As I watched this miniature world self-assemble on my windshield like an alien landscape, I wondered about the physics behind these patterns. I learned later that these patterns of ice are related to a rich and very active current area of research in math and physics known as universality. The key mathematical principles that belie these intricate patterns lead us to some unexpected places, such as coffee rings, growth patterns in bacterial colonies, and the wake of a flame as it burns through cigarette paper.

Let’s start with a simple example. Imagine a game similar to Tetris, but where you only have one kind of block – a 1 x 1 square. These identical blocks fall at random, like raindrops. Here’s a question for you. What pattern of blocks would you expect to see building up at the bottom of the screen?

You might guess that since the blocks are falling randomly, you should end up with a smooth, uniform pile of blocks, like the piles of sand that collect on a beach. But this isn’t what happens. Instead, in our make-believe Tetris world, you end up with a rough, jagged skyline, where tall towers sit next to deep gaps. A tall stack of blocks is just as likely to sit next to a short stack as it is to sit next to another tall stack.

This doesn’t look much like what I saw on my windshield. For one thing, there aren’t any gaps or holes. But we’ll get to that later.

This Tetris world is an example of what’s known as a Poisson process, and I’ve written about these processes before. The main point is that randomness doesn’t mean uniformity. Instead, randomness is typically clumpy, just like the jagged skyline of Tetris blocks that you see above, or like the clusters of buzzbombs dropped over London in World War II.

This Tetris example might seem a bit abstract, so let me introduce you to a guy who takes abstract ideas and connects them to real-world examples. His name is Peter Yunker, and he’s a physicist at Harvard who’s also really into his coffee.

Yunker was curious about what causes these ring shaped coffee stains. In 1997, a group of physicists worked out the reason that coffee forms this ring. As a drop of coffee evaporates, liquid from the center rushes outwards to the edge of the drop, carrying coffee particles with it. The drop starts to flatten. Eventually, all you’re left with is a thin ring, as the coffee particles have all rushed to the edge of the drop. Here’s a (wonderfully trippy) video of work by Yunker’s team, showing what this process looks like.

What Yunker demonstrated is really pretty neat. He discovered that the reason that coffee makes a ring has to do with the shape of the coffee particles. Look at a drop of coffee under a microscope, and you’ll find tiny, round coffee particles suspended in water. If you zoom into the edge of an evaporating coffee drop, you’ll see coffee particles sliding past each other, just like the blocks in our Tetris world. In fact, Yunker demonstrated mathematically that the pattern of growth of these coffee particles exactly mirrors that of our randomly falling Tetris blocks!

And here’s the crazy thing. Yunker and his colleagues also discovered that if you replaced all the spherical coffee particles with new particles that are more elongated, sort of like ovals, then you get an entirely different pattern. Instead of a ring, you get a solid blotch. You can see this happening in the video above.

In one case you get a coffee ring, and in the other case you get a solid blotch. So why does tweaking the shape of the particle change the overall pattern of growth? To understand why the oval particles behave differently from the spherical ones, we first need to tweak our Tetris game. Let’s call the new version Sticky Tetris.

In sticky Tetris, a block keeps falling until it touches another block. As soon as the falling block touches another block, even if only from the side, it immediately sticks into place.

It’s a small modification to the rules, but it has a pretty big consequence. In regular Tetris, it takes very many blocks to fill a deep gap, in sticky Tetris, you can fill a gap with a single block. Very quickly, the height differences between towers start to even out. Instead of the jagged, rough skyline of our regular Tetris world, the skyline in the sticky Tetris world is more smooth.

That looks a lot more like the pattern on my windshield!

And here’s the point. While the spherical coffee particles behave like regular Tetris pieces, the oval shaped particles behave just like these sticky Tetris pieces. The moment an oval coffee particle touches another one, it sticks in place. Instead of the jagged skyline from before, you get this Swiss cheese like pattern, an intricate structures of sprawling filaments separated by holes and gaps.

So here we have essentially two distinct kinds of growth processes. On the one hand we have things that accumulate like Tetris blocks, or like particle of coffee in a coffee ring. Here’s an animation of real data from Yunker’s lab showing what this looks like.

On the other hand, we have things that accumulate like Sticky Tetris blocks or like oval shaped coffee particles. The growth of these particles looks like this (again, this is real data).

It’s clear that these are two qualitatively different kinds of patterns.

But it’s also a quantitative difference. Remember that in the Tetris world, you end up with a jagged skyline, while in the sticky Tetris world, the skyline is more smooth. By studying how the topmost layer of particles (the skyline) widens over time, physicists can classify growth processes into different categories. In the jargon of the field, processes that grow at different rates really belong into different Universality Classes.

You can think of universality classes like a sort of mathematical filing cabinet. Say that you’re studying how ice particles clunk together on your windshield. If the rate at which the skyline widens matches the blue curve above, ice clunking is in the same universality class as Tetris. If it matches the purple curve, then ice clunking is in the same universality class as Sticky Tetris. Now, there are other universality classes out there, and not all growth processes can be neatly filed into a universality class. But the key point is that many seemingly different physical systems, when analyzed mathematically, show identical patterns of growth. This slightly mysterious tendency for very different things to behave in very similar ways is the essence of universality.

What’s more, there is a rich mathematical theory behind this sticky Tetris universality class, described by an equation known as the Kardar–Parisi–Zhang (KPZ) equation. To give you a sense of how current this research is, it was as late as 2010 that mathematicians managed to prove that this KPZ equation is in the same universality class as sticky Tetris.

These deep connections between coffee rings and the KPZ equation took Peter Yunker by surprise. In Yunker’s words, “Alexei Borodin, a mathematician from MIT, contacted us after we published a paper on how particle shape affects particle deposition regarding the coffee-ring effect. He saw our experimental videos online and was reminded of simulations that he has performed. I think this is a great example of the value of reaching out across disciplines – we never would have studied this topic without Alexei bringing it to our attention.”

And this sticky Tetris universality class has turned up in all sorts of odd places. One example involves burning paper. A physics experiment in 1997 took sheets of copier paper, carefully lit them on fire from one end, and recorded the flame front as it burnt through the paper. Here’s a sketch of what they saw. You’re looking at multiple snapshots of the flame, as it burns through the paper.

