Shape of Nature

Course No. 1460
Professor Satyan L. Devadoss, Ph.D.
Williams College
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Course Overview

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
 |  Average 31 minutes each
  • 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

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Your professor

Satyan L. Devadoss

About Your Professor

Satyan L. Devadoss, Ph.D.
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...
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Reviews

Shape of Nature is rated 3.9 out of 5 by 55.
Rated 2 out of 5 by from Highly-Specialised Course for Mathematicians Regret to say I could not finish this course as it deals with very high-level, abstract mathematical theories which, as far as I could determine, have no application or relevance to my world or life. Professor Devadoss is clearly very enthusiastic & excited about his subject; he is eminently qualified, earned his doctorate at an early age. He comes across as a little smug, which is not exactly endearing or encouraging. The course should, I feel, be re-named: perhaps "The Shapes of Mathematical Theories" OR "Topology: Mathematical Shapes" would be more descriptive of the actual content. The inclusion of "Nature" strikes me as somewhat misleading unfortunately. For those with a very profound, absorbing interest in higher mathematics (and preferably with at least a basic degree in maths or a directly-related field), this course may well be a treat and a compelling intellectual excercise, but I found it inscrutable and bewildering. I have bought more than 70 Great Courses and have benefitted greatly from most, but this particular course is so extremely specialised that it does not seem to fit in. Sorry.
Date published: 2012-07-21
Rated 1 out of 5 by from Shape of Nature I have completed 7 other "Great Courses" in mathematics, and all were good-to-excellent. This topology course is awful, and I could not finish it. Perhaps it is my problem, but I could not stand the professor, particularly his repeated (and I mean repeated and repeated) self-congratulatory pap about the superiority of mathematicians. The content was fine, but I honestly could no longer bring myself to watch because so much time was wasted on the professor's repetitive auto-ego-stroking. Truly obnoxious.
Date published: 2012-07-05
Rated 5 out of 5 by from perceptual gymnastics Brace yourself for and engaging and accessible introduction to the history and theory of geometry and topology. Dr. Devadoss covers an expanse of subject material that leaves one hungry for more. The course guide provides a bibliography and other suggested reading which is well worth exploring to plumb deeper into the depths of the mathematics and its theorists (Euler, Poincare, Weeks, Yau). This course is a joy to watch if you are not a mathematician but are looking for something beyond the superficial; an intellectual challenge in one, two, three or more dimensions.
Date published: 2012-06-09
Rated 4 out of 5 by from Great course but difficult for average learner I bought this course expecting that this course would be lot about exploring mathematical designs in flowers, plants, animals and human structures. The course has something on that line, but most of the content is on topology. I am a medical doctor and I did not understand most of what the professor was talking, may be it is above my head, I don't know. But overall I recommend this course because it gives a good introduction to topology. The professor in one lecture says that mathematics is not just numbers, not just equations, it is about ideas. So, universe built on mathematics is built upon ideas. Ideas point us to a Mind, This one point I quickly noted and used in essay I have been writing on Design Argument, God as the Author of all ideas on which universe is built upon. Overall, this is a great course. Thanks to Teaching Company and the Professor Devadoss. He reminds me Ramanujan, a great Indian mathematician.
Date published: 2012-05-15
Rated 2 out of 5 by from An Honorable Effort I like this guy and I want him to succeed--for his sake and mine together--but I don't think he's solved his very challenging problem of presentation here. You could tell he'd given a lot of thought to making his material accessible but at any number of points, insights that he thought self-evident or proven came across as not at all obvious to me. I found myself wishing for his breakout discussion sessions. Aside from that, I'd love to sit with him through a frame-by-frame analysis as we worked to make a potentially important course worth his time and effort. I hope GC gives him another chance but this isn't there yet.
Date published: 2012-05-03
