Everyone loves the Eiffel Tower. Its classic, iconic shape is an instantly recognizable symbol of Paris. So you might be surprised to learn that while the tower was being built, art critics were not quite as glowing in their praise. Here are some of the more colorful phrases they used to describe it.

“this truly tragic street lamp” (Léon Bloy)

“this belfry skeleton” (Paul Verlaine)

“this mast of iron gymnasium apparatus, incomplete, confused and deformed” (François Coppée)

“this giant ungainly skeleton upon a base that looks built to carry a colossal monument of Cyclops, but which just peters out into a ridiculous thin shape like a factory chimney” (Maupassant)

“a half-built factory pipe, a carcass waiting to be fleshed out with freestone or brick, a funnel-shaped grill, a hole-riddled suppository” (Joris-Karl Huysmans)”

To modern eyes, the tower’s shape is elegant and graceful, perhaps even timeless. But to contemporary critics it was a monstrosity. The tower represented a new kind of aesthetic, and it took people a while to appreciate this. Eiffel was going after a deeper kind of beauty, a kind that wasn’t just skin deep. His notion of beauty had to do with economy and structural efficiency, with achieving the greatest strength with the least possible material. It had to do with seeing pure, efficient, well-engineered structures as works of art.

Hidden Rules of Harmony

Here’s Eiffel describing his new aesthetic, in response to his critics.

“Are we to believe that because one is an engineer, one is not preoccupied by beauty in one’s constructions, or that one does not seek to create elegance as well as solidity and durability? Is it not true that the very conditions which give strength also conform to the hidden rules of harmony? [..] there is an attraction in the colossal, and a singular delight to which ordinary theories of art are scarcely applicable.”

The Eiffel tower is incredibly well optimized to do what it was designed to do, to stand tall and stand strong, while using a minimum of material. Rather than hide its inner workings with a facade, Eiffel exposed the skeleton of his masterpiece. In doing so, he revealed its “hidden rules of harmony”, many of the same rules that give your skeleton its lightweight strength.

To understand Eiffel’s ingenious design, let’s start with a little puzzle. Imagine that someone melted all of the iron in the tower into a solid ball. How big do you think that ball would be?

Each of the balls shown in the image are drawn to scale, next to their diameters.

Before reading any further, take a moment to guess your answer.

The correct choice is D (here’s the math to prove it). If you melted all the iron in the Eiffel Tower into a ball, it would be just 12 meters (less than 40 feet) in diameter. The tower’s immense height (324 meters, or over a thousand feet) belies the fact that it’s incredibly light for its size. To see it another way, if you were to melt the Eiffel Tower’s iron into a rectangular block as big as its base, then that block of iron would be only 6 centimeters (2.4 inches) tall. It wouldn’t even be visible in the image above.

One last way to picture the Eiffel Tower’s lightness. Imagine the smallest cylinder that completely wraps around the Eiffel Tower.

Now think about this. The air in this tube outweighs all the iron in the tower. (Don’t take my word for it, here’s the math.)

So how did Eiffel design a structure that’s strong enough to withstand the elements, and yet weighs about as much as the air surrounding it?

The secret lies in understanding the shapes of strength. It’s a lesson we can learn by looking inwards… literally. By studying our bones, we can discover some of the same principles that Eiffel used in designing his tower.

Shapes Within Shapes Within Shapes…

If you slice a bone open, you’ll find that it’s kind of like a baguette – hard on the outside and spongy on the inside. The outer bone material is hard and compact. This compact bone does most of the heavy lifting for the bone. On the interior is a spongier bone material. This spongy bone also plays an important role in carrying the pushing and pulling forces that our bones constantly endure.

Now let’s zoom into the crust of that bone baguette – the compact bone. It’s made up of tiny tubes called osteons, each just 2 tenths of a millimeter across, with a blood vessel running down the middle. Zooming further into the walls of these osteons, we find that they’re made out of tinier bundles called fibrils. Zoom further still, into one of these fibrils, and we see that they’re really a bundle of fibers, and each fiber is really three interwoven strands. Pull these strands apart, and we’ve unweaved our bones into its most fundamental unit, a long chain-like molecule called collagen.

This fractalesque way of putting things together, building with materials that are self-similar as you keep zooming in, is known as structural hierarchy. And it’s this structural hierarchy – tubes within tubes within tubes within tubes – that gives our bones their lightweight strength. (The spongy bone also has a fractalesque, self-similar design. If you look at a piece of it under an electron microscope, you’ll find that it looks just as spongy.)

Bamboo exploits the same idea. This ultra-fast growing grass needs a way to minimize material and stay very light, so it can grow tall and not collapse under its own weight. Bamboo’s hollow tube shape is a very efficient way to create stiffness. And like bone, bamboo is made out of tinier tubes, which in turn are made out of bundles of fibers, that are each made of out even smaller bundles of fibers, and so on. When you unweave a bamboo down to its tiniest thread, at the scale of a nanometer, you arrive at another long chain-like molecule – cellulose.

Bamboo and bone are both natural nano-engineered materials that use structural hierarchy to achieve their lightness and strength. The Eiffel Tower uses a similar idea. Eiffel borrowed this notion from bamboo and bone (although he probably arrived at it independently), and put it to use on a colossal scale.

Like many modern structures, the Eiffel Tower uses an arrangement of criss-crossing ‘X-shaped’ beams known as a truss. This is a very efficient way to engineer structures by relying on the inherent strength and stability of triangles. If you zoom into one of the Eiffel Tower’s trusses, you’ll find that they aren’t as solid as they seem – each of them are made up out of smaller, similar trusses. The material has more holes than it has iron. This hollow form contributes to the tower’s mind-boggling lightness. The next time you go over a bridge, look carefully, and you’re likely to see the same idea at play.

Shaped by the Wind

Once you’ve figured out how to build a lightweight tower, how do you ensure that it stays standing? The Eiffel Tower has to contend not just with gravity but with the considerable toppling force of the wind. To counter this, its sloping curve closely follows the most efficient shape for resisting the wind.

The trick to building a well engineered structure lies in transferring the forces from where you don’t want them to act to where you want them to act. Eiffel understood this. The shape of his tower has the special property that the combined force of the wind and the tower’s own weight will flow down the legs of the tower, all the way down to the strong foundations. (In physics terms, the tower has just the right shape so that the torque, or toppling tendency, generated by the wind is balanced by the torque due to its own weight.)

In the interview where he responds to his art critics, Eiffel describes this idea.

“Now to what phenomenon did I have to give primary concern in designing the Tower? It was wind resistance. Well then! I hold that the curvature of the monument’s four outer edges, which is as mathematical calculation dictated it should be […] will give a great impression of strength and beauty [..]”

That’s My Crane!

By understanding how forces flow, Eiffel’s engineers could design an optimal structure, putting stuff only where it’s needed, and leaving it out where it isn’t. The method that they used to visualize the flow of forces has a curious connection with the science of bones. It’s described in D’Arcy Thompson’s On Growth and Form, a delightful and insight-packed 1000+ page treatise on the mathematical laws that govern biology, published in 1917.

“A great engineer, Professor Culmann of Zürich, to whom by the way we owe the whole modern method of “graphic statics,” happened (in the year 1866) to come into his colleague Meyer’s dissecting-room, where the anatomist was contemplating the section of a bone. The engineer, who had been busy designing a new and powerful crane, saw in a moment that the arrangement of the bony trabeculae [spongy bone] was nothing more nor less than a diagram of the lines of stress, or directions of tension and compression, in the loaded structure: in short, that Nature was strengthening the bone in precisely the manner and direction in which strength was required; and he is said to have cried out, “That’s my crane!””

When an engineer looks at a structure, she looks beyond the material and sees the forces that act on it – it’s a bit like owning a pair of X-ray goggles. These forces come in two types – pushing forces that squeeze an object inwards, and pulling forces that stretch an object outwards. Every physical object that you encounter, from a table or a chair, to a bridge or a skyscraper, is basically a big party of these pulling and pushing forces (or as engineers refer to them, tension and compression forces).