As the flame burns through the paper, it develops a smooth, wavy pattern. And when the physicists studied the growth of this flame front in detail, they found that it exactly matches the predictions of the KPZ equation. They repeated their experiment using cigarette paper as well as copier paper, and saw the same results. In their words, “The second set of experiments on the cigarette paper gave results consistent with those for the copier paper despite the fact that the cigarette paper is strongly anisotropic and may contain nontrivial correlations.” (Always gotta watch out for those nontrivial correlations in cigarette paper.)

And another example that’s pretty neat and unexpected – bacterial colonies. A team of Japanese physicists showed in 1997 that in certain nutrient conditions, the edge of a bacterial colony grows outwards in exactly the manner predicted by the KPZ (sticky Tetris) universality class. Here’s an animated gif of this in action, adapted from their paper. What you’re looking at is a zoomed in photograph of the edge of a bacterial colony, as it grows in a petri dish.

Now, if you think about it, there’s something deeply puzzling here. Bacterial colonies, travelling flames, and coffee particles are all totally different systems, and there’s no reason to expect that they should obey the same mathematical laws of growth. So what’s behind this mysterious universality? Why do such different beasts play by the same rules?

You might have noticed that all these examples look a little, well, fractal-esque. It turns out that the phenomenon of universality is intricately connected to the fact that these systems are each self-similar, like fractals. As I zoomed my camera into the ice particles on my windshield, the overall pattern looked basically the same. The same is true for the front of the flame, the edge of the bacterial colony, or the skyline of sticky Tetris. Here’s an example of a curve that’s self-similar (or scale-invariant, as physicists like to call it).

Surprisingly, this self-similarity implies that many of the nitty-gritty physics details of bacteria, flames, or coffee turn out to be irrelevant. According to Peter, “the fractal nature of these growth processes is essential to their universality. In order to be universal, a system cannot depend on its microscopic details, like particle size or typical interaction lengthscale. Thus, a universal system should be scale-invariant.”

Which brings me back to the ice particles on my windshield. They clumped together in these wonderfully fractal-esque patterns that, to my eye, looked a lot like sticky Tetris. I wanted to know if there’s a connection between these ice particles and the KPZ universality class. I put the question to Peter Yunker.

He responded, “These videos are fantastic. I agree with you that the underlying process occurring here appears quite similar to a KPZ process. However, this may be a great example of why it is difficult to identify KPZ processes in real experiments. The rearrangements of these structures have a strong effect on how the interface is developing. Thus, it is very unlikely that this system exhibits the same growth exponents as a KPZ process.”

It seems that the very piece of physics that makes these ice patterns short-lived is also what makes them so hard to study. And so, let me end with a very short video, a tiny meditation on the theme of growth and longevity.

References

Coffee Stains Test Universal Equation. Physics 6, 7 (2013) – an excellent readable account on the research of Yunker, Yodh, Borodin and colleagues

In Mysterious Pattern, Math and Nature Converge. Natalie Wolchover does a really great job of covering Universality from a totally different angle. If you’re not reading her stuff, you ought to!

Ace mathematician Terrence Tao has written a good explainer on Universality. It’s a long read that’s packed with insights.

Animated gifs of Tetris simulations and coffee deposition data were made with permission from data by Yunker et al. (2013)

Academic References

Effects of Particle Shape on Growth Dynamics at Edges of Evaporating Drops of Colloidal Suspensions. Yunker, Lohr, Still, Borodin, Durian and Yodh, Phys. Rev. Lett. 110, 035501 (2013)

Suppression of the coffee-ring effect by shape-dependent capillary interactions. Yunker, Still, Lohr and Yodh, Nature 476, 308–311 (2011)

The Kardar-Parisi-Zhang equation and universality class by Ivan Corwin – Although very mathematical, this an excellent and clearly written review of the KPZ equation and its connection to Universality, written by one of the experts in the field.

Self-Affinity for the Growing Interface of Bacterial Colonies. Wakita, Itoh, Matsuyama and Matsushita, J. Phys. Soc. Jpn. 66 (1997)

Kinetic Roughening in Slow Combustion of Paper. Maunuksela, Myllys, Kähkönen, Timonen, Provatas, Alava and Ala-Nissila, Phys. Rev. Lett. 79, 1515–1518 (1997)

Boeing recently launched a new line of aircraft, the 787 Dreamliner, that they claim uses 20% less fuel than existing, similarly sized planes.

How did they pull off this sizeable bump in fuel efficiency? And can you always build a more fuel-efficient aircraft? Imagine a hypothetical news story, where a rival company came up with a new type of airplane that used half the fuel of its current day counterparts. Should you believe their claim?

More generally, do the laws of physics impose any limits on the efficiency of flight? The answer, it turns out, is yes.

There’s something about flying that doesn’t sit well with us. If we never saw a bird fly, it may never have occurred to us to build flying machines of our own.

Here’s where I think this sense of unease comes from. It takes stuff to support stuff. Everyday objects fall unless other things get in their way. Take the floor away, and you’ll plummet to your doom – the air below your feet isn’t going to do much for you. We move through air so effortlessly, that we barely notice it’s there. So what keeps a plane up? There doesn’t seem to be enough ‘stuff’ there to hold up a bird, let alone a Boeing aircraft weighing up to 500,000 pounds. To put that last number in context, its more than the weight of an adult blue whale!

Why is it that planes fly and whales typically don’t? The answer is easy to state, but its consequences are rather surprising. Planes fly by throwing air down. That’s basically it. It’s an important point, so I’ll say it again. Planes fly by throwing air down.

As a plane hurtles through the air, it carves out a tube of air, much of which is deflected downwards by the wings. Throw down enough air fast enough, and you can stay afloat, just as the downwards thrust of a rocket pushes it up. The key is that you have to throw down a lot of air (like a glider or an albatross), or throw it down really fast (like a helicopter or a hummingbird).

A physicist’s two-step guide to flight (it’s simple, really!)

Let’s make this idea more quantitative. Following David MacKay’s wonderful book on Sustainable Energy, I’m going to build a toy model of flight. A good model should give you a lot of bang for the buck. The means being able to predict relevant quantities about the real world while making a minimum of assumptions.

Step 1: Sweep out a tube of air

As a plane moves, it carves out a tube of air. This air was stationary, minding its own business, until the airplane rammed into it. This costs energy, for the same reason your car’s fuel efficiency drops when you speed up on the highway. Your car has to shove air out of its way.

Exactly how much energy does this cost? You might remember from high school physics that it takes an amount of energy equal to to bring stuff with mass up to a speed .

