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.
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?
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!
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.
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.
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 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.) Continue reading How to Dance with a Tree: Visualizing Fractals With Dance
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 characteristicshape 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.
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.
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.
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.
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.”
Freeman et al. Active learning increases student performance in science, engineering, and mathematics. PNAS.