Rated 5 out of 5 by from Superb, absolutely wonderful, fantastic I have picked up books on Topology before and not gotten very much out of them. The Shape of Nature has gone way beyond my expectations. I wish I could give even more stars for this course - it is outstanding. The Professor is everything I've ever wanted in a math instructor and even more. His discussions are clear and the enthusiasm and love of math is so delightful. His ability to convey information that I have been looking for in a very understandable way is thrilling, totally exciting. Now when I look back at the topology books they make more sense. I will look for more math courses by Devadoss. Thank you so much for providing this course.
Date published: 2012-02-08
Rated 5 out of 5 by from Excellent Topology Course Just finished "Shape of Nature" and was completely fascinated by the material and the excellent presentation by the lecturer, Dr.Devadoss. 5 Stars. But . . . As has been pointed out by others, this is a challenging MATH course and anyone who is not interested in doing math at a fairly high level should steer clear of this offering. The course description is somewhat misleading and does not suggest the level of abstract mathematical thought needed to absorb the difficult material. Despite having a BA in math, I had little previous experience with Topology and was amazed at the variety of new topics introduced and also the excellent presentation of recent proofs and developments from 1980-2005. Very well done. Perhaps the title should be changed to "Topology: The Shape of Nature" as to not mislead. Dr Devadoss and Dr. Ed Burger both teach at Williams College and I envy the students who get to study with these excellent teachers. I would welcome any new courses offered by either of these men.
Date published: 2011-06-04
Rated 4 out of 5 by from Lots of shapes, not much nature I'm considering returning this course. Not because the lecturer is good--he's a terrific lecturer and charming to boot, Not because I didn't learning anything--I learned a lot about things I didn't know exist, Not because the course and visuals are not well presented--they are exquisitely presented. So why would I return it? Because I think it is falsely advertised. The connections between the math and nature are never completed. Most lectures start with a series of scientific areas to which the math to follow is applicable. Then comes the math. But I am always frustrated that true application of the math to the science problem is never completed. For example, Lecture 31 poses the problem of avoiding collisions between robots--a fascinating question. It then goes on to describe configuration spaces, but it never comes back to how one actually solves the problem of how robot collisions are avoided. He's left me hanging. I'm convinced that this problem can be solved using the idea of configuration, I just don't know how. Another example: The shortest distance between two points, the geodesic, a powerful concept in general relativity. But I never actually saw the application to issues in space-time. In the end, coitus interuptus.
Date published: 2011-05-16
Rated 5 out of 5 by from SUPER COURSE! This is a fun and comprehensive mathematics course and as some reviewers have said, may have benefited from a more explicit title, perhaps something with the word MATH in it, explaining and summarizing its' lecture titles. We're really glad the Teaching Company offers this delightful math course, especially since their courses are in general decidedly math-averse. That aside, Professor Devadoss is brilliant and he covers an encyclopedic range of topics in the only coherent shorthand possible given the time constraints.Some of us really do love to play with math the way jazz musicians jam and this course understand us. If you do, and aren't a closet torus, then you'll relish this review and renew that takes math far beyond basic accounting principles into the SciFi realms of H:O:H versus Ice Ih, or for un-chemists, tepid bath water versus Mandelbrot snowflakes. If I've lost you, try another course first.
Date published: 2010-10-19
Rated 2 out of 5 by from Should be retitled to "The Mathematics of Shape" I purchased this course thinking it would give me greater insight into the shapes and patterns that we see in nature and our universe. Instead, about 75% of the course was on painful mathematical proofs of abstract theories and ideas that have very little to do with nature. Maybe 15% of the course actually had some examples in reality. All others were mostly speculative theories. If you're looking for a purely mathematics course on shapes with very little to do with reality, this is a good course. Needless to say, I returned the course. On a positive note, the professor is quite dynamic and enthusiastic about the subject.
Date published: 2010-09-07
Rated 5 out of 5 by from shape of nature is mostly math i thought that the course name pointed more to the physics of nature and the nature of things in respect to reality and real properties of functional shapes (like proteins functionality) but it's mostly pure math and the attempt to understand it and reason with it about the world. it is boring to the people that math isn't their subject of interest but a real pleasure to the math-wiz!!!