In our case, we have

There’s still this mysterious factor of the mass of the air tube. To work this out, we can use a favorite trick in the toolbox of a physicist – unit cancellation. We can re-write the humble kilogram as a seemingly complicated product of terms.

What we’ve done here is to express an unknown mass of air in terms of other quantities that we do know. Each of these terms makes sense. Air that’s more dense will weigh more. A fatter plane (larger cross-sectional area) sweeps out more air, as does a faster plane. We’ve arrived at a meaningful result, just by playing around with units. In the words of Randall Munroe, unit cancellation is weird.

Put these two ideas together and here’s what you find:

Here’s a graph of what that looks like.

If you’re with me so far, we just found that for a plane to plow through air, it has to expend an amount of energy proportional to the speed of the plane to third power. (The extra factor of v comes from the fact that faster planes sweep out a larger mass of air.) If you want to go twice as fast, you need to work 8 times as hard to shove air out of your way.

We’ve arrived at a general rule about the physics of drag. This holds true for a car on the highway, or for a swimmer or cyclist in a race. It’s why drag racing cars get only about 0.05 miles to a gallon! If we want to reduce overall energy consumption by cars, one option is to lower the speed limits on highways.

What does this mean for our toy plane? It would seem that the slower the plane, the higher its efficiency. So are airplane speed limits also in order? Absolutely not! To see why, read on to the second half the story..

Step 2: Throw the air down

In order to fly, a plane must throw air downwards. This generates the lift that a plane needs to stay up. It turns out that slower planes have to throw air harder to stay afloat. That’s why slow moving hummingbirds and pigeons have to flap their wings frenetically. It’s also why planes extend flaps while landing – they’re not throwing the air fast enough, so they compensate by throwing more of it.

More precisely, for a plane to stay afloat, the speed of the air jettisoned downwards must be inversely proportional to the speed of the plane. (You can take my word for this, although if you want to see where it comes from, take a look at David MacKay’s book.)

So we can now work out the second part of the puzzle. How much energy does it take to throw air down? As before, this is given by

Just as we did in the first step, let’s express things in terms of the speed of the plane.

In words, the energy spent in generating lift is inversely proportional to the speed of the plane. Here’s what this looks like on a graph.

You can see from the plot that, as far as lift is concerned, slower flight is less efficient than faster flight, because you have to work harder in throwing air downwards.

There’s a lot to chew on here. To summarize, we’ve discovered that in making a machine fly, you have to spend energy (really fuel) in two ways.

1. Drag: You need to spend fuel to push air away. This keeps you from slowing down.
2. Lift: You need to spend fuel to throw air down. This is what keeps the plane afloat.

The total fuel consumption is the sum of these two parts.

If you fly too fast, you’ll spend too much fuel on drag (think of a drag racer or an F-16). Fly too slow, and you’ll have to spend too much fuel on generating lift, like a hummingbird furiously flapping its wings, powered by high calorie nectar. However, at the bottom of this curve there is a happy minimum, an ideal speed that resolves this tradeoff. This is the speed at which a plane is most efficient with its fuel. Be it through the ingenuity of aircraft engineers, or the ruthless efficiency of natural selection,  airplanes and birds are often fine-tuned to be as energy efficient as possible.

Here’s a plot of experimental data of the power consumption of different birds, as their flight speed varies.

You can see that it matches the qualitative predictions of the toy model.

But we can do more than this, and actually extract quantitative predictions from the model. An undergraduate schooled in calculus should be able to work out that special optimal speed at which energy consumption is a minimum. David MacKay plugs in the numbers in  his book, and finds that the optimal speed of an albatross is about 32 mph, and for a Boeing 747 is about 540 mph. Both these numbers are remarkably close to the real values. Albatrosses fly at about 30-55 mph, and the cruise speed of a Boeing 747 is about 567 mph. 

That’s a lot of mileage from a toy model!

And with this physicsy interlude into the world of albatrosses, hummingbirds, and jet planes, we come back to the question of the fuel efficiency of Boeing’s new aircraft.

You can actually use the model to work out the fuel efficiency of a plane. What you find is that it really just depends on a few factors: the shape and surface of the plane, and the efficiency of its engine. And of these factors, the engine efficiency plays the biggest role. So we would predict that engine efficiency, followed by improvements in body design might drive Boeing’s fuel savings.

This agrees with Boeing’s own assessment.

New engines from General Electric and Rolls-Royce are used on the 787. Advances in engine technology are the biggest contributor to overall fuel efficiency improvements.

New technologies and processes have been developed to help Boeing and its supplier partners achieve the efficiency gains. For example, manufacturing a one-piece fuselage section has eliminated 1,500 aluminum sheets and 40,000 – 50,000 fasteners.

Try as we like, we can’t squeeze a lot of improvement out of airplanes. Engines are already remarkably efficient, and you certainly can’t shrink the size of a plane by much, as economy class passengers can well attest. New manufacturing techniques could cut the amount of drag on the plane’s surface, but these improvements would only raise fuel efficiency by about 10%.

To quote David Mackay,

The only way to make a plane consume fuel more efficiently is to put it on the ground and stop it. Planes have been fantastically optimized, and there is no prospect of significant improvements in plane efficiency.

A 10% improvement? Yes, possible. A doubling of efficiency? I’d eat my complimentary socks.

References

I based this blog post on material I learnt from David MacKay’s fantastically clear book, Sustainable Energy without the Hot Air. It’s available online for free, and is highly recommended for anybody looking to use numbers to understand energy.

David MacKay (2009). Sustainable Energy – Without the Hot Air UIT Cambridge Ltd

I used this tip to make those XKCD style plots.

Whereas the rigid universe is notable for its strict adherence to a few basic principles, the squishy universe is a different beast altogether.

I was recently out paddling, and noticed that as you move the paddle through water, tiny whirlpools begin to develop along its sides. The whirlpools grow in size, become self-sustaining, and break off and float away. Eventually they die out, as they lose their energy to the fluid around them.

You could also watch the spirals and vortices created by rising smoke. Or notice the strange shapes made by the wind as it sweeps through the clouds. It’s as if fluids have a life of their own, often wondrous and beautiful, and other times surprising and counter-intuitive.

