How do you mathematically describe the world around you? Discover the fascinating answer with The Shape of Nature, a visually stunning and authoritative guide to the mathematical shapes around you: how they're formed, how they're studied, and how they relate to your life.
The Shape of Nature
Discover the intricate relationship between nature and mathematics with this visually stunning guide to the mathematical shapes all around you.
Overview
About
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.
01: 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.
32 min
02: 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.
30 min
03: 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.
30 min
04: 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.
30 min
05: 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).
29 min
06: 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.
32 min
07: 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.
31 min
08: 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.
31 min
09: 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.
33 min
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.
32 min
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.
31 min
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.
32 min
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?
32 min
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.
33 min
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.
33 min
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.
32 min
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).
31 min
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.
31 min
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.
32 min
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.
32 min
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.
31 min
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?
30 min
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.
31 min
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.
29 min
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).
31 min
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.
31 min
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.
32 min
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.
32 min
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.
32 min
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.
31 min
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.
31 min
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.
30 min
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.
30 min
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).
31 min
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.
29 min
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.
31 min
