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Shape of Nature

Shape of Nature

Course No.  1460
Course No.  1460
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Course Overview

About This Course

36 lectures  |  31 minutes per lecture

How do you measure the size of a black hole? The motion of individual particles at the subatomic level? The possible shape of space-time itself? In short: How do you mathematically describe the world around you? The answer lies in the fascinating ways mathematicians use geometry and topology to study and understand the shape of nature, whether leaf formations, DNA entanglements, or quantum fields. Mysterious, complex, and undeniably captivating, the study of the shape of nature lies at the forefront of current research in both mathematics and science. What's more: It's provided us with previously unimaginable scientific and technological advances, including our ability to:

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How do you measure the size of a black hole? The motion of individual particles at the subatomic level? The possible shape of space-time itself? In short: How do you mathematically describe the world around you? The answer lies in the fascinating ways mathematicians use geometry and topology to study and understand the shape of nature, whether leaf formations, DNA entanglements, or quantum fields. Mysterious, complex, and undeniably captivating, the study of the shape of nature lies at the forefront of current research in both mathematics and science. What's more: It's provided us with previously unimaginable scientific and technological advances, including our ability to:

Read genetic data to better determine the relationships between species; Closely study how proteins are built through the intricate process of folding; Model and predict wind currents around the globe; Map the seemingly random terrain of vast mountain ranges; Develop facial recognition software for cameras and high-level security systems; and Design and improve the way that robots move and behave.

While the mathematics involved in the study of shapes and nature is important to how we grasp and live in the world, it remains a mystery to many of us. But these concepts and ideas are not completely inaccessible. All you need is the right guide and an engaging way to approach the subject—both of which are available in The Shape of Nature. This visually stunning course is your authoritative guide to the mathematical shapes around us: how they're formed, how they're studied, and how they're applied to our everyday lives.

In 36 lectures, you'll discover the intricate relationship between mathematics and nature, get a pointed introduction to the language mathematicians use to study shapes and dimensions, and learn how to finally make sense of this abstract—yet undeniably intriguing—subject. And it's all brought to you by award-winning Professor Satyan L. Devadoss of Williams College, a dynamic instructor with an abiding and contagious passion for the worlds of geometry and topology. His clear and engaging presentation style, accompanied with eye-catching animations and graphics, will make your journey into the world of shapes both insightful and unforgettable.

Explore Fascinating Shapes

The vibrant heart of The Shape of Nature lies in its spirited exploration of the world of shapes. And the secret to understanding how shapes are created and how they work involves two powerful mathematical fields:

Geometry, the ancient discipline that focuses on quantitative notions such as the length, area, and volume of a particular shape

Topology, the modern field that focuses on qualitative notions such as connectivity, underlying structures, and the relationships between shapes.

After an engaging introduction to these two fields and their critical role in understanding shapes, Professor Devadoss takes you deep into the four main categories of shapes. Each category occupies its own particular dimension, has its own unique characteristics, and plays an important role in the worlds of physics, biology, and chemistry.

Knots: Begin your journey by learning about the simplest of shapes, the knot. Defined as circles placed in 3-D, knots appear throughout the world in DNA, in string theory, in knotted molecules, and in genetic mutations. These shapes serve as a jumping-off point from which you examine other simple shapes, including tangles, braids, and links.

Surfaces: Surfaces are the most common shapes in nature; essentially everything you see is the surface of some object, and these shapes are involved in everything from origami designs to wind flows to colored patterns on animals. Among the surfaces you study in this group of lectures are spheres, the five Platonic solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), the Klein bottle, and the projective plane.

Manifolds: Also known as 3-manifolds, these shapes are fascinating 3-D objects that can be finite or infinite in volume; can have boundaries or be without them; and can be orientable or non-orientable. As you move through these beautiful and complex shapes, you learn how to build them using simple operations such as multiplication and gluing and discover how they are instrumental in understanding the topology of the universe itself.

Higher dimensions: Leave the comfort of three dimensions and enter worlds that stretch our imagination. The 4-D polytopes are considered higher-dimensional analogs to the Platonic solids, including the famous 120-cell polytope (made of dodecahedra), the 600-cell polytope (made of tetrahedra), and the associahedron—the most influential shape of the last 25 years. Then move to arbitrary dimensions and enter the inner workings of particle collisions, robotic motions, genetic evolution, and more; even fractals and chaos theory make an appearance.