So when Culmann was designing his crane, he was using his newly devised method of ‘graphic statics’ to map out these push and pull forces. And this is what he drew.

On the left is a drawing of the push and pull forces in the crane he was studying. And on the right is a similar drawing of push and pull forces in the top of the thigh bone (the femur). These images, adapted from Culmann and Wolff’s publication in 1870, represent the first collaboration between an engineer and an anatomist.

So when Culmann saw the pattern of the spongy bone in the top of the thigh bone, it reminded him of his crane. He was immediately struck by how clearly he could see the criss-crossing lines of forces in the bone.

The spongy interior of your thigh bone is efficiently arranged so that the material is present where the forces are the greatest, and absent where there aren’t any forces. In bone, this process occurs gradually over its development. The spongy bone hardens and aligns in directions where it experiences the greatest force, and atrophies in places where it isn’t used. There’s an analogy here to how those impressive sandstone arches are carved by the wind. The wind carves away places where the stone is least stressed, leaving in place a three dimensional outline of the lines of force, where the stone is most densely compacted.

In recent years, the mathematical exactness of this relation between bone and force has been called into question. But the general principle, that bone adapts to its functional demands, and that bone structure corresponds to the forces it experiences, is still widely accepted.

What does this have to do with Eiffel? Well, Culmann’s approach of graphically representing the push and pull forces was a powerful new tool, one that’s still used today. One of Culmann’s students, Maurice Koechlin, worked for Eiffel. And it was Koechlin who sketched the original concept of the Eiffel Tower, drawing from his training in visualizing forces. The same tools that Culmann developed and used to understand bone were later used by Eiffel’s engineers to design a tower that minimizes the use of material.

So while the critics who called Eiffel’s tower a skeleton meant it as insult, it’s actually quite the compliment. When it comes to engineering, we still have a lot that we can learn from our bones.

References

On the shape of the Eiffel Tower

Sundaram, M. Meenakshi, and G. K. Ananthasuresh. “Gustave Eiffel and his optimal structures.” Resonance 14.9 (2009): 849-865.

Gallant, Joseph. “The shape of the Eiffel Tower.” American Journal of Physics 70.2 (2002): 160-162.

Weidman, Patrick, and Iosif Pinelis. “Model equations for the Eiffel Tower profile: historical perspective and new results.” Comptes Rendus Mecanique332.7 (2004): 571-584.

On Structural Hierarchy

Lakes, Roderic. “Materials with structural hierarchy.” Nature 361.6412 (1993): 511-515.

Wegst, Ulrike GK, et al. “Bioinspired structural materials.” Nature materials (2014).

Rayneau-Kirkhope, Daniel, Yong Mao, and Robert Farr. “Ultralight fractal structures from hollow tubes.” Physical review letters 109.20 (2012): 204301.

Aizenberg, Joanna, et al. “Skeleton of Euplectella sp.: structural hierarchy from the nanoscale to the macroscale.” Science 309.5732 (2005): 275-278.

On the Forces in the Femur

Thompson, Darcy Wentworth. “On growth and form.” On growth and form.(1942).

Gray’s Anatomy on the Femur

Ruff, Christopher, Brigitte Holt, and Erik Trinkaus. “Who’s afraid of the big bad Wolff?:“Wolff’s law” and bone functional adaptation.” American journal of physical anthropology 129.4 (2006): 484-498.

Phillips, Andrew. “Structural optimisation: biomechanics of the femur.” arXiv preprint arXiv:1110.1286 (2011).

On the link between Culmann and the Eiffel Tower

Skedros, John G., and Richard A. Brand. “Biographical sketch: Georg Hermann von Meyer (1815–1892).” Clinical Orthopaedics and Related Research 469.11 (2011): 3072-3076.

As we celebrate the Earth completing another lap around the Sun, let’s take a moment to imagine what life would be like in a world without years – a world that somehow ceased to orbit its star. Admittedly, it’s a strange question, but its’s one that I’ve been obsessively wondering about lately. Not because it’s of any particular relevance, but simply because it’s amusing (at least to me) and fun to think about.

What would happen to us if a giant space finger were to gently stop the Earth in its orbit?

Nothing good.

Here, try it out for yourself. Press ‘start’ in the simulation below (created by Michael Dubson and the folks at Phet Interactive Simulations / University of Colorado). You should see a planet orbiting the Sun.

Now press ‘reset’, and drag the circle with the letter ‘v’ to shrink the planet’s speed . Then press ‘start’ again. What happens? (While you’re playing with this, you might enjoy trying out some of the different scenarios in the drop-down menus, and watching the gravitational ballet that ensues.)

If you slowed down the planet enough, you should see it crash into the Sun.

To see why, let’s first remember why things stay in orbit. Every child looking at the sky has at some point wondered, “why doesn’t the moon fall down?” The answer is beautifully simple, yet it took a mind as brilliant as Isaac Newton’s to work it out. (Perhaps a sign of genius is coming up with simple answers to children’s questions.)

Newtons’ response to the child’s question would have been – the moon does fall. It’s constantly falling. Being in orbit is a state of always falling, and always missing what you’re falling towards. In The Hitchiker’s Guide to the Galaxy, Douglas Adams describes the secret to flight. “The knack”, he writes, “lies in learning how to throw yourself at the ground and miss”. As it turns out, this is also a great description of what it means to orbit something.

Here’s how Newton explained it. Imagine a cannonball is fired from a height. If you fire the cannonball with more speed, it’ll travel further before it hits the ground. The faster the cannonball, the further it travels.

But wait – the Earth is round. That means that if you shoot the cannonball with enough speed, then by the time it would have hit the ground, it’s travelled far enough that the ground has curved away beneath it. So the cannonball continues to fall towards the ground, and the ground continues to curve away from it. It’s now in a state of perpetual free fall – the cannonball is in orbit!

(Newton’s idea is masterfully explained in this wonderful Radiolab segment.)

So the only thing that makes an orbit different from plain-old falling is having enough speed to miss the thing you’re falling towards. Think dropping a cannonball with zero speed versus shooting it into orbit. And for the same reason, if the Earth were robbed of all of its orbital speed, it would fall straight into the Sun. It would no longer have the speed it needs to miss the Sun.

How long would this ‘Earthfall’ take?

(If you remember some high school physics and want to work out the answer for yourself, here’s a hint for solving it without any calculus.)

I’ll skip the math, but it turns out that we’d have 64 and a half days before we plunged into the fiery depths of the center of the Sun. But don’t worry, we’d be quite dead before that happens.

As the Earth falls towards the Sun, it picks up speed. The further it falls, the more intense the sunlight, and so Earth  starts to heat up.

Here’s a plot of the Earth’s average temperature over these 64 and a half days.

If we zoom out, we see that most of the action happens in Earth’s last day.

You can see that things are going to get pretty uncomfortable fairly soon.

Let’s take this flight of fancy a step further, and imagine what things would be like on Earth as it descends into the Sun. What follows is my attempt at a science-based play-by-play of Earth’s final 64 and a half days.

Day 0

We begin our plunge towards the Sun.

Day 6

After 6 days of falling to the Sun, the Earth’s temperature has risen by about 0.8 degrees celsius. That’s the same amount by which we’ve increased our planet’s temperature since 1880. You may not feel the heat as yet, but you will soon.

Day 21

The average global temperature has now risen by about 10 degrees celsius, to 35 C (95 F). The planet is now experiencing an extremely intense global heat wave, whose temperature rise rivals the record-breaking 2003 European heat wave. Crops are beginning to fail.

Day 35

It’s been over a month of Earthfall, and we’re now 20% of the way to the Sun. The Sun in unbearably bright and intense, and noticeably larger in the sky. At 58 C (137 F), the average global temperature now exceeds the historic hottest temperature recorded on Earth, which was 56.7 C (134 F) measured in Death Valley, CA.