But the motion of fluids is notoriously hard to predict. It’s so difficult that if you can solve the equations of fluid flow, there are people willing to offer you a million dollars. The difficulty comes from a mathematical property of the equations known as non-linearity. Simply put, a non-linear system is one where a small change can lead to a large effect. The same thing that makes these equations difficult to solve is also what makes fluids surprising and interesting. It’s why the weather is so hard to predict – tiny changes in local temperatures and pressures can have a large effect.

At this point, most reasonable people would throw their arms up in despair. But physicists are an unreasonably persistent bunch, and when faced with an equation that they can’t solve, they try to get some insight by looking at what happens at extremes. For example, thick and syrupy fluids like glycerine behave in a surprisingly orderly fashion. Take a look at this video (watch through to the end, it’s worth it).

I bet you’ve never seen a fluid do that before. So what’s going on here? And what does this have to do with swimming sperm?

Let’s take a step back. Picture a flowing river. If there is an obstruction to the water’s path, like a rock jutting out of the surface, the water will move around it and swirl back upstream. Behind the rock, the water remains relatively calm. What you get is a spot on a moving river where the water is remarkably still. These calm spots are called eddies, and kayakers treat them as parking spots on the river.

But fluids don’t always behave like this. If you replace all the water in a river with a viscous fluid like glycerine, there won’t be any eddies. The syrup will simply follow the contours of the rock and smoothly flow around it.

In one case we have smooth, orderly flow, and in the other case we have eddies and turbulent flow. The question arises, is there any way to know what kind of flow will result in a given situation? This question was answered by the physicist Osborne Reynolds in 1883, and he answered it in style.

Here is Reynolds’ elegant experiment. He sent fluid flowing through a thin pipe (analogous to the river), and injected colored dye in a small section of the flow. He watched the dye flow down the tube, and could plainly see whether the flow was smooth or disorderly. By tweaking the parameters in this experiment, he was able to discover the conditions that ensure an orderly flow.

What he found is that there is one simple, magic number that can predict what is going to happen. It neatly ties together all the different physical quantities involved. It’s been named Reynolds number (Re for short), and is given by

These are all quantities that you can directly measure. The viscosity of a fluid is a measure of how slowly it flows. Thick and syrupy fluids like honey and corn syrup have a high viscosity, gases like air have a very low viscosity, and water is somewhere in between. The length in the above equation is a length that describes the object that you are studying (say the width of the rock). Reynolds used the diameter of the pipe. And the speed is that of the fluid.

The Reynolds number has the nice property of being dimensionless, meaning that the number is the same in whatever system of units you choose to measure the above quantities (dimension-full quantities are things like speed, which you could measure in km/h or mph). What Reynolds found is that as this number exceeds 2000, you suddenly get turbulent flow. In fact, this week’s issue of Science magazine mentions a new experiment that verifies this surprising result, and puts the turning point at Re = 2040. (The specifics of this number has to do with a fluid moving through a cylindrical tube with smooth walls. In a different situations, the number will change, but the principle is the same. There is a sudden jump from order to turbulence.)

The above figure gives you an idea of what happens as you increase Reynolds number. Here’s an analogy. The low Reynolds number world is like a collectivist ideal, where water moves along uniformly like soldiers marching in step. The high Reynolds number world is the individualist nightmare, where everyone looks out for themselves. Think of a march versus a mob.

We can arrive at this number from another route. There are two fundamentally different type of forces that act on an object immersed in a fluid. The first kind are inertial forces. This is like the push you give to the water when you take a stroke while swimming. Inertia is what allows water particles to keep moving undisturbed. On the other hand, you have viscous forces which measure the tendency for the fluid to smooth out any irregularities. To use the above analogy, inertial forces reflect the individuality of bits of fluid, and viscous forces are like a communist government enforcing conformity. And when you take the ratio of these forces, you get back the Reynolds number.

This number is of immense importance to aeronautical engineers and to biologists interested in locomotion.

Let’s say you want to simulate the effect of wind on a new wing design. You build a scale model in the lab that is one tenth the size of the actual wing.

But remember how the Reynolds number is defined.

If you shrink the size of the wing by a factor of 10, you have to increase the windspeed by the same amount in order to keep the number fixed. The key point is that systems with the same Reynolds number have essentially the same nature of flow. If you didn’t account for this, your wing would be quite a disaster.

How would a biologist use this idea? Well, nature presents us with organisms that cover an incredible range of sizes, from the tiniest microbes to the blue whales. Here is a table of Reynolds numbers across this range.

The list covers 14 orders of magnitude. A whale swims at a huge Reynolds number. This means that inertial forces completely dominate. If it flaps its tail once, it can coast ahead for an incredible distance. Bacteria live at the other extreme. In a delightful paper entitled Life at low Reynolds number, the physicist Edward Purcell calculated that if you a push a bacteria and then let go, it will coast for a distance equal to one tenth the diameter of a hydrogen atom before coming to a stop. And it will do this in 3 millionths of a second. Bacteria clearly inhabit a world where inertia is utterly irrelevant.

Eels and sperms may look similar, but their method of moving is very different, as their Reynolds numbers are far apart. In fact, we can now answer the question, what would it feel like to swim like a sperm or a bacteria? To do this, you have to somehow get down to their Reynolds number. We can’t change our size, but we can shrink our Reynolds number by swimming in a very viscous fluid. Purcell estimated that you would have to submerge yourself in a swimming pool full of molasses, and move your arms at the speed of the hands of a clock. (Don’t try this at home. Swimming in molasses is not a good idea.) Under these conditions, if you managed to cover a few meters in a few weeks, then you qualify as a low Reynolds number swimmer.

This clearly isn’t a hospitable environment for denizens of our Middle World. But yet this is the scale of the task that microbes face simply to get around.

Except, it’s even harder. Remember the youtube video of the colored dye swirling in the glycerine? The reason that the colors come back to where they start is because at low Reynolds number, flow is reversible. Because inertial forces are so small, certain terms drop out of the complicated fluid flow equations. The equations simplify considerably, and not only are they now solvable, they don’t depend on time any more. If you took the youtube video and played it backwards, you wouldn’t be able to tell the difference.

But this reversibility has a surprising consequence. It means that anything that swims using a repeating flapping motion can’t get anywhere. If it moves forward in one stroke, the other stroke will bring it right back to where it started. Scallops swim by opening their jaws and snapping it shut. In low Reynolds number, scallops can’t get anywhere.

Don’t believe me? See it for yourself. Here’s a rubber band powered toy that paddles forward when in water.