With each of these categories, you learn methods for building these shapes, the ways they relate to one another, the important theorems and ideas that advanced our understanding of how they look, and more. Professor Devadoss, an expert at making the theoretical practical, takes care to guide you through this challenging and rewarding mathematical territory with detailed explanations, stunning examples, in-studio demonstrations, and helpful summaries, so that while the shapes are challenging and require a deep interest in mathematics, you'll never feel overwhelmed by what you're learning.

Discover New Mathematical Ideas and Tools

Because The Shape of Nature tackles pioneering concepts in mathematics, it also serves as a powerful introduction to the revolutionary ideas and tools that modern mathematicians use to understand and work with shapes. In addition to helping you make sense of concepts such as equivalence, isotopy, and homeomorphism, Professor Devadoss demonstrates some of the many intriguing theorems and devices that have vastly expanded our understanding of shapes and the natural world.

These include:

The Poincaré conjecture, the milestone in mathematical thought that helps mathematicians distinguish spheres from other manifolds;

Voronoi diagrams, which aid in analyzing and interpreting the areas of influence that emerge from clusters of objects known as point clouds;

The Jones polynomial, a powerful polynomial that, so far, has been able to distinguish any knot from the unknot and is related to the ideas in string theory;

The Seifert algorithm, a famous algorithm that, using any given knot, is able to build an orientable surface whose boundary is that knot; and

Dehn surgery, a method of cutting, twisting, and gluing manifolds to form new ones.

Learning about the techniques involved in the study of nature's shapes will not only give you a thorough grasp of the world of topology, it will round out your understanding of the mathematical world by introducing you to the mathematical issues and concepts of the last quarter-century.

Enter a World of Mathematical Mystery

What makes The Shape of Nature so engrossing and accessible despite the complexity of its subject matter is the undeniable passion and teaching skills of Professor Devadoss. Adept at bringing to life the fascinating world of shapes and explaining the mathematics behind them, he makes each lecture a joy to listen to. In addition, the eye-grabbing animations and visual demonstrations that flavor these lectures—many crafted by Professor Devadoss himself—bring the multidimensional wonders of the world to vivid life.

"It's amazing how much nature holds in her mysteries," Professor Devadoss remarks with his characteristic excitement and enthusiasm. And by the concluding lecture of The Shape of Nature, as you bring to a close your journey through this mathematical territory rarely charted for the average individual, you'll undoubtedly find yourself nodding in agreement.

