C. Their best estimate is around 2.2 or 2.3 °C.

If true, this is good news, because other studies suggest 3 °C as the best estimate, 2 to 4.5 °C as the “very likely” range, and a chance of even higher figures.

On the other hand, Nathan and his collaborators predict a significantly higher climate sensitivity on land. Here’s a graph of the probability density for various possible values

As you can see, their analysis easily allows for warming of 3 to 4 °C on land if we double the amount of CO₂.

The best summary of the paper is this new interview of Nathan Urban by the blogger ‘thingsbreak’:

• Thingsbreak, Interview with Nathan Urban on his new paper “Climate sensitivity estimated from temperature reconstructions of the last glacial maximum”, Planet 3.0, 24 November 2010.

So, check that out if you want more details but aren’t quite ready for the actual paper! There’s a lot of important stuff I haven’t said here.

Lynn Margulis, 1938-2011

Lynn Margulis died on Tuesday, the 22nd of November.

She is most famous for her theory of endosymbiosis, which says that eukaryotic cells—the complicated cells of protozoa, animals, plants, and fungi—were formed in a series of stages where one organism swallowed another but, instead of digesting it, took it on as a symbiotic partner!

As a young faculty member at Boston University, she wrote a paper on this theory. It was rejected by about fifteen scientific journals, but eventually accepted:

• Lynn Sagan, On the origin of mitosing cells, J. Theor. Bio. 14 (1967), 255–274.

(At the time she was married to the well-known astronomer Carl Sagan; she later wrote a number of books with their son Dorion.)

While Lynn Margulis was not the first to suggest that endosymbiosis played a major role in evolution, her brilliant and endlessly energetic advocacy of this idea helped trigger a scientific revolution. By now, almost all biologists agree that chloroplasts, the little green packages where photosynthesis happens in plant cells, were originally free-living bacteria of their own. Here’s a model of a chloroplast:

As you can see, it’s quite a complex world of its own. Biologists also believe that mitochondria, the ‘powerhouses’ of eukaryotic cells:

were originally bacteria.

Indeed, both chloroplasts and mitochondria still have their own DNA, and they reproduce like bacteria, through fission. This is very useful in research on human populations, since mitochondrial DNA is passed down, not through the sperm cells, but only through the egg cells—which means it’s passed down from mothers to daughters, with sons as a mere sideshow. On the other hand, Y chromosomes go down from father to son.

There’s a lot more evidence for endosymbiosis as the origin of mitochondria and chloroplasts… and in fact, we can see this process going on today in various organisms. There are also many other ways in which organisms pass genes to each other.

This led Margulis to advocate a worldview in which the ‘tree’ of life is replaced by something more like a ‘thicket’:

It also suggests that competition between organisms is part of a bigger story where the formation of partnerships plays a major role. She has written many popular books on this, and they make great reading, even though some of her ideas are still controversial.

In 1995, the famous evolutionary biologist Richard Dawkins wrote:

I greatly admire Lynn Margulis’s sheer courage and stamina in sticking by the endosymbiosis theory, and carrying it through from being an unorthodoxy to an orthodoxy. I’m referring to the theory that the eukaryotic cell is a symbiotic union of primitive prokaryotic cells. This is one of the great achievements of twentieth-century evolutionary biology, and I greatly admire her for it.

I’ll end by quoting an excerpt from this essay, which you can read in its entirety online:

• Lynn Margulis, Gaia is a tough bitch, in The Third Culture: Beyond the Scientific Revolution, ed. John Brockman, Simon & Schuster, 1995.

I think it’s a good sample of how she wasn’t afraid to stir up controversy. Indeed, you’ll see her poking Dawkins with a sharp stick at the same time he was complimenting her!

More importantly, you’ll get a sense of how her work helped push evolutionary biology away from neo-Darwinism, also known as the Modern Synthesis. This approach tried to explain all evolution as a process of populations slowly changing through random mutation and natural selection. Now biologists are struggling to formulate an Extended Synthesis which takes many more processes into account.

Her remarks also some contain good warnings for mathematicians and physicists, such as myself, who want to dabble in biology. Physics may be defined as ‘the study of natural systems that can be accurately modeled by beautiful mathematics’. That’s why we can learn a surprising amount about physics just by sitting around scribbling. But living systems always surprise us, since they’re always more rich and complex than our models. So, biology requires more emphasis on experiment, observation, and perhaps a feeling for the organism:

I work in evolutionary biology, but with cells and microorganisms. Richard Dawkins, John Maynard Smith, George Williams, Richard Lewontin, Niles Eldredge, and Stephen Jay Gould all come out of the zoological tradition, which suggests to me that, in the words of our colleague Simon Robson, they deal with a data set some three billion years out of date. Eldredge and Gould and their many colleagues tend to codify an incredible ignorance of where the real action is in evolution, as they limit the domain of interest to animals—including, of course, people. All very interesting, but animals are very tardy on the evolutionary scene, and they give us little real insight into the major sources of evolution’s creativity. It’s as if you wrote a four-volume tome supposedly on world history but beginning in the year 1800 at Fort Dearborn and the founding of Chicago. You might be entirely correct about the nineteenth-century transformation of Fort Dearborn into a thriving lakeside metropolis, but it would hardly be world history.