Woohoo! Look at it go. Now, take the same toy and place it in a vat of viscous corn syrup.

The reversibility of the flow ensures that the boat can’t make any progress.

So how, then, do microbes manage to get anywhere? Well, many don’t bother swimming at all, they just let the food drift to them. This is somewhat like a lazy cow that waits for the grass under its mouth to to grow back. But many microbes do swim, and they make use of remarkable adaptations to get around in an environment that is entirely alien to us.

One trick they can use is to deform the shape of their paddle. By cleverly contorting the paddle create more drag on the power stroke than on the recovery stroke, single cell organisms like paramecia break the symmetry of their stroke and thus elude the scallop conundrum. Indeed, this is how the flapping structures known as cilia thrust a cell forward: they flex.

There is an even more ingenious solution that has been hit upon by bacteria, sperm and other cells. Rather than having a cilia, which is essentially a flexible paddle, these cells adopt a different strategy: they use a corkscrew for a propeller. Just as a corkscrew used on a wine bottle converts winding motion into motion along its axis, these organisms spin their helical tails (flagellum) to push themselves forward.

But don’t expect to see human swimmers doing ‘the corkscrew’ anytime soon. This strategy works only at low Reynolds number, where water ‘feels’ as thick as cork, so you can push against it effectively.

And here’s proof. Whereas our rubber band powered stiff paddle couldn’t make any headway in the corn syrup, take a look at what happens if you instead have a helical propeller.

It winds its way into the fluid and inches forwards.

Motion in this viscous world is counter-intuitive and puzzling. By applying science, we can imagine what it must feel like to be very small. And we can work out how to build tiny ships in such a world. But evolution has beaten us to the punchline, and microorganisms have evolved intricate and wonderful structures that pulsate rhythmically and take advantage of the quirks of physics at this scale.

References

Purcell, E. (1977). Life at low Reynolds number American Journal of Physics, 45 (1) DOI: 10.1119/1.10903

Avila K, Moxey D, de Lozar A, Avila M, Barkley D, & Hof B (2011). The onset of turbulence in pipe flow. Science (New York, N.Y.), 333 (6039), 192-6 PMID: 21737736

Reynolds, O. (1883). An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels. Proceedings of the Royal Society of London, 35 (224-226), 84-99 DOI: 10.1098/rspl.1883.0018

In addition to the above papers, I learnt a lot about this subject from the following excellent book, from which many of the figures in this post are taken:
Life in moving fluids: the physical biology of flow by Steven Vogel (1996)

The theme of this post came from reading a following wonderful out-of-print book that I discovered in the basement of Strand bookstore in NYC:
On Size and Life (Scientific American Library) (1983)

Image Credits

Figures from the cited papers or from Life in moving fluids by Steven Vogel are attributed in place.

Slinky by Aaron Steele

Paddle Prints by deanspic

Cartoon of eddies was lifted from Whitewater kayaking: the ultimate guide by Ken Whiting & Kevin Varette

An airfoil wing in a wind tunnel courtesy NASA Langley Research Center

Cilia on a Paramecium courtesy Yellow Tang Moodle

Difference of beating pattern of flagellum and cilia courtesy Wikimedia Commons

Update: Added discussion on launch angle at the end of the post.

Edit: The final numbers in this post went through a few rounds of revision. What is the world coming to, when you have to track down missing factors of 2 in your blog posts?!

This week, I’m looking at the strategies and mechanisms by which different animals solve the problem of getting around. I started off by writing about how birds and aquatic animals conserve energy on-the-go. This post is another spinoff on the theme of locomotion.

Here’s a clip from one of my favorite documentaries, David Attenborough’s Life of Mammals. It shows the incredible sifaka lemur of Madagascar, a primate that has a really remarkable way of getting around. (If the embed doesn’t work, you can watch it here)

As they launch out from the trees, they almost look like they’re defying gravity. And so, taking inspiration from Dot Physics, I thought it might be interesting to put physics to use and analyze the flight of the sifaka.

I loaded the above video into Tracker, a handy open source video analysis software. I can then use Tracker to plot the motion of the sifaka. I chose to analyze the jump at about 21 seconds in. I like this shot because it isn’t in slow motion (that messes up the physics), the camera is perfectly still (we expect no less from Attenborough’s crew), and the lemur is leaping in the plane of the camera (there are no skewed perspective issues that would be a pain to deal with). The whole jump lasts under a second, but at 30 frames per second, there should be plenty of data points.

This is what it looks like when you track the sifaka’s motion:

The red dots are the position of the sifaka at every frame. That’s the data. In order to analyze it, we need to set a scale on the video. I drew this yellow line as a reference for 1 unit of size (call it 1 sifaka long). And how big is that?

If we believe this picture that I found on the National Geographic website, then a sifaka is about half the size of this folded arms dude.

Now, to the physics..

While the sifaka flies through the air, the only force acting on it is gravity, which points downwards. So the acceleration of the lemur should also be downwards. (I’m ignoring air resistance. We’ll find out if this is a good idea.)

If we plot its horizontal motion, it should be moving at a fixed speed, with no acceleration. But its vertical motion will give away its acceleration.

This is what we get if we plot at the horizontal position of all the points with respect to time.

The squares are the data points, and the line is a plot of the equation of a straight line

I was amazed by how well they agree, since I expected air resistance to matter a little more. I guess ignoring air resistance is a pretty good approximation.

We find that there’s a straight line relationship between position and time, which implies that the sifaka moves at a constant speed in the horizontal direction. The slope of this line () has units of meters/second (or in our case sifaka/second) and is the speed of the sifaka.

What about the vertical direction? Well, it certainly can’t be a straight line relationship with time, because at some point the sifaka turns and comes back down. Here is what the plot looks like:

The small squares are the vertical positions of the dots plotted versus time, and the red curve is the plot of an equation for a parabola

Here is the vertical launch speed, is acceleration, and is time.

So, over time, the vertical position traces out a parabola, which is a characteristic shape for motion under a fixed acceleration (in this case, the earth is accelerating the lemur downwards). The nice thing about analyzing motion is that we can analyze the horizontal and vertical motion independently of each other.

The fit to the parabola is not great, but it’s not too shabby either. I suspect the main reason for the discrepancy is that its hard to track the center of mass of the sifaka, and if you choose any other place on the sifaka, you’ll also be tracking the spin of the sifaka about its center of mass.