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36 Lectures
  • 1
    Understanding Nature
    Start your investigation with this introductory lecture that gives you an overview of the course, explains the critical importance of studying shapes at all scales of nature, and reveals why mathematics is the key to understanding these complexities. x
  • 2
    The Language of Shapes
    Discover why geometry and topology (the study of shapes through their relationships) are the best mathematical tools for grasping the shape of nature. Also, explore two important and recurring concepts: equivalence and dimension. x
  • 3
    Knots and Strings
    Knots and strings appear almost everywhere in nature: in chemistry (as knotted molecules), in biology (as the shape of DNA), and much more. In addition to examining strings, learn how three mathematical moves—known as the Reidmeister moves—allow us to study a knot's 2-D projection without altering the knot itself. x
  • 4
    Creating New Knots from Old
    Learn how to manipulate knots through addition and why the subtraction of knots is impossible. Then, see how two classic knot invariants—properties assigned to knots that don't change with deformation—play roles in two of knot theory's biggest unsolved problems. x
  • 5
    DNA Entanglement
    Two or more knots tangled together are called links. Professor Devadoss introduces you to fascinating examples of links (such as the Borromean rings), as well as two tools used to measure their different aspects: the linking number (the amount two knots are tangled) and the writhe (how twisted an individual knot is). x
  • 6
    The Jones Revolution
    Create from scratch the Jones polynomial, a powerful invariant that assigns not a number but an entire polynomial to any knot. As you construct the Jones polynomial, you gain insight into how mathematicians attack complex problems and the critical role of algebra in differentiating knots and links. x
  • 7
    Symmetries of Molecules
    One of the broadest issues in the study of shapes is symmetry. Investigate why the Jones polynomial is beautifully designed to help us examine the mirror images of knots, and how chemists use this power in their work with molecular compounds and topological stereoisomers. x
  • 8
    The Messy Business of Tangles and Mutations
    Move from the world of chemistry to the world of biology; specifically, the structure of DNA. Here, uncover how tangles—parts of a projection of a knot or link around a circle crossing it exactly four times—allow scientists to explore and discuss genetic mutation from a mathematical point of view. x
  • 9
    Braids and the Language of Groups
    Braiding is one of the oldest forms of pattern making and one of the most basic shapes related to knots and tangles. Learn how the concept of a group—one of the most important algebraic structures in mathematics—is a key tool for understanding the deeper structure of braids. x
  • 10
    Platonic Solids and Euler's Masterpiece
    Turn from one-dimensional shapes to two-dimensional shapes with this illuminating look at surfaces: the most important of all shapes. This lecture focuses on the five Platonic solids—the tetrahedron, cube, octahedron, dodecahedron, and icosahedron—as well as the impact of Euler's formula on the study of these and other polyhedra. x
  • 11
    Surfaces and a New Notion of Equivalence
    You've learned how to compare and distinguish knots—but how can you study surfaces? Motivated by an intriguing question about the Earth, explore the notion of homeomorphism (similarity of form) and discover why mathematics is not the study of objects but of the relationships between them. x
  • 12
    Reaching Boundaries and Losing Orientations
    Explore the wildest surfaces in topology: those that are non-orientable and possessing only one side. Professor Devadoss demonstrates how to build surfaces of any genus from gluing together simple polygons, shows how to classify every possible homeomorphically equivalent surface imaginable, and more. x
  • 13
    Knots and Surfaces
    Using your newfound understanding of surfaces, attack problems about knots. Is there an invariant of the genus of a surface? If so, how do you find it? How does addition make knotting more complicated? What are the distinctions between classic mutant knots? x
  • 14
    Wind Flows and Currents
    Turn back to the Earth and study more about its properties, this time focusing on how wind flows along its surface. Investigate how the relationship between vector fields and shapes helps us model the flow of wind and uncover the properties of how wind currents behave. x
  • 15
    Curvature and Gauss's Geometric Gem
    Curvature is one of the most important geometric notions ever discovered. Here, learn what it means for curves and surfaces to have curvature, and encounter the profound Gauss-Bonnet theorem, which shows that there is a deep relationship between the curvature of rigid geometrical surfaces and the flexible surfaces of topology. x
  • 16
    Playing with Scissors and Polygons
    Get a taste of discrete geometry in this lecture on its building block: the polygon. See how scissors congruency (when smaller pieces of one polygon can be rearranged to form another polygon) helps us grasp the perplexing shapes and designs found in the natural world. x
  • 17
    Bending Chains and Folding Origami
    Instead of cutting and gluing, focus on the problem of folding. In this lecture, explore how folding works in both the one-dimensional world (through linkages that can be seen in protein folding and robot motion) and the two-dimensional world (through origami, which is found in plant leaves and package design). x
  • 18
    Cauchy's Rigidity and Connelly's Flexibility
    Move from folding to the opposite side of the spectrum: rigidity. Here, Professor Devadoss leads a guided tour of Cauchy's Rigidity Theorem—which states that a convex polyhedron with rigid faces and flexible edges cannot be deformed—and Robert Connelly's result about the existence of flexible polyhedra. x
  • 19
    Mountain Terrains and Surface Reconstruction