By “codifying ignorance” I refer in part to the fact that they miss four out of the five kingdoms of life. Animals are only one of these kingdoms. They miss bacteria, protoctista, fungi, and plants. They take a small and interesting chapter in the book of evolution and extrapolate it into the entire encyclopedia of life. Skewed and limited in their perspective, they are not wrong so much as grossly uninformed.

Of what are they ignorant? Chemistry, primarily, because the language of evolutionary biology is the language of chemistry, and most of them ignore chemistry. I don’t want to lump them all together, because, first of all, Gould and Eldredge have found out very clearly that gradual evolutionary changes through time, expected by Darwin to be documented in the fossil record, are not the way it happened. Fossil morphologies persist for long periods of time, and after stasis, discontinuities are observed. I don’t think these observations are even debatable. John Maynard Smith, an engineer by training, knows much of his biology secondhand. He seldom deals with live organisms. He computes and he reads. I suspect that it’s very hard for him to have insight into any group of organisms when he does not deal with them directly. Biologists, especially, need direct sensory communication with the live beings they study and about which they write.

Reconstructing evolutionary history through fossils—paleontology—is a valid approach, in my opinion, but paleontologists must work simultaneously with modern-counterpart organisms and with “neontologists”—that is, biologists. Gould, Eldredge, and Lewontin have made very valuable contributions. But the Dawkins–Williams–Maynard Smith tradition emerges from a history that I doubt they see in its Anglophone social context. Darwin claimed that populations of organisms change gradually through time as their members are weeded out, which is his basic idea of evolution through natural selection. Mendel, who developed the rules for genetic traits passing from one generation to another, made it very clear that while those traits reassort, they don’t change over time. A white flower mated to a red flower has pink offspring, and if that pink flower is crossed with another pink flower the offspring that result are just as red or just as white or just as pink as the original parent or grandparent. Species of organisms, Mendel insisted, don’t change through time. The mixture or blending that produced the pink is superficial. The genes are simply shuffled around to come out in different combinations, but those same combinations generate exactly the same types. Mendel’s observations are incontrovertible.

So J. B. S. Haldane, without a doubt a brilliant person, and R. A. Fisher, a mathematician, generated an entire school of English-speaking evolutionists, as they developed the neo-Darwinist population-genetic analysis to reconcile two unreconcilable views: Darwin’s evolutionary view with Mendel’s pragmatic, anti-evolutionary concept. They invented a language of population genetics in the 1920s to 1950s called neo-Darwinism, to rationalize these two fields. They mathematized their work and began to believe in it, spreading the word widely in Great Britain, the United States, and beyond. France and other countries resisted neo-Darwinism, but some Japanese and other investigators joined in the “explanation” activity.

Both Dawkins and Lewontin, who consider themselves far apart from each other in many respects, belong to this tradition. Lewontin visited an economics class at the University of Massachusetts a few years ago to talk to the students. In a kind of neo-Darwinian jockeying, he said that evolutionary changes are due to the Fisher–Haldane mechanisms: mutation, emigration, immigration, and the like. At the end of the hour, he said that none of the consequences of the details of his analysis had been shown empirically. His elaborate cost-benefit mathematical treatment was devoid of chemistry and biology. I asked him why, if none of it could be shown experimentally or in the field, he was so wedded to presenting a cost-benefit explanation derived from phony human social-economic “theory.” Why, when he himself was pointing to serious flaws related to the fundamental assumptions, did he want to teach this nonsense? His response was that there were two reasons: the first was “P.E.” “P.E.?,” I asked. “What is P.E.? Population explosion? Punctuated equilibrium? Physical education?” “No,” he replied, “P.E. is `physics envy,'” which is a syndrome in which scientists in other disciplines yearn for the mathematically explicit models of physics. His second reason was even more insidious: if he didn’t couch his studies in the neo-Darwinist thought style (archaic and totally inappropriate language, in my opinion), he wouldn’t be able to obtain grant money that was set up to support this kind of work.

The neo-Darwinist population-genetics tradition is reminiscent of phrenology, I think, and is a kind of science that can expect exactly the same fate. It will look ridiculous in retrospect, because it is ridiculous. I’ve always felt that way, even as a more-than-adequate student of population genetics with a superb teacher—James F. Crow, at the University of Wisconsin, Madison. At the very end of the semester, the last week was spent on discussing the actual observational and experimental studies related to the models, but none of the outcomes of the experiments matched the theory.

This passage shows her tough side—these are the top names in evolutionary biology that she’s criticizing here, after all. But when I saw her speak, she was engaging and fun! You can see that yourself in these interviews. Hear how she started as a bad student in 4th grade, why her laboratory budget got cut to $0 in 2004… and get a sense of her career, personality, and ideas.