For most people on the planet, it’s now impossible to stay alive without air conditioning, and the electricity infrastructure is either tapped out or failing. Forest fires are ravaging through the wilderness. Land animals that can’t burrow in to the soil to get respite from the heat are going extinct. The insects, too, are feeling the heat and dying out. The increasing water temperature will cause fish to start dying out, because warmer water holds less oxygen and more ammonia (which is toxic to fish), and because the entire marine food chain would be disrupted and collapsing.

It’s so hot that even the Saharan silver ant, one of the most heat resistant land animals on Earth, can no longer survive the heat (for it can stay alive up to 53.6 C). However, the Sahara desert ant is thriving – it can survive surface temperatures of up to 70 C. As scavengers, these ants feed on the corpses of other creatures that have died from the heat, and there’s now plenty of food to go around.

Day 41

We’ve now crossed Venus’s orbit. The average temperature exceeds 76 C (169 F), a temperature too hot for even the Sahara desert ant.

The Pompeii Worm, however, is still holding on. These amazing creatures grow up to 13 centimeters (5 inches) long, and are known to survive temperatures of up to 80 C. It’s thought that they owe this heat-resisting superpower to a protective “fleece-like” layer of bacteria on their backs, which insulates them from the heat. These worms are “the most heat-tolerant complex animal known to science”, with the exception of tardigrades (whom we’ll hear from soon).

Day 47

We just left the habitable zone, that Goldilocks region of a solar system (not too hot, not too cold) capable of sustaining life as we know it.

At 103 C (217 F), the ambient temperature now exceeds the boiling point of water. The Earth’s oceans are boiling. Liquid water can no longer exist on much of Earth’s surface and steam envelopes the planet. Most life on Earth is extinct by now, particularly complex life forms, even the amazingly heat-resistant Pompeii Worm. Hyperthermophiles (such as heat-resistant bacteria) are surviving (perhaps even thriving) deeper in the ocean where the water pressure prevents boiling. Fire tolerant plants are still holding on.

Tardigrades (or water bears) take the prize for the toughest known living things. Heck, these creatures have even survived in the vacuum, extreme cold and high radiation of outer space for a whopping 10 days. Truly, they are among Earth’s extreme survivors.

At this point, the Tardigrades are perhaps just beginning to notice that something is awry. They’re probably bunkering down by suspending their metabolism, curling up and dehydrating themselves into a desiccated state that contains only 1% of their original water. In this dehydrated state, called a tun, they can stay alive and dormant for nearly a decade.

Day 54

Farewell, dear tardigrades. You outlived us all. Although you can endure a crazily impressive temperature range, from near absolute zero to 151 C, Earth’s temperature now exceeds 160 C, too hot even for you.

Day 57

We’ve crossed Mercury’s orbit, and are now the closest planet to the Sun, a distinction that we will hold for another week. The ambient temperature exceeds 200 C.

Day 64

The Earth is now in its final day. Because of the Earth’s immense accumulated speed, and the intense gravitational force of the nearby Sun, we’ll cover the last 7 percent of our journey’s length by 1 pm today. The Sun’s gravity is now so extreme, that it pulls the front of the Earth with significantly more force than the back of the Earth. This differential gravity, or tidal force, is stretching Earth out into an oval shape. Magma erupts throughout cracks and fissures in the planet’s crust.

The day starts off at a balmy 800 C, with the Sun fourteen times its regular size in our sky. By noon the temperature hits 2000 C, more than hot enough to melt rock. Earth’s surface melts into magma.

At half past noon, we’ve nearly arrived. The Sun is so close that it fills most of the day sky. The Earth is crossing an imaginary line of no return called the Roche Limit. This is the point where the tidal forces pulling Earth apart exceed Earth’s ability to hold itself together. As it crosses this critical radius, the tidal effect of gravity rips the Earth into smaller balls of magma and melting rock.

And this is how our disintegrating planet finally meets the end of its journey to the Sun. I hope you enjoyed the trip.

P.S. Before you begin to fret about Earth falling into the Sun, consider this. Earth’s speed in our orbit is about 30 kilometers/second. That’s a lot of speed that we’d have to shed for the scenario in this blog post. It would actually be (very slightly) easier for the giant space finger to give us a shove and increase our speed to 42.1 kilometers/second, the escape velocity at which Earth would break free of the Sun’s gravity and become a rogue planet. I really did start off this paragraph trying to help. Sigh.

Nerdy Footnotes

To calculate the temperature of the planet, I used the rule of thumb for the Sun that the equilibrium temperature of a planet is 250/sqrt(d) where d is the distance to the Sun in astronomical units. You can find a derivation of that here. Additionally, I used a simplified model of the atmosphere as a single layer to account for the greenhouse effect – this boils down to multiplying the temperature by the fourth root of 2. I used Mathematica to numerically solve for the relation between time in Earthfall and distance covered by the Earth. Combining this piece with the above relation of temperature vs. distance, I arrived at the plot of temperature versus time in this post.

There are tons of reasons why this calculation is a massive simplification – for one, the greenhouse effect will increase considerably as the melting Earth’s atmosphere probably contains way more greenhouse gases from all the burning carbon. But, hopefully, this post should give a reasonable order-of-magnitude estimate of life (and lack thereof) during Earthfall. Also, I’m totally sweeping under the rug the force that would arise during Earth’s deceleration as it’s being slowed down by the giant finger, because life’s too short.

People learn best by doing. That’s a simple idea, backed by reams of evidence. And yet I always struggle with this idea when I’m writing. Online science communication is by-and-large a passive medium, where the writer tells a story, and the reader listens. It might be an incredibly compelling and engaging story, but it’s ultimately one where the writer is at the wheel and the reader is taken along for the ride. Sometimes this limitation frustrates me, because I recognize that it isn’t the most effective way to communicate ideas.

But today I came across something that made me see a different way of communicating online, one that whole-heartedly adopts this ‘learn by doing’ philosophy and puts the reader in the driving seat. It’s called Parable of the Polygons, and was built by Vi Hart and Nicky Case. It’s what that they call a playable blog post, part story and part game, set in an imaginary world of squares and triangles. While it might at first seem like an odd mathematical game, it delivers a lucid and very relevant lesson on real-world segregation.

The goal of the game is to move the squares and triangles around until they’re all happy. These shapes like living in a diverse world inhabited by squares and triangles alike – in fact they prefer diversity. But there’s a small problem. Each shape is slightly ‘shapist’. Here’s how Hart and Case put it,

“These little cuties are 50% Triangles, 50% Squares, and 100% slightly shapist. But only slightly! In fact, every polygon prefers being in a diverse crowd:

You can only move them if they’re unhappy with their immediate neighborhood. Once they’re OK where they are, you can’t move them until they’re unhappy with their neighbors again. They’ve got one, simple rule:

“I wanna move if less than 1/3 of my neighbors are like me.”

Harmless, right? Every polygon would be happy with a mixed neighborhood. Surely their small bias can’t affect the larger shape of society that much? Well…”

By playing around with these squares and triangles, you’ll discover how even slight biases towards similarly shaped neighbors can lead to total segregation. It’s a tour of the counter-intuitive math of segregation, first spelt out by the Nobel Prize winning game theorist Thomas Schelling.

But it isn’t all gloom, for the post also teaches us that if all shapes demand even the smallest bit of diversity in their neighborhoods (a slight anti-bias, if you will), then segregation plummets. The lesson here is that small individual preferences can create a large societal effect. It’s up to us to determine which direction we want that effect to go – towards a diverse world or a completely segregated one.

The Parable of the Polygons is a truly interactive way of communicating an idea. And, perhaps just as important, it’s incredibly well designed. The disarmingly charming cast of characters – delightfully animated circles and squares – playfully distill the essence of the idea, and allow Hart and Case to deliver an effective lesson about race and equality without getting embroiled in a heated political debate.

That’s enough talk. Now go check out the Parable of the Polygons.