    How can polygons help us approximate the geography of our planet? Find out the answer in this lecture on how graphing triangulations made from terrain data (in the form of point clouds) allow us to actually reconstruct mountain ranges and other large-scale terrains of the Earth. x
  • 20
    Voronoi's Regions of Influence
    Voronoi diagrams (regions of influence that emanate from two-dimensional point clouds) are useful for planning cities and studying populations, rainfall, and cell growth. Explore a powerful method for constructing these diagrams and discover the intriguing relationship that Voronoi diagrams have with triangulations. x
  • 21
    Convex Hulls and Computational Complexity
    Given a point set, the convex hull is the smallest convex set that contains the point set. Study how to compute the convex hull; use it to understand the similarities and differences in the ways mathematicians and computer scientists think; and examine what happens when this idea is pushed into data points in three dimensions. x
  • 22
    Patterns and Colors
    Investigate a series of theorems underlying the structure of patterns and the topology of coloring on both planes and spheres through an engaging question: If two adjacent countries on a map must have different colors, how many colors are needed to successfully color the map? x
  • 23
    Orange Stackings and Bubble Partitions
    What's the most efficient way to pack identical spheres as tightly as possible within a given space? What's the best way to partition space into regions of equal volume with the least surface area between them? Delve into the world of geometric optimization: geometry problems that deal with getting the most out of a situation. x
  • 24
    The Topology of the Universe
    Embark on your first full adventure into the three-dimensional realm with this lecture on 3-D objects known as manifolds. With an understanding of how to build manifolds through the familiar tool of multiplication, you'll be able to better grasp some of the possible shapes of our universe. x
  • 25
    Tetrahedra and Mathematical Surgery
    Explore in detail three different ways to construct all the possible shapes in the universe: gluing together the sides of some polyhedron; gluing together the boundaries of two solid surfaces (also known as handlebodies); and cutting and gluing knot and link complements (a process known as Dehn surgery). x
  • 26
    The Fundamental Group
    Look at one of the greatest and most useful invariants that assigns an algebraic group structure to shapes: the fundamental group. Also, get an introduction to a new form of equivalence—homotopy—that helps define the elements of this group, and learn how to calculate some fundamental groups of simple surfaces. x
  • 27
    Poincaré's Question and Perelman's Answer
    Now apply your newfound knowledge of the fundamental group to knots and 3-manifolds, specifically the 3-sphere. Bringing these concepts together, Professor Devadoss takes you through the greatest problem in the history of topology: the Poincaré conjecture, whose recent solution became a milestone in mathematical thought. x
  • 28
    The Geometry of the Universe
    Enter the world of space-time, the four-dimensional universe we inhabit that is bound by Einstein's revolutionary theories. As you look at the cosmos through a geometric lens, you learn how our understanding of two-dimensional and three-dimensional geometries grew and how these ideas were woven into Einstein's notions of special and general relativity. x
  • 29
    Visualizing in Higher Dimensions
    Venture into the frontiers of higher dimensions and discover how to understand them not only mathematically, but visually as well. How can colors and "movies" help you to actually see higher dimensions? What does the now-familiar concept of knots reveal about the powers of four dimensions? Uncover the eye-opening answers here. x
  • 30
    Polyhedra in Higher Dimensions
    Encounter two of the most famous polytopes (which describe polyhedra in arbitrary dimensions): the 120-cell made of dodecahedra and the 600-cell made of tetrahedra. Learn how to explore these fascinating objects and visualize them with Shlegel diagrams—a tool that lets us draw 4-D objects using 3-D tools. x
  • 31
    Particle Motions
    The key to grasping the world of higher dimensions, you'll find, can be based on simple notions such as particles moving back and forth along an interval. Here, use the language of a configuration space (a space containing all possible movements) and Shlegel diagrams to study particle motions on lines and circles. x
  • 32
    Particle Collisions
    Turn from particle motions to particle collisions. As you explore ways to manipulate and alter the configuration space of particles to study these collisions, you'll find yourself coming face to face with the associahedron—the most famous and influential polyhedron of the last 25 years. x
  • 33
    Evolutionary Trees
    What happens when you apply the idea of configuration spaces to theoretical biology and the study of genetics? In this lecture, learn how techniques of higher-dimensional study—in the form of phylogenetic "tree" structures—help reveal the relationship between certain organisms based on their genetic data. x
  • 34
    Chaos and Fractals
    Take a brief, enlightening excursion into the mysterious worlds of chaos theory and fractals. A highlight of this lecture: Professor Devadoss's engaging explanations of famous fractals such as the Sierpinski Triangle (a fractal built from infinite removals) and Koch's Snowflake (a fractal built from infinite additions). x
  • 35
    Reclaiming Leonardo da Vinci
    Many people believe that mathematics has always been connected with the sciences, not the humanities. Dispel that notion with this penultimate lecture that discusses how both iconic artists (like Leonardo and Dali) and contemporary artists (like Sol LeWitt and Julie Mehretu) push their masterpieces against the boundaries of shape. x
  • 36
    Pushing the Forefront
    Finish the course with a look back at all the fascinating mathematical territory you've charted in the previous 35 lectures. And as a final coda to The Shape of Nature, Professor Devadoss gives you a peek into what fruits current and future research in this revolutionary field may yield. x