Date published: 2010-08-25
Rated 4 out of 5 by from Very Interesting Professor Devadoss clearly knows and loves his subject. That said, this course is not for the faint of heart or those who have difficulty with mathematical concepts. The wide ranging topics are well presented but many suffer from a lack of depth. My preference would have been to cover fewer topics but in more detail. I would also have liked to have seen a more thorough linkage between the mathematical topics and natural shapes or objects. Too often the link was mentioned but not really illustrated. Overall, however, I found the course very interesting. An aside having more to do with production values than course content is that the numerous closely spaced parallel lines in the new set create distracting visual illusions when the lecturer moves around constantly as does Dr. Devadoss.
Date published: 2010-07-19
Rated 5 out of 5 by from very advanced content Professor Satyan Devadoss presented very advanced material from Knot Theory, Algebraic Topology, etc... in about as clear a manner as is humanly possible. Using many visual aids, animations, and physical models, Professor Devadoss explains the mathematics of knots, braids, manifolds, ... including applications in physics and biology. BEWARE: Sometimes sketching out the proofs of some of the famous (if you're a mathematician) theorems, this course is not for the mathematically faint of heart.
Date published: 2010-07-05
Rated 5 out of 5 by from Awesome I have a background in theoretical mathematics, but never studied knot theory or the topology of surfaces, etc. Professor Devadoss manages to distill complex subjects into very comprehensible images, and to document his proofs in an informal but very convincing manner. The illustrations are breathtaking and add much to the discussions. Thank you.
Date published: 2010-06-25
Rated 5 out of 5 by from Absolutely Delightful. One of the very best. I have over 300 TC courses (including the old VHS tapes) and I enjoy virtually all of them. Even with that said, this one has become one of my top-5 favorites. First, Professor Devadoss is very comfortable in front of a camera. This seems like a point that goes without saying but many advanced math professors of this caliber have a tendency to be introverted. They often stare at their shoes or their notes while they talk instead of making eye-contact with the camera (the students). It's a unfortunate cosmic fact of life that engaging math teachers are just not as easy to find as engaging history teachers. Hats off to TC for recruiting a dynamic and captivating speaker for this course. Dr Devadoss' enthusiasm and excitement often made me smile as I watched the presentations. Second, this course is jam packed with mind-bending mathematics. There’s very little overlap with other TC math courses. Lectures #1 - #5 sets you up with background material… some terminology… and then BAM in lecture #6, you get a healthy dose of the Jones Polynomial. Wow! This is concrete math. He shows algebraic thinking and manipulation applied to the categorization of knots. Fabulous stuff. Lectures #6 to #34 are the meat of this course covering amazing territories of topology and geometry. The last two lectures #35 and #36 do not introduce new math topics. In lecture #35, he offers his take on the intersection of math and art. The last lecture #36 has 2 parts: the first half is a recap and the second half emphasizes the many mathematical questions that are still unsolved. Throughout the lectures, 2 points constantly struck me: #1-- The math topics assembled by professor Devadoss are accessible to any motivated student with knowledge of high-school algebra. However, the material is not diluted down to just pretty pictures or pseudomath. Most lectures have equations that require multiple viewings to appreciate. Very challenging. I think Dr. Devadoss found the right balance between lightweight metaphors that just talk around the math vs impenetrable proofs that require graduate training to understand. This course continues the TC tradition of carving out various pieces from advanced mathematics and making them accessible to a wider audience. #2 -- The modern math in this course absolutely requires modern presentation techniques. You could imagine that simpler mathematics such as the Pythagorean theorem could have be disseminated in ancient times by drawing lines and triangles in the sand or sketching on cave walls. However, teaching the concepts of multi-dimensional geometry would be tedious if not impossible without the aid of computer generated graphics. This type of material requires the combination of a skilled teacher along with dynamic visuals which plays perfectly into The Teaching Company’s strengths of delivering multimedia learning products.
Date published: 2010-05-19
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