By solving for the values of , and that best match the data, we get the launch speed and acceleration of the lemur.

To be a little more empirical about things, I did this analysis twice, and averaged the results. Here’s what I got:

Horizontal launch speed:
Vertical launch speed:
Vertical acceleration:

The negative sign on the acceleration indicates that gravity is pulling the sifaka downwards  (in the negative y direction). So far things look good qualitatively, but do the numbers work out?

Well, according to National Geographic, the tail of a sifaka monkey is 46 cm, whereas according to wikipedia it is 50 to 60 cm. Let’s go with 50 cm on average. The length scale I drew in Tracker is about the length of the Sifaka’s tail. So we can set 1 sifaka = 0.5 meters.

That gives us a value of for the acceleration caused by gravity, which is within 16% of the known result of . I think that’s pretty darn good for a first stab at video analysis, especially as the sifaka was a blur in each frame and often obscured by trees.

Next, we can use Pythagoras’ theorem in the above velocity triangle to solve for the total launch speed

where and are the horizontal and vertical components of velocity.

This gives a launch speed of 4.25 meters per second or 9.5 miles per hour (15.3 km/h). This speed sounds reasonable to me, as it’s about how fast your typical bicycle moves. If we include a fudge factor that fixes our acceleration to the known result, then the launch speed is actually faster by 16%.

Update: added discussion on launch angle.

We can also solve for the launch angle of the sifaka, by using some high-school trigonometry on the triangle:

Solving for the angle gives 34.7 degrees.

Is this angle correct? Fortunately, Tracker has a handy built in protractor, so we can check it. Marking out the initial leap for both runs, I get an average launch angle of 34.5 degrees.

Which agrees to within half a percent of our result inferred from the physics!! Eerily accurate..

It’s a bit of a coincidence that the result is as close as it is, given the many possible sources of error. However, one reason why this result is so accurate is that the angle comes from a ratio , and so common sources of error (such as error in estimating the length of a sifaka) end up cancelling out. This is also why physicists prefer to measure ratios, rather than numbers that have units (they call such quantities dimensionless).

And there you have it folks, SCIENCE being put to use to answer the burning questions that keep you up at night.

If you want to read more about how the sifakas glide, Darren Naish has a detailed post describing research on the physics of this.

The story is about the nature of light and matter. And it is driven by a fervent battle of ideas between some of the greatest minds in science. It begins at the turn of the eighteenth century.

By then, Isaac Newton had already made a name for himself as the biggest badass in science. He invented calculus (edit: although the origins of calculus are somewhat mired in controversy), devised the law of gravity and formulated the laws that govern how things move. That’s pretty eventful for a few decades (in fact, he did much of this work in a single year), and it’s almost inhuman that all this came from a single person.

And things were just getting started. By the turn of the century, Newton had turned his considerable attention towards the problem of light. How does it work? What is it made of? Using a series of simple, methodical experiments, he argued that if you stripped light down to its tiniest constituents, you would end up with particles that he called corpuscles. This idea was widely adopted, and became the mainstream scientific opinion for over a hundred years.

There were always doubters to this idea, but they weren’t many of them, and they weren’t popular. It was another brilliant English scientist, Thomas Young, who would take the next step in understanding light.

Young was quite the Renaissance man. In addition to being a physicist, he made significant contributions to fields as diverse as music, language (he compared the vocabulary and grammar of 400 different languages), Egyptology (he partly deciphered Egyptian hieroglyphics from the Rosetta stone) and the physiology of vision.

But what Young considered his greatest achievement (and he had a few) was overthrowing Newton’s century-old notions of light. In its place, he argued that light was not made up of particles, but was instead a wave, quite like the ripples on the surface of water.

At first, he met with huge resistance to his ideas. But in 1803, Young convinced his skeptics with a simple, game-changing experiment.

Here is how it works. Imagine light is composed of tiny particles, like droplets of water, that are being spit out from a lamp. If this light falls on a barrier that has two thin slits sliced into it, it will shine through these slits. If this ‘garden hose’ theory of light is correct, you would expect to see something like the following picture.

What you see here is the drops passing through the slits and striking the wall in basically two places. At the right is a plot of how many drops hits the wall, and there are two piles of water drops that are directly behind the two slits.

Now imagine the same experimental setup, but instead of spraying droplets, we fill the room ankle-deep in water. Things now look something like this:

At one end, you start sending out ripples in the water. Just like when you drop a pebble into a pond, these ripples move rhythmically outwards in circles, peak followed by trough followed by peak.. and so on. When the ripple hits the barrier, you now have two ripples coming out, one from each slit.

And as these ripples start moving towards the screen on the right, something entirely new happens – they interfere. That is, there are places on the screen where the crest of one wave hits the trough of another, and the two waves cancel each other out. And there are other places where two peaks or the two troughs line up – the waves reinforce. If you were to look along the screen, you would find places where the water stays completely still, next to places where it splashes wildly.

If you wanted, you could do the same experiment with sound. Instead of slits, you could have two speakers. Sound is a wave, it’s a vibration in the air that wiggles our eardrums. As you move your ear along the screen, you would find places where the two sound waves reinforce, and you would hear a louder sound. And there would be also places where you would hear nothing as the sound waves cancel each other out (no vibration). The overall picture you get is this striped interference pattern shown above, of alternating highs and lows.

So Young performed this experiment with light. To everyone’s surprise (but his), he found that light doesn’t act like the bullets of a machine gun. What he saw on the screen was an interference pattern – alternating bands of light and dark. The interpretation was unambiguous – light behaves like a wave, not like a bunch of particles.

And for the next century, the wave theory reigned supreme, until no less a figure than Albert Einstein came onto the scene. In his amazing year 1905, Einstein explained a famous experiment – the photoelectric effect – by invoking the idea that light is made of particles that carry energy. He would later win the Nobel Prize for this achievement. Somewhat embarrassed by Newton’s corpuscles, physicists rebranded these particles with a new name – photons.

And soon after, engineers were building devices that could make noises whenever they detected light. Rather than hearing some kind of continuous splish-splosh that you may expect from a wave, they would hear a sound like individual raindrops – tick, tick, tick. Each of those ticks was a photon striking the detector.

Now, if you’re with me so far, this is a point where you can stop and scratch your head. On the one hand, Young proved that light is a wave. But then you have Einstein and these detectors. They’re practically screaming in our ears that light is a particle. So what’s really going on here?