And once you’re done with exploring that, if you live in the US, you might also be interested in this racial map of the US which shows you how diverse or segregated your neighborhood is.

Here’s a fun project that my friend Upasana and I put together some weekends ago. It’s a visual exploration of fractals through dance, a piece of generative art that’s part performance and part mathematical exploration.

The two ingredients that went into creating this were the Microsoft Kinect sensor, which lets your computer track how your body moves, and Processing, a programming language that lets you create interactive visuals with code. Put the two together, and you can use your body to control virtual shapes and objects.

The idea for this project came about while I was walking home from work late October, idly watching the recently bare tree branches swaying in the wind. And for some reason that made me wonder, what would it be like to be a tree for an evening? Imagine lifting your arms, and a tree waves its branches.

And then I remembered reading about fractals in Daniel Shiffman’s book Nature of Code. Fractals are those wonderfully intricate structures that look the same as you keep zooming in to them. Benoit B. Mandelbrot was one of the earliest explorers of the fractal world. He coined the word fractal to mean a kind of geometric shape whose parts resemble “a reduced-size copy of the whole.”  (Some fractal humor: What does the B in Benoit B. Mandelbrot stand for? Benoit B. Mandelbrot.)

At the heart of being a fractal is self-similarity, the idea that each piece appears similar to the whole. Think of how a coastline on a map appears similarly wrinkly across different levels of zoom. The same could be said of the jagged terrain of a mountain.

Or picture the ever branching paths that lightning follows as it travels down to the Earth.

Or the nested arrangement of leaves within a fern.

Or the buds in a head of Romanesco broccoli. Each bud contains smaller buds upon it, arranged in the same spiraling pattern.

From coastlines to broccoli, and lightning to trees, many of nature’s patterns are better described by fractals than by the usual cast of shapes like lines, circles, and triangles. (In the real world, objects can only be roughly fractal, at some level of zoom the repetition will end. But in mathematics, the self-similarity of fractals continues forever.)

While shapes like circles and triangles are the essence of simplicity, fractals, on the other hand, capture a kind of organic complexity in their infinite fuzziness. There’s a world of difference between a triangle and a mountain.

Which brings me back to the tree. You can think of a tree as being fractal-like. Here’s a sketch of a tree.

But look closely, and each branch is kind of a “reduced-size copy of the whole”, to use Mandelbrot’s words.

For example, take a look at this picture.

You might think you’re looking at an entire tree, but actually, it’s a picture of a branch sticking out sideways. (I was lying down and looking up when I took the picture.) The branch looks like a smaller-scale tree, and it’s this self-similarity that we’re trying to capture.

In his book, Daniel Shiffman implements a simple algorithm in Processing to draw such fractal trees.

I slightly tweaked his code so that the angle at which the branches form is controlled by the position of my mouse. So by moving my mouse around, I could animate the tree. You can play with this for yourself.

I tried adding in some leaves to make the tree a little prettier.

Here’s an interactive version of that for you to play with.

The next step was to incorporate the Kinect sensor bar, so you can control this tree with your body. To do this, I used the Simple-OpenNI library (you can read more about how this works in my earlier post on Kinect hacking). My friend Upasana Roy volunteered to play the role of dancer and puppeteer (video at the top of this post).

And voila, by waving her hands about, she animates the tree. The angle of her hands (measured from her hip) controls the branching angles in the fractal. The physical space that her body inhabits is mapped onto an abstract mathematical space (in this case, the space of all fractals you get by varying the branching angle of this tree).

We stopped there, but there are endless other virtual shapes, objects, and spaces that you might be interested in exploring.

I asked Upasana how she felt as she started to play with this program.

What did you experience when you first started using it?

“It was frustrating at first. A lot of times my moves didn’t get any response from the fractal, and the few times it did the changes were jerky and sudden. Even when I moved my arm in one smooth motion, the fractal tree would spasm so I didn’t know what was going on.”

What changed as you spent more time using it?

“I realized that when I dance I use all the space around me with my arms. But in this case the fractal responded to motion in some parts of space more than others. For example, moving my arms up and down made the fractals change much more than moving them side-to-side.”

How did you go about choreographing the dance? What was challenging about it and what was fun about it?

“After spending some time I could figure out a relationship between my arm positions and the fractal design. So then I could move it in a predictable way. After that it was about playing around with the fractal patterns and the music – that was the fun part. The hard part was keeping my arms up the entire time!”

“I remember feeling a bit like a hi-tech puppeteer, because I was moving my hands around in the air but you weren’t seeing THAT movement, but a transformed version of it. That felt pretty cool, to control the tree with abstract hand movements in the air.”

References

Here’s my Processing code for the visualization, based on modifying code by Max Rheiner (who wrote the OpenNI library that lets you access the Kinect in Processing) and Dan Shiffman (who wrote the code for the recursive tree).

Fractals Chapter in Dan Shiffman’s great book Nature of Code

Making Things See by Greg Borenstein is a very handy guide to Kinect hacking with Processing

A previous post by me on using the Kinect to make a dance video

And a big thanks to Upasana Roy for being willing to dance till she couldn’t keep her arms up any more.

When I was about 10, I broke my first skateboard by riding it into a ditch. A decade later, in college, I broke another skateboard within an hour of owning it (surely a record) in a short-lived attempt at doing an ollie. (Surprisingly, the store accepted a return on that board even though it was in two pieces.) Then I was gifted a really nice, high-quality skateboard. The first thing I did with it was ride it down a big hill, a valiant but ill-fated adventure which ended with me jumping off the skateboard, rolling down the grass, and arriving scraped up, deflated, and rather disoriented near the entrance to my college cafeteria. (In my defense, the wheels and ball-bearings on that skateboard had been pre-lubricated to minimize friction, and why would anyone do that, that’s just crazy.)

So believe me when I tell you that I am incredibly envious of skaters who can pull off tricks like this.

Now, I might not be able to skate to save my life, but I can do a little physics. So here’s a thought – maybe I can use physics to learn how to do an ollie. Here’s the plan. I’m going to open up the above video of skateboarder Adam Shomsky doing an ollie, filmed in glorious 1000 frames-per-second slow motion, and analyze it in the open source physics video analysis tool Tracker.

The first thing I did was track the motion of the front and back wheels (Tracker has a very convenient autotracker feature that can do this for you.)

One useful physics trick here is to track the center of mass of the skateboard, i.e. the average of the positions of the front and back wheels. Here is that curve overlapped in green.

Now, if you were to do the same tracking exercise for a soccer ball that’s been kicked, you’d get a neat arc-like shape called a parabola. This is the characteristic shape you get when the only force influencing an object’s motion is gravity.*

But the green curve in the above gif — the motion of the center of mass of the skateboard — is nowhere close to being a parabola. It’s lumpy and weird. This means that gravity isn’t the only force affecting the skateboard. Unlike a soccer ball in mid-flight, a skateboard mid-ollie is being actively steered.

This is exactly what makes doing an ollie so hard. It’s not enough to get the skateboard up into the air – you also have to steer it while it’s in the air.

In fact, we can work out how you need to steer the skateboard. Tracker has a nice feature that we’ll call ‘force arrows’. These arrows show you how much force acts on an object at every instant, and in which direction the force acts. So for example, if you were to kick a ball into the air, while the ball was mid-flight, this arrow would always point down and be the same length, even though the ball is moving forward. That’s because the only force acting on the ball is gravity, which pulls it straight down, and acts with a constant strength. (For those of you who’ve studied physics, these arrows denote the acceleration of the center of mass, which by Newton’s second law is proportional to the net force acting on the skateboard.)

Here’s what we find when we work out the force arrows for the skateboard.

Or, if you prefer to see all the arrows overlaid,

It’s a neat piece of science art, and it also tells us something interesting. The arrows show us that the force on the skateboard is constantly changing, both in magnitude as well as in direction. Now the force of gravity obviously isn’t changing, so the reason that these force arrows are shrinking and growing and tumbling around is that the skater is changing how their feet pushes and pulls against the board. By applying a variable force that changes both in strength and direction, they’re steering the board.

