## The Universal laws behind growth patterns, or what Tetris can teach us about coffee stains

The morning after a big snowstorm swept through the US northeast, I sat in my car, ready to brave hazardous road conditions and drive to the local coffee shop. My home in New Jersey was outside of the storm’s central path, so instead of piles of snow, we were greeted with a delightful wintry mix of sleet and freezing rain. And sitting in my car, I couldn’t help but be mesmerized by these strange patterns of ice particles forming on my windshield. Here’s what I saw:

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? Continue reading The Universal laws behind growth patterns, or what Tetris can teach us about coffee stains

## Can we build a more efficient airplane? Not really, says physics.

Update (13 October):  I emailed David MacKay to get his opinion on some of the critical comments responding to this blog post. David is a physicist at Cambridge University, author of the book ‘Sustainable Energy – Without the Hot Air’, and is the chief scientific adviser to the UK Department of Energy and Climate Change. You can read his response in the comments below. There’s also a interesting discussion of this post over at hacker news.

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 $1/2 m v^2$ to bring stuff with mass $m$ up to a speed $v$.

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 so our model teaches us that flying machines should never have speed limits. Whether made of metal or meat, every plane has an ideal speed. If you stray from this value, you have to pay for it in fuel cost. Slowing a car down may improve your mileage, but for a plane, the mileage actually gets worse.

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.

## What it feels like for a sperm, or how to get around when you are really, really small

We don’t usually learn about the physics of squishy things. Physics textbooks are filled with solid objects such as incompressible blocks, inclined planes and inelastic strings. This is the rigid world that obeys Newton’s laws of motion. Here, squishiness is an exception and drag is routinely ignored. The only elastic object around is a spring, and it is perfectly elastic. It will never bend too far and lose its shape. But any child who has played vigorously with a Slinky has stretched past the limits of this Newtonian world.

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?

## Launch speed of the leaping sifaka

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..

## Why a quantum particle is not like a water drop. A tale of two slits, part 1

I want to describe a certain beautiful experiment, perhaps the most beautiful experiment in science. This is an experiment that has captivated me from the time that I first heard about it in high school. That’s because it’s simple to understand, and yet it captures the essence of what is truly messed up about quantum mechanics. This is a tale of two slits. And it would be no exaggeration to say that through these slits, we encounter a word that is so strange, it is beyond our human capacity to imagine.

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.

## Using flies to sniff out a new theory of smell

Our sense of smell is really quite incredible. Every time we take in a breath or taste food, countless molecules swarm into our nasal passages. As they move up the nasal tract, these visitors arrive at a patch of cells on which there are over 10,000 different kinds of docking stations. These cells are odor receptors, and each of them can register a different odor. Together they make up a chemical detector that is much more sensitive and versatile that anything we can come close to building.

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.