This is the dilemma that gave rise to quantum mechanics – depending on what experiment you do, light seems to behave like a wave, or like a particle. It turns out, as physicists later discovered, that this is true for any kind of stuff, not just light. If you take atoms or electrons and send them through the double slit, they’ll behave like the interfering ripples, not like water drops.

My teachers made a big deal about this in high school, and I used to always wonder what the fuss is all about. After all, water ripples are waves, but water is made of particles (H20 molecules). No one talks about the wave-particle duality of water. What’s different about these subatomic particles?

Well, I’m going to tell you now what my school teachers never taught me, but I learned from popular science books instead. Let’s say you’re a rogue physicist, and you think that everyone’s having you on with all this wave-particle mumbo jumbo. Here’s a simple experiment you can perform to prove them wrong. What if you repeat the double slit experiment, with one small change – this time, you send stuff in one particle at a time. Before you read on, take a minute to think about what you would expect to see on the screen.

If you haven’t come across this before, then your intuition is probably telling you that the particle couldn’t possible interfere, because there aren’t any other particles around to bump into.

Now, here’s what actually happens. Remember, at any given time there is only one particle in the picture. (the video you’re seeing was done for electrons, but the same thing happens for light)

How wild is that? Our rogue physicist’s idea fell flat. Even though you send in the electrons one at a time, they still manage to interfere. If they behaved like billiard balls, they would be going through one slit or the other, in which case you would just see two bright bands that are behind the slits. Instead they produce these alternating dark and light bands. How could this be?

“Aha!”, says the rogue physicist. “I’ve figured out what must be happening. The reason I was thwarted is because I must have been wrong about the electron in the first place.”

“Maybe what’s really happening is that the electron somehow splits into two pieces, goes through both slits, and then interferes with itself.” And so the physicist hatches a plan to vindicate their idea.. by looking to see which slit the electron really goes through.

And here comes the mind-bogglingly bizarre thing about quantum mechanics. Let’s say you shine light on the electron to see which slit it goes through. What you see is that it always goes through one slit or the other, nothing weird is going on. But, once you do this, the interference pattern disappears and instead you get just the two bands!

If you detect which slit the electron goes through, then it behaves like a plain-old billiard ball. But when you turn off the light, and can’t tell which door the electron goes through, it interferes and you get the pattern of light and dark fringes. So, just by looking at the particles, we are changing the outcome of the experiment.

There is no picture that you can hold in your head that will explain these results. It’s not like a billiard ball with some strange gears and clockwork inside it. It’s something fundamentally different.

But they have gotten used to the idea that measurements matter. In the quantum world, a measurement fundamentally alters the thing that you are studying.

Rather than finding this strangeness unsatisfying, I find it incredibly exciting. In a way, I feel that we’ve been pretty lucky so far. Our brains have evolved to understand the African jungles and savannah in which we grew up. Everything that makes sense to us is not too alien from this environment. But when we start to push our understanding to this sub-atomic level, there is no good reason for our macroscopic intuition to apply here. Like Alice’s looking glass, these two slits are our door into a strange new world.

And if you thought things couldn’t get any weirder, you’re wrong. Here’s the clincher. Let’s say you repeat the double slit experiment, but this time you put a detector only behind one slit. So I know if the electron goes through slit B, but not if it goes through slit A. What would you expect to happen now?

What happens is not that you get half an interference pattern. Instead, this is no different from the previous experiment with two detectors. The electrons pile up behind the slits.

Think about what this means. If you are still clinging in your head to a picture of an electron as a ball flying through one of the slits, then you are forced to conclude that the electron going through slit A somehow knew about the detector in slit B, and decided not to interfere. This is absurd (even by quantum standards) and many a crackpot have travelled down this road. Instead, what we should learn from this experiment is that our classical picture is fundamentally broken. The electron does not behave like a ball, or anything else we can readily imagine.

Well, then, what the hell is it? To find out, tune in to part 2 of a tale of two slits.

That’s it for now. I’ll let this madness settle in. In part 2 of this post, I will explain the rules of the new game in town: quantum mechanics. I’ll revisit the double slit experiment, but this time I’ll describe the elegant methods that physicists use to make predictions in the quantum world. And I’ll comment on an interesting new experiment that revisits the double slit.

References:

If you want to understand how quantum mechanics works without getting into the mathematical details, you could do no better than read Feynman on the subject. His book QED: The Strange Theory of Light and Matter has my vote for one of the finest popular science books of all time.

Richard P. Feynman (1988). QED: The Strange Theory of Light and Matter Princeton University Press ISBN-13: 978-0691024172

Young, T. (1804). The Bakerian Lecture: Experiments and Calculations Relative to Physical Optics Philosophical Transactions of the Royal Society of London, 94, 1-16 DOI: 10.1098/rstl.1804.0001

The experiment that I mentioned with a detector behind just one slit is a special case of the Quantum Eraser experimental setup.

Here’s the writeup by Hitachi that goes along with the video embedded above.

After being very pleased with myself for coming up with  ‘a tale of two slits’, I googled it to find that it had already been coined. However, my mild annoyance quickly turned to delight as I started reading Ethan Siegel’s excellent and entertaining blog.

Edit: Richard Dawkins has a way with words. In this fascinating TED talk from 2005 (one of the first talks on TED.com), he speaks eloquently about the idea that our brains have evolved to find the universe intuitive at a given middle-sized scale. He calls this our Middle World, and we can step out of middle world by looking at things that are at a different scales in size and speed.

Footnotes:

I remember that the double slit experiment, one photon at a time, is the first experiment I ever did in college. It was my first semester, and at the time, I didn’t expect it to work. It just seemed too far out to be true. Sitting in that dark room watching the interference pattern slowly build up on the computer screen essentially shook the classical physics out of me.

In a paper published in the journal PNAS in February, the authors demonstrate through a series of ingenious experiments that smell can be sensitive enough to pick up on tiny differences in atomic vibrations.

The conventional theory of smell works somewhat like a lock and a key. The molecules are the key, and they ‘lock in’ to receptors that fit their exact shape and size. This is the shape theory of smell, and the basic idea had been suggested in the 1st century BCE by the Epicurean philosopher Lucretius. The idea has since garnered substantial evidence with the discovery of odor receptors, leading to the 2004 Nobel Prize in Medicine for working out the overall picture of how smell works.