In fact, we can go back and see how much force each wheel experiences.

Crucially, at any instant, each foot applies a different amount of force. These unequal forces at each end is what causes the skateboard to turn (in physics lingo, it creates a torque). It’s how the skater steers the board.

We can see this more clearly if we subtract away the motion of the center of mass (i.e. subtract the green arrows above from the red and the blue arrows). Now, we’re only looking at how the wheels accelerate relative to the center of the board, not relative to the ground.

You can see there how the skater uses unequal forces to turn the board, shifting their weight from their front foot while moving up, to their back foot while descending.

To summarize, a skateboarder’s feet need to do two things successfully to complete an ollie. They need to provide a changing force to move the board correctly (so that the combined force of gravity and the skater’s feet add up to the green arrows above), and they need to provide different amounts of force with each foot (shown by the red and blue arrows above) to steer and turn the board into the right orientation.

Sadly, after all this geeking out, I’m no more successful in my attempts to do an ollie. But at least now I can explain why I suck at it.

Footnotes

Thanks to Robin Wylie for helping me headline this post.

*Technically this curve is a (segment of an) ellipse, but so long as you aren’t kicking the football into orbit, it’s close enough to a parabola.

I want to tell you about one of the most beautiful ideas that I know.

It’s a physics experiment, and it’s beautiful because in one elegant stroke, it expands our consciousness, forcing us to realize that objects can behave in ways that are impossible for us to picture (but remarkably, possible for us to calculate). It’s beautiful because it calls into question the bedrock of logic on which we’ve built our understanding of the world. It’s beautiful because it’s deceivingly simple to understand, and yet its consequences are deeply unsettling. And it’s beautiful because I refused to accept it until I ran the experiment for myself, and I distinctly remember watching my worldview shatter as the picture slowly built up on the computer monitor.

This was eleven years ago. I was a college freshman, sitting in a physics lab with all the lights turned out, staring at a blank computer screen, and for reasons that I won’t go into here, listening to a best-of compilation of 80s pop hits.

Here’s the setup. On the table in front of me there’s a box with two thin slit-like openings at one end. We’re shooting particles into this box through these slits. I did the experiment with photons, i.e. chunks of light, but others have done it with electrons and, in principle, it could be done with any kind of stuff. It’s even been done with buckyballs, which are soccer ball shaped arrangements of 60 carbon atoms that are positively ginormous compared to electrons. For convenience, I’m going to call the objects in this experiment electrons but think of that word as a stand-in for any kind of stuff that comes in chunks, really.

At the other end of the box is a CCD camera, that takes a snapshot of whatever hits it. Every time a particle makes it to the other side of the box, I see a dot light up at the corresponding point on my computer screen.

Just to be extra careful, we’ve set up the experiment so that there is only one particle inside the box at any given time. Picture, if you like, very tiny baseballs being flung into the box, one at a time. The 80s music plays on, and we sit and wait.

What would you expect to see on the other side of the box? Well, if electrons behaved like waves, you’d expect to see fringes of bright and dark bands, like ripples in a tank of water. That’s because waves can interfere with each other, canceling out when the peak of one wave meets the trough of another, and getting reinforced when the peaks line up.

But electrons aren’t waves – they come in chunks. I know this, because I can see them arriving at the screen one at a time, and they strike at a single place, like raindrops falling on dry pavement. And if electrons are chunk-like, then you’d expect to see them piling up behind the slits and nowhere else. In short, you’d expect them to behave like baseballs.

And indeed, if you do this experiment with only one slit open, they behave just like baseballs, hitting the wall in a single band behind the open slit. A reasonable prediction, then, is that when we run the experiment with both slits open, we should see two bands – one behind each slit.

So what do the electrons do?

Here, see for yourself. You can watch the electrons coming in one at a time in this video produced by scientists at Hitachi in 1989. The video has been sped up around 30X.

It takes a moment to realize just how odd this is. Somehow, the electrons created this interference pattern of bright and dark stripes. But they were sent in one at a time, so what could they possibly have interfered with? If you’re picturing the electron as a tiny baseball, you’re forced to conclude that an electron going through one slit ‘sniffs out’ that the other slit is open, and adjusts its behavior accordingly. Or that it’s somehow taking both paths and interfering with itself. This doesn’t make any kind of sense.

So let’s take a step back, and try to piece together what happened. Here’s an obvious question that you might ask. Think about an electron that arrives at the screen. Which slit did it go through?

Did the electron go through the left slit?

No! Because when you cover up the right slit, the stripey pattern disappears and you get a boring single band instead.

Did the electron go through the right slit?

No! For the same reason as above. When you cover up the left slit, instead of the stripey pattern you get a single band.

Does the electron go through both slits?

No! Because if that were true, we’d expect to see the electron split into two, and one electron (or maybe half) would go through each slit. But if you place detectors at the slits you find that this never happens. You always see only one electron at a time. It never, ever splits into two.

Did the electron go through neither slit?

No! Of course not, that’s just silly. If you cover both slits, nothing happens.

At this point you’re probably thinking that this is getting a bit ridiculous. Why can’t we just look at the damn electron and see which path it took? The problem with this idea is that looking at something means shining light on it, and shining light on it means bumping it with a photon. If you’re a tiny electron, this bump disturbs your original path.

Well, wait. Maybe you could make the bump really, really gentle, so you don’t disturb the electron much. Indeed, you can do that, but as you make the light more gentle (lower momentum), you also make it more spread out (larger wavelength), and you end up not being able to tell which slit the electron went through.

There’s no way to win here. Any scheme that you can invent to determine which route the electron took will also destroy the interference pattern.

To summarize, we’ve arrived at a pattern of fringes that’s built up one particle at a time. But when you try to work out exactly how those particles got there, you find that they didn’t take the left route, they didn’t take the right route, they didn’t take both routes, and they didn’t take neither routes. As MIT professor Allan Adams puts it, that pretty much exhausts all the logical possibilities!

An electron is not like a wave of water, because unlike a wave, it hits a screen at a single location. An electron is not like a baseball, because when you throw in a bunch of them through a double slit, they interfere and create patterns of fringes. There is no satisfying analogy that anyone can give you for what an electron is. As Allan Adams points out in his introductory lecture on quantum mechanics,

These electrons are doing something we’ve just never thought of before, something we’ve never dreamt of before, something for which we don’t really have good words in the English language.

Apparently, empirically, electrons have a way of moving, [..] a way of being, which is unlike anything that we’re used to thinking about.

And so do molecules.

And so do bacteria.

So does chalk.

It’s just harder to detect in those objects.

Physicists have a name for this new mode of being. We call it superposition.

It’s sometimes useful to think of an electron as behaving like a particle, and it’s sometimes useful to think of an electron as behaving like a wave. But these are just conveniences of language, and they’re both incomplete pictures. An electron is neither a wave nor a particle. An electron is an electron. The same goes for a photon, an atom, a buckyball, a giant molecule, or what have you. (The larger the object, the harder it is to see these fringes.)

Werner Heisenberg, one of the creators of quantum mechanics, understood this clearly. In 1930, he wrote,

The solution of the difficulty is that the two mental pictures which experiment lead us to form—the one of the particles, the other of the waves—are both incomplete and have only the validity of analogies which are accurate only in limiting cases. [..] the apparent duality arises in the limitations of our language.

As Heisenberg and others taught us, although language fails us, it’s possible to come up with rules that correctly predict how tiny things behave. Those rules are quantum mechanics. You can learn these rules for yourself by reading Richard Feynman’s classic book QED, or watching his lectures on the subject. Using these rules, physicists can throw around fancy sounding sentences like ‘the electron wavefunction is in a superposition of going through the left slit and going through the right slit’. These sentences have well defined mathematical meanings, and they make predictions that match the data. But what’s missing here is a coherent picture that you can hold in your head that will explain which path the electron took, and what’s more, we can be fairly confident that such a picture can never exist.