Lecture Titles

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Satyan L. Devadoss
Ph.D. Satyan L. Devadoss
Williams College
Dr. Satyan L. Devadoss is Associate Professor of Mathematics at Williams College, where he has taught for more than eight years. Before joining the faculty of the Mathematics and Statistics Department at Williams, Professor Devadoss was a Ross Assistant Professor at The Ohio State University. He holds a Ph.D. in Mathematics from The Johns Hopkins University. Professor Devadoss has earned accolades for both his scholarship and his teaching. Among these awards are the William Kelso Morrill Award for excellence in teaching mathematics and the Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member, a national honor given by the Mathematical Association of America to honor faculty whose teaching has been extraordinarily successful and whose effectiveness in teaching undergraduate mathematics is shown to have influence beyond their own classrooms. Professor Devadoss's work in topology and computational geometry has also earned him numerous grants from the National Science Foundation, visiting positions at the University of California, Berkeley and The Ohio State University, and a position as a research member of the Mathematical Sciences Research Institute. In addition, he has published more than a dozen scholarly papers on mathematical subjects ranging from configuration spaces and cartography to origami and juggling.
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Reviews

Rated 4.1 out of 5 by 25 reviewers.
Rated 5 out of 5 by Fantastic Look at How Mathematicians Think I have a background in academic mathematics, and one of the things that struck me about this course was how well Prof. Devadoss incorporated not just the basics, but cutting-edge research (including some of his own research). In addition, Prof. Devadoss talks about how mathematicians think and attack problems generally, and what Prof. Devadoss said rang true to me. In terms of the other content, I thought it was fantastic: the presentation was easy-going enough for this format, yet detailed enough to satisfy someone looking for real mathematical details. I would recommend it to anyone with a mathematical bent. March 8, 2014
Rated 4 out of 5 by Where'd this Come From? In the beginning of this course I was totally confused, having explored nothing beyond beginning calculus in school. However, once I got used to the idea that everything in this course is totally new to me, and interesting in it's own right, I thoroughly enjoyed the new concepts. Lots of visuals helped enormously to a rudimentary understanding of some concepts, and the material was very well presented. If you like new ideas, go for this course. August 7, 2013
Rated 5 out of 5 by Astonishing and Wonderful. Give It a Chance! If you are into mathematics, and already know you have a liking for topology and higher-dimensional geometry, you will love this course. If you are not into math, you may *still* love this course! I urge you, very strongly and very sincerely, to give it a chance. It will introduce you to ways of thinking about the shape of the world, and especially about shape in mathematics, that you likely have never encountered before. More importantly, it will encourage your sense of wonder at the beauty of mathematics and at the incredible creativity of the human mind. The course is primarily an introduction to areas of advanced mathematics - topology and higher-dimensional geometry - with which most of us (including me) have zero experience. It is taught at a very elementary level, so that any reasonably intelligent, reasonably interested viewer will be able to get the basic ideas. And, Professor Devadoss is outstanding. He speaks beautifully, in crisp, clear, enthusiastic sentences that convey this complex material as well as I think humanly possible. He is the first science/math professor at The Great Courses - or anywhere else, for that matter - that I would consider the equal of my favorite prof, Alex Filippenko. The visuals are remarkably excellent as well, plentiful and wonderfully done, despite their illustration of very complex material. The Course Guidebook is the only disappointment - it is so brief and superficial as to be almost useless, with a very inadequate glossary. But - do not expect to come away with anything more than a very general, qualitative impression. If you think you understand any lecture fully, you are either a genius or a math professor, or you weren't listening. And do *not* think this is a course about nature. Despite the title, it is not. The few sentences connecting the math with the natural world in each lecture add little to the course, and were clearly encouraged by the editors as a selling point, since calling the course "An Introduction to Topology" would probably not have been a good marketing move. This course is pure, abstract math. It is also wonderful. If you take it with an open mind, not expecting it to give insight into anything in the "real" world of our everyday experience, but simply to appreciate the magnificence of the ideas and of the human mind, you may find it very, very rewarding. For these reasons - whether you think you like math or not - I highly recommend it. June 13, 2013
Rated 5 out of 5 by Passionate Topologist I have to say, when I took this course I knew nothing of topology, despite having a broad math background. And while the course was a bit hard in places, the professor kept the subject moving pretty well. What was most impressive was the professor's passion for topology and how well he could speak about topology with only a few misspeaks here and there, as well as his great use of body language and movement to communicate. In short a great lecturer. I wish many of the other instructors for these Great Courses would watch his presentation style and learn from him. Although I probably learned more about knots than I cared to, it was amazing to think how math could be used for such an application. Perhaps the most interesting thing was to learn and understand more about dimensions better. This course is probably not for those who just want an introductory lesson with few details, but if you really want to delve into the topic and be challenged, this course will provide a lot of fun. May 31, 2013
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