An alternative hypothesis is the vibration theory. This proposes that smell works not by detecting the shape of molecules, but by measuring how the atoms in a molecule are vibrating.

Molecules are groups of atoms that are held together by chemical bonds. These bonds are somewhat elastic, causing the atoms in the molecules to constantly jiggle about. This is analogous to what would happen if you were to connect balls together with springs (something that physicists love to do). But the analogy breaks down at this microscopic scale, and one needs to resort to the laws of quantum mechanics to understand what is happening. It turns out that, similar to the balls and springs, molecules have certain ways in which they prefer to jiggle. They can stretch, rock, wag and twist around.

So, which is it? Does smell work via shape or vibration? The authors set out to address this question with flies.

The best way to distinguish the two theories would be to find two chemicals that are identical in shape, but vibrate in different ways. This is exactly what the experimenters did, by taking a molecule, and replacing some of its hydrogen atoms with deuterium. Deuterium is a sort of heavier sibling of hydrogen that behaves very similarly, but is about twice as heavy. And a heavy atom is harder to wiggle – so the molecule and its counterpart will now vibrate at different rates, but their shape and size will remain the same.

In their experiments, flies were sent down a T shaped corridor. At the junction, they were presented with different odors from their left and right. If they could not distinguish between the odors, you would expect no more flies going left than right – it would just be based on chance.

In the first experiment, the authors presented the flies with acetophenone at one exit. Acetophenone is a colorless sweet smelling liquid with a fairly simple molecular structure. It’s the stuff that’s added to give that cherry or strawberry smell to chewing gum. The other exit just had plain old air. They counted how many flies went through each exit.

They the repeated this experiment with the ‘deuterized’ versions of acetophenone – same shape, different vibrations. If the ability to smell relies only on the shape of the molecule, and not on the vibrations, then one would expect nothing to change.

Instead, here is what they saw.

In figure A, the bars show the percentage of flies that chose that chose one exit over the other. The flies were initially attracted to the acetophenone (ACP). However, as the researchers replaced more and more of the hydrogen atoms with deuterium in this molecule (3, 5 or all 8 hydrogen atoms), they found that the instead of being attracted by the scent, the flies were repelled by it.

If the flies were presented with acetophenone at one exit, and its heavier counterpart on the other, they strongly preferred the former scent (first bar in figure B). This implies that they can distinguish between odors whose molecules differ in vibration but are identical in shape! They repeated this experiment with two different chemicals (octanol and benzaldehyde) to ensure the results were robust.

The experimenters then systematically went on to rule out alternate explanations for their results.

First, is this just about scent? Could the deuterium be affecting the flies in some other way altogether, one that has nothing to do with odor? To answer this, they repeated the experiment with genetic mutant flies who lacked a crucial part of their odor receptors. These flies couldn’t smell, and neither did they have a preference between acetophenone or it’s heavier counterpart. So, this is all about the smell.

Now, it could still be possible that perhaps some impurity crept into all the deuterium version. To rule out this explanation, the authors conducted a beautiful set of experiments. The set up is the same as before, acetophenone on one side, and the heavy version on the other. But now, they zapped the flies with an electric shock whenever they made a particular choice – say, for choosing the heavy molecule. In this way, they could condition the flies to reliably prefer either of the two compounds.

What they did next is quite ingenious. They took the flies trained on acetophenone (ACP), and put them back in the T shaped corridor. Only this time, they were distinguishing between versions of a different chemical – the heavy and regular versions of benzaldehyde (BZA). In other words, these flies were trained with the scent of strawberry, and were now faced with the smell of bitter almond – completely unrelated molecules that were synthesized differently. Here is what happened:

The flies could apply their lessons from one scent to the other. When zapped on regular ACP, they avoided regular BZA, and went zapped on heavy ACP, they avoided heavy BZA. (The experiments were repeated across 3 pairs of molecules to ensure robustness). Since these chemicals are synthesized in entirely different ways, this means that it’s not the impurities, but that the flies can somehow sniff out the ‘deuterium-ness’ of a molecule.

But this raises another question. Are the flies somehow sniffing out the deuterium, or is it the vibrations? In other words, could the deuterium be causing some subtle non-vibrational change in the chemistry that the flies can detect?

In order to answer this, the authors made a clear prediction. Using computer simulations, they showed that when you replace hydrogen by deuterium, the vibrations of ACP change in one essential way – a particular kind of wiggle of the atoms known as a stretch is slowed down. (demonstrated below with hydrogen in blue)

They then identified a pair of chemicals that differed in this precise way. It’s like finding two different sets of musical chords, each set differing in just one note. The pair they found had a citrus lemongrass smell, and had no atoms of deuterium. The flies had no preference for either chemical in the pair.

Based on their theory – that it is the vibrations, not the presence of deuterium, that is being used by the flies to discern smells – the authors made the following prediction. Flies that are trained to differentiate between the deuterium based pair should also distinguish between this new pair of chemicals that have the same difference in vibrations.

And they were right! The preferences the flies had been conditioned with were generalized to this chemically novel setting. So the odor receptors of flies are essentially incorporating a biological version of a spectrograph – an instrument that can tune in to vibrations at different frequencies.

A working model of this ‘tuning in’ might work was first proposed in 1996 by Luca Turin, one of the authors of the current paper. His idea relies on a bizarre but well understood feature of the quantum world, where a subatomic particle like an electron can ‘tunnel’ through a solid barrier. Turin’s theory has remained fairly controversial and is not adopted by the community at large, but his idea has since been checked by physicists and shown to be a consistent, workable model.

I find this work remarkable for a number of reasons. First, it’s science at it’s best – the authors address a fascinating and fundamental question through clear, cleverly designed and simple to understand experiments. Secondly, if Turin’s model is correct, then it is incredible to imagine that natural selection has driven this system to such extreme precision that it is making use of atomic physics. And finally, this work is a great example of what goes by the awful name of interdisciplinary research. This study would not have been possible had the authors not possessed a thorough understanding of biology, chemistry and physics. Such research bridges the arbitrary distinctions between departments, and focuses on what is truly exciting – nature herself.

References:
Franco MI, Turin L, Mershin A, & Skoulakis EM (2011). Molecular vibration-sensing component in Drosophila melanogaster olfaction. Proceedings of the National Academy of Sciences of the United States of America, 108 (9), 3797-802 PMID: 21321219 Link