It’s perhaps not surprising that our ape-brains, which evolved in a middle-sized world throwing rocks and spears, can’t visualize the behavior of very small things. What’s surprising to me is that even though we’re unable to picture this quantum world, we’ve managed to work out the rules of the game.

References

This post has largely been inspired by watching Allan Adams’s excellent introduction to quantum physics made available by MIT OpenCourseWare. The first lecture is an fascinating and often hilarious look at the principle of superposition explained in a non-technical way. I highly recommend checking it out – he’s a very engaging lecturer.

We’ve all been there. You pick up a slice of pizza and you’re about to take a bite, but it flops over and dangles limply from your fingers instead. The crust isn’t stiff enough to support the weight of the slice. Maybe you should have gone for fewer toppings. But there’s no need to despair, for years of pizza eating experience have taught you how to deal with this situation. Just fold the pizza slice into a U shape (aka the fold hold). This keeps the slice from flopping over, and you can proceed to enjoy your meal. (If you don’t have a slice of pizza handy, you can try this out with a sheet of paper.)

Behind this pizza trick lies a powerful mathematical result about curved surfaces, one that’s so startling that its discoverer, the mathematical genius Carl Friedrich Gauss, named it Theorema Egregium, Latin for excellent or remarkable theorem.

Take a sheet of paper and roll it into a cylinder. It might seem obvious that the paper is flat, while the cylinder is curved. But Gauss thought about this differently. He wanted to define the curvature of a surface in a way that doesn’t change when you bend the surface.

If you zoom in on an ant that lives on the cylinder, there are many possible paths the ant could take. It could decide to walk down the curved path, tracing out a circle, or it could walk along the flat path, tracing out a straight line. Or it might do something in between, tracing out a helix.

Gauss’s brilliant insight was to define the curvature of a surface in a way that takes all these choices into account. Here’s how it works. Starting at any point, find the two most extreme paths that an ant can choose (i.e. the most concave path and the most convex path). Then multiply the curvature of those paths together (curvature is positive for concave paths, zero for flat paths, and negative for convex paths). And, voila, the number you get is Gauss’s definition of the curvature at that point.

Let’s try some examples. For the ant on the cylinder, the two extreme paths available to it are the curved, circle-shaped path, and the flat, straight-line path. But since the flat path has zero curvature, when you multiply the two curvatures together you get zero. As mathematicians would say, a cylinder is flat — it has zero Gaussian curvature. Which reflects the fact that you can roll one out of a sheet of paper.

If, instead, the ant lived on a ball, there would be no flat paths available to it. Now every path curves by the same amount, and so the Gaussian curvature is some positive number. So spheres are curved while cylinders are flat. You can bend a sheet of paper into a tube, but you can never bend it into a ball.

Gauss’s remarkable theorem, the one which I like to imagine made him giggle with joy, is that an ant living on a surface can work out its curvature without ever having to step outside the surface, just by measuring distances and doing some math. This, by the way, is what allows us to determine whether our universe is curved without ever having to step outside of the universe (as far as we can tell, it’s flat).

A surprising consequence of this result is that you can take a surface and bend it any way you like, so long as you don’t stretch, shrink or tear it, and the Gaussian curvature stays the same. That’s because bending doesn’t change any distances on the surface, and so the ant living on the surface would still calculate the same Gaussian curvature as before.

This might sound a little abstract, but it has real-life consequences. Cut an orange in half, eat the insides (yum), then place the dome-shaped peel on the ground and stomp on it. The peel will never flatten out into a circle. Instead, it’ll tear itself apart. That’s because a sphere and a flat surface have different Gaussian curvatures, so there’s no way to flatten a sphere without distorting or tearing it. Ever tried gift wrapping a basketball? Same problem. No matter how you bend a sheet of paper, it’ll always retain a trace of its original flatness, so you end up with a crinkled mess.

Another consequence of Gauss’s theorem is that it’s impossible to accurately depict a map on paper. The map of the world that you’re used to seeing depicts angles correctly, but it grossly distorts areas. The Museum of Math points out that clothing designers have a similar challenge — they design patterns on a flat surface that have to fit our curved bodies.

What does any of this have to do with pizza? Well, the pizza slice was flat before you picked it up (in math speak, it has zero Gaussian curvature). Gauss’s remarkable theorem assures us that one direction of the slice must always remain flat — no matter how you bend it, the pizza must retain a trace of its original flatness. When the slice flops over, the flat direction (shown in red below) is pointed sideways, which isn’t helpful for eating it. But by folding the pizza slice sideways, you’re forcing it to become flat in the other direction – the one that points towards your mouth. Theorema egregium, indeed.

By curving a sheet in one direction, you force it to become stiff in the other direction. Once you recognize this idea, you start seeing it everywhere. Look closely at a blade of grass. It’s often folded along its central vein, which adds stiffness and prevents it from flopping over. Engineers frequently use curvature to add strength to structures. In the Zarzuela race track in Madrid, the Spanish structural engineer Eduardo Torroja designed an innovative concrete roof that stretches out from the stadium, covering a large area while remaining just a few inches thick. It’s the pizza trick in disguise.

Curvature creates strength. Think about this: you can stand on an empty soda can, and it’ll easily carry your weight. Yet the wall of this can is just a few thousandths of an inch thick, or about as thick as a sheet of paper. The secret to a soda can’s incredible stiffness is its curvature. You can demonstrate this dramatically if someone pokes the can with a pencil while you’re standing on it. With even just a tiny dent, it’ll catastrophically buckle under your weight.

Perhaps the most mundane example of strength through curvature are the ubiquitous corrugated building materials (corrugate comes from ruga, Latin for wrinkle). You could hardly get more bland than a corrugated cardboard box. Tear one of these boxes apart, and you’ll find a familiar, undulating wave of cardboard inside the walls. The wrinkles aren’t there for any aesthetic reasons. They’re an ingenious way to keep a material thin and lightweight, yet stiff enough to resist bending under considerable loads.

Corrugated metal sheets use the same idea. These humble, unpretentious materials are a manifestation of pure utility, their form perfectly matched with their function. Their high strength and relatively low cost has blended them into the background of our modern world.

Today, we hardly give these wrinkled sheets of metal a second thought. But when it was first introduced, many saw corrugated iron as a wonder material. It was patented in 1829 by Henry Palmer, an English engineer in charge of the construction of the London Docks. Palmer built the world’s first corrugated iron structure, the Turpentine Shed at the London Docks, and although it might not seem remarkable to modern eyes, just listen to how an an architectural magazine of the time described it.

“On passing through the London Docks a short time ago, we were much gratified in meeting with a practical application of Mr Palmer’s newly invented roofing. […] Every observing person, on passing by it, cannot fail being struck (considering it as a shed) with its elegance and simplicity, and a little reflection will, we think, convince them of its effectiveness and economy. It is, we should think, the lightest and strongest roof (for its weight), that has been constructed by man, since the time of Adam. The total thickness of this said roof, appeared to us from a close inspection (and we climbed over sundry casks of sticky turpentine for that purpose,) to be, certainly not more, than a tenth of an inch!” [1]

They just don’t write architectural magazines like they used to.

While corrugated materials and soda cans are pretty strong, there’s a way to make materials even stronger. To discover it for yourself, go to your fridge and take out an egg. Put it in the palm of your hand, wrap your fingers around the egg, and squeeze. (Make sure you aren’t wearing a ring if you attempt this.) You’ll be amazed at its strength. I wasn’t able to crush the egg, and I gave it everything I had. (Seriously, you need to try this to believe it.)

What makes eggs so strong? Well, soda cans and corrugated metal sheets are curved in one direction but flat in the other. This curvature buys them some stiffness, but they can still potentially be flattened out into the flat sheets that they came from.

In contrast, egg shells are curved in both directions. This is the key to an egg’s strength. Expressed in math terms, these doubly curved surface have non-zero Gaussian curvature. Like the orange peel we encountered earlier, this means that they can never be flattened without tearing or stretching — Gauss’s theorem assures us of this fact. To crack an egg open, you first need to dent it. When the egg loses its curvature, it loses its strength.

The iconic shape for a nuclear power plant cooling tower also incorporates curvature in both directions. This shape, called a hyperboloid, minimizes the amount of material required to build it. Regular chimneys are a lot like giant soda cans – they’re strong, but they can also buckle easily. A hyperboloid shaped chimney solves this problem by curving in both directions. This double curvature locks the shape into place, giving it extra rigidity that a regular chimney lacks.

Another shape that gets its strength from double curvature is the Pringles potato chip*, or as mathematicians tend to call it, a hyperbolic paraboloid (say that three times fast).

Nature exploits the strength of this shape in a mind-blowingly impressive way. The mantis shrimp is infamous for having one of the fastest punches in the animal kingdom, a punch so strong that it vaporizes water, creating a shock wave and a flash of light. To deliver its impressive death blow, the mantis shrimp uses a hyperbolic paraboloid shaped spring. It compresses this spring to store up this immense energy, which it releases in one lethal blow.

You can watch biologist Sheila Patek describe her discovery of this amazing phenomenon. Or have Destin explain it to you in his brilliant Youtube channel Smarter Every Day.

The strength of this Pringles shape was well understood by the Spanish-Mexican architect and engineer Félix Candela. Candela was one of Eduardo Torroja’s students, and he built structures that took the hyperbolic paraboloid to new heights (literally). When you hear the word concrete, you might think of dreary, boxy constructions. Yet Candela was able to use the hyperbolic paraboloid shape to build huge structures that expressed the incredible thinness that concrete can provide. A true master of his medium, he was equal parts an innovative builder and a structural artist. His lightweight, graceful structures might seem delicate, but in fact they’re immensely strong, and built to last.

So what makes this Pringles shape so strong? It has to do with how it balances pushes and pulls. All structures have to support weight, and ultimately transfer this weight down to the ground. They can do this in two different ways. There’s compression, where the weight squeezes an object by pushing inwards. An arch is an example of a structure that exists in pure compression. And then there’s tension, where the weight pulls at the ends of an object, stretching it apart. Dangle a chain from its ends, and every part of it will be in pure tension. The hyperbolic paraboloid combines the best of both worlds. The concave U-shaped part is stretched in tension (shown in black) while the convex arch-shaped part is squeezed in compression (shown in red). Through double curvature, this shape strikes a delicate balance between these push and pull forces, allowing it to remain thin yet surprisingly strong.

Strength through curvature is an idea that shapes our world, and it has its roots in geometry. So the next time that you grab a slice, take a moment to look around, and appreciate the vast legacy behind this simple pizza trick.

Update: Via twitter, Rose Eveleth shared this really nice TED-Ed animation on the math & physics of pizza bending.

References

Reid, Esmond. Understanding buildings: a multidisciplinary approach. MIT Press, 1984.

[1]  Mornement, Adam, and Simon Holloway. Corrugated iron: building on the frontier. WW Norton & Company, 2007.

Garlock, Maria E. Moreyra, David P. Billington, and Noah Burger. Félix Candela: engineer, builder, structural artist. Princeton University Art Museum, 2008.

*According to an FDA ruling, Pringles aren’t legally potato chips because they’re made from dried potato flakes.

Huge thanks to Upasana Roy, Yusra Naqvi, Steven Strogatz, and Jordan Ellenberg for their helpful feedback on this piece.

Home Page Photo: m10229 / CC

My friend and colleague Teresa Riordan invited me to something truly awesome that she has been working hard to put together over the years. Art of Science is an annual art competition at Princeton University, where students, faculty, and alumni submit artistic works created in the process of doing science. I wandered through the exhibit during the opening reception and found it to be a great draw, bringing people together in sharing their excitement about science. It goes beyond the test tubes, graphs, and equations that are the bread and butter of everyday science, and instead showcases science as a vivid, technicolor, human experience.

One of my favorite pieces (at the top of this post) is a stunning image of a baby squid taken using a fluorescence microscope. Seen from this perspective, the squid embryo looks like an alien inhabitant of the microcosmos. “Even sea monsters start as babies”, writes chemistry professor and microcosmic explorer Celeste Nelson.

Among the entries are some wonderful ‘oops’ moments, where an experiment goes beautifully wrong, revealing art where you might not have expected to see it. Take this piece by Jason Wexler, Ian Jacobi, and Howard Stone. “This beautiful pattern, contrasting the relative order of the structured posts to the apparent chaos of the flowing blobs, would never have been seen had the experiment succeeded”, they write.

But most of these submissions aren’t accidents. Many of these pieces reveal form, structure, and beauty hidden at a scale that our eyes can’t perceive. There’s the self-assembled, intricate microscopic sculptures in electron microscope images of lab-grown crystals, or the mesmerizing 12-fold symmetry of quasicrystals, mirroring patterns from Islamic art. There’s the graceful dance of vortices inside a flickering flame, or the cavernous crystalline structures deposited in a dried up drop of (a protein extracted from) cow’s blood. “Watch any liquid – from tap water to the richest coffee – evaporate off a surface and you will see it leave a unique, ghostly mark”, write Hyoungsoo Kim, François Boulogne, and Howard Stone.

These works highlight that beauty doesn’t just exist at the human-sized scale that we encounter everyday, but is also hiding out of sight, from the scale of the universe to inside a drop of blood. It’s waiting for us to discover it, if only we can sharpen our instruments and take a closer look. And through these works, we see that science is very much a human experience, brimming with beauty in every drop.

For more, either head to the image and video gallery at Art of Science, drop by the Friend Center Gallery (free and open to the public) on the Princeton University campus, or catch their highlights on display in the New York Hall of Science.

I was heading out from home to get lunch, when I caught a glint of light out of the corner of my eye. I saw what looked like tiny drops of mercury, sitting on the leaves of a plant in my backyard.

Huh. Those balls of mercury were really just very reflective drops of water. But something about this plant mesmerized me, and I stopped to take a closer look. The plant, by the way, is a plume poppy (Macleaya cordata). It’s got these lovely fractalesque, large green leaves and is native to China, Japan, and Southeast Asia.

Do you notice what struck me as odd about this scene?

Those water drops are just so.. round. They’re like tiny, glass marbles, gently sitting in place. Give the leaf the lightest flick, and they’ll roll away.

That’s not how water usually behaves. Water wets things. It clings to the surface and flattens out like a pancake. It doesn’t roll around like a glass bead. This leaf must have some kind of natural water-repelling surface that prevents it from getting wet.

A couple of days later, I snapped off a leaf and brought it to my friend Janine Nunes. Janine is a postdoctoral researcher at Princeton University. She’s a super-skilled researcher, and she also has access to some of the coolest toys in existence. Among these impressive devices is this Phantom ultra-high speed camera.

She mounted the leaf on a stand, and had the camera ready to go.

Now for some fun. Here’s what happens when a water drop hits a plume poppy leaf.

See how the water drop bounces off the leaf instead of splashing?

If the water hits the leaf harder, it’ll splash. But it still doesn’t wet the surface.

You can watch the water drops merge into one big, wobbly drop.

So how does this leaf repel water?

To understand this, we first need to know what it means to get wet. Since water molecules attract each other, a blob of water wants to shrink inwards. That’s why a water blob floating in space is round, like a sphere (it’s the most ‘shrunken-in’ shape). But down here on Earth, water isn’t floating in mid-air. It’s sitting on some surface, like your table, your bathtub, or a leaf. This surface pulls down on the water, and squishes the sphere into a pancake. So it looks more like this.

In fact, you can measure just how ‘wettable’ a surface is by calculating its contact angle.

The more a surface attracts the water, the more it squishes the ball into a pancake, and the wetter the surface.

On the other hand, hydrophobic (water-hating) surfaces attract the water less, so the drop is more round.

And then you have superhydrophobic surfaces, like Never Wet, which barely attract the water at all. On these surfaces, water drops are almost spherical. It’s nearly impossible to get these surfaces wet – the water just rolls off them.

To find out what’s going on with our leaf, we need to measure its contact angle. Janine put a tiny drop of water on the leaf.

BOOM. The contact angle works out to about 175 degrees. The leaf is extremely water repellent – it’s superhydrophobic.

But how does a leaf become superhydrophobic? The trick to this, Janine explained, is that the water isn’t really sitting on the surface. A superhydrophobic surface is a little like a bed of nails. The nails touch the water, but there are gaps in between them. So there’s fewer points of contact, which means the surface can’t tug on the water as much, and so the drop stays round.

If this explanation is correct, then the surface of the plume poppy’s leaf must be coated in tiny needles. To find out, we stuck one of these leaves inside an electron microscope (didn’t I tell you she has access to the coolest toys? I wasn’t lying.)

And, just as we expected, we saw this field of tiny wax needles, each needle just a few microns in length!

Here’s another look at these tiny spikes. You can see the ripples on the leaf behind it.

Zooming in further…

The water drops are suspended on these ultra-microscopic wax needles, and that keeps it from wetting the surface.

Next, we looked at the underside of this leaf with the microscope. We’d noticed earlier that the underside of the leaf was also superhydrophobic, and you could see it was covered with tiny hair-like filaments. But we were blown away with what we saw through the microscope.

Here’s a closer look at those fibers.

At this scale, they look like claws reaching out from the veins. To give you a sense of scale, each of these fibers is about as thick as a regular human hair. Let’s land on one of them.

Once again, you see a fine mesh of tiny, ultra-microscopic wax needles coating each of these fibers, each needle being only a few microns in length. These needles are way smaller than your eye can see. This ability to touch without really touching, by resting the water on a bed of nails, is the secret to the incredible water-repelling powers of this leaf.

There’s one last thing I wanted to know. Why did this plant, and so many others, evolve this incredible ability to keep water at bay? One common explanation is that this allows the leaves to clean themselves. You see, as water rolls around on a superhydrophobic surface, it scoops up dirt and sand with it. Here’s Janine demonstrating this neat self-cleaning property of the leaf with genuine Jersey Shore (TM) sand.

However, I’m not sure that I buy this explanation. Why would a plant evolve a method that cleans the under-side of its leaves? Maybe it produces the wax for some other reason, and as an accidental benefit, this wax just happens to keep the leaves clean? Is there a clear evolutionary advantage for these leaves to be superhydrophobic? I don’t know the answer, but I’d love to find out. If you have any leads, drop me a note in the comments.

Oh, and when something interesting catches the corner of your eye, don’t forget to stop and check it out.

Update: There’s a good discussion of this post brewing at Hacker News, with some thoughtful points about the benefits of being superhydrophobic.

Shoutouts

A big thanks to Janine Nunes and to Howard Stone’s lab at Princeton U. for indulging me with their time and sharing their equipment. This post wouldn’t have been possible without their extensive resources and immense help.

And a shoutout to my colleague Jaclyn and to Ed Moran for identifying the plant in my pictures.

(I used an ImageJ plugin called DropSnake to calculate the contact angle. It’s freely available, and you can learn how to use it here.)

Think back to when you learned how to ride a bike. You probably didn’t master this skill by listening to a series of riveting lectures on bike riding. Instead, you tried it out for yourself, made mistakes, fell down a few times, picked yourself back up, and tried again. When mastering an activity, there’s no substitute for the interaction and feedback that comes from practice.

What if classroom learning was a little more active? Would university instruction be more effective if students spent some of their class time on active forms of learning like activities, discussions, or group work, instead of spending all of their class time listening?

A new study in the Proceedings of the National Academy of Sciences addressed this question by conducting the largest and most comprehensive review of the effect of active learning on STEM (Science, Technology, Engineering and Mathematics) education. Their answer is a resounding yes. According to Scott Freeman, one of the authors of the new study, “The impact of these data should be like the Surgeon General’s report on “Smoking and Health” in 1964–they should put to rest any debate about whether active learning is more effective than lecturing.”

Before you study something quantitatively, you have to define it. The authors combined 338 different written responses to arrive at the following definition of active learning:

Active learning engages students in the process of learning through activities and/or discussion in class, as opposed to passively listening to an expert. It emphasizes higher-order thinking and often involves group work.

They then searched for classroom experiments where students in a STEM class were divided into two groups – one group engaged in some form of active learning, while the other group participated in a traditional lecture. At the end of the class, both groups took essentially identical exams.

The authors looked at studies where both groups were taught by the same instructor and the students were assigned at random to each group, as well as less ideal experimental conditions, where the instructors differed, or the students weren’t assigned to groups at random. They evaluated the performance of these studies using two metrics – their scores on identical exams, and the percentage of students that failed (receiving a D, F or withdrawing from the class). In all, they identified 228 studies matching their criteria, to analyze further.

Here’s what they found.

1. Students in a traditional lecture course are 1.5 times more likely to fail, compared to students in courses with active learning

The authors found that 34% of students failed their course under traditional lecturing, compared to 22% of students under active learning. This suggests that, just in the studies that they analyzed, 3,500 more students would have passed their courses if taught with active learning. By conservative estimates, this would have saved the students about 3.5 million dollars in tuition. The authors point out that, were this a medical study, an effect size this large and statistically significant would warrant stopping the study and administering the treatment to everyone in the study.

A large drop in the number of failing students meets a demonstrated need to increase the retention of STEM students, and should be taken very seriously. Nearly a third of all students entering US colleges and universities intend to major in STEM fields, and more than half of these students eventually either switch their majors to a non-STEM field or drop out of college without a degree. This attrition problem is particularly acute for minorities, as only 20% of under-represented minority students who are interested in the STEM fields finish university with a STEM degree.

2. Students in active learning classes outperform those in traditional lectures on identical exams

On average, students taught with active learning outperformed those taught by lectures by 6 percentage points on their exam. That’s the difference between bumping a B- to a B or a B to a B+. Here’s another way that the authors describe this result. Picture a student in a traditional lecture class who scored higher than 50% of the students on the exam. If the same student were taught with active learning instead, they would score higher than 68% of the students in this lecture class.

Both these results were incredibly robust. They held up for all of the STEM subjects for which there was sufficient data. They held in large and small classes (although the impact of active learning was larger in small classes), and they held in introductory as well as upper-level courses. The exam performance results also held up irrespective of how the students were split into the two groups – whether the groups had the same or different instructors, or whether the student were randomly assigned to courses or not. The authors were also careful to account for whether their study was affected by publication biases (the bias to publish positive results over negative ones) and they found that this did not significantly impact their findings.

I asked Scott Freeman whether star lectures with strong teaching evaluations should be interested in these findings as well. He responded,

“Most of the studies we analyzed were based on data from identical instructors teaching active learning v lecturing sections; some studies (e.g. Van Heuvelen in Am. J. Physics; Deslauriers et al. in Science) have purposely matched award-winning lecturers with inexperienced teachers who do active learning and found that the students did worse when given “brilliant lectures.” We’ve yet to see any evidence that celebrated lecturers can help students more than even 1st-generation active learning does.”

I’ll leave the last word with Scott, who makes a strong case for active learning.

“[Under active learning,] students learn more, which means we’re doing our job better. They get higher grades and fail less, meaning that they are more likely to stay in STEM majors, which should help solve a major national problem. Finally, there is a strong ethical component. There is a growing body of evidence showing that active learning differentially benefits students of color and/or students from disadvantaged backgrounds and/or women in male-dominated fields. It’s not a stretch to claim that lecturing actively discriminates against underrepresented students.”

 References

Freeman et al. Active learning increases student performance in science, engineering, and mathematics. PNAS.

Data on STEM attrition rates from the Department of Education’s STEM attrition report and the President’s Council of Advisors on Science and Technology report.