Mathematics from the Visual World

Course No. 1447
Professor Michael Starbird, Ph.D.
The University of Texas at Austin
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Course No. 1447
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Course Overview

Geometry has long been recognized not only as a fascinating skill, but as a gateway to the highest realms of human thought. Mathematics from the Visual World, taught by distinguished Professor Michael Starbird, introduces you to the terms, concepts, and astonishing power of geometry, including topology, conic sections, non-Euclidian geometry, congruence, and much more. In 24 richly illustrated lectures, you discover the important role this profound mathematical field plays in everything from algebra and calculus to cosmology and chemistry to art and architecture. This delightful, invigorating, and enlightening journey will allow you to master one of the most glorious inventions of the human mind.

Plato's Academy in Athens was the think tank of the ancient world and bore this motto over its door: "Let no one ignorant of geometry enter here." Ever since, geometry has been recognized as not only a useful and fascinating skill, but also as a gateway to the highest realms of human thought. Seemingly simple geometric ideas such as the Pythagorean theorem turn out to have profound implications in unexpected places, including our modern conception of space and time.

Mathematics from the Visual World, taught by veteran Teaching Company Professor Michael Starbird of The University of Texas at Austin, takes Plato's dictum to heart and introduces you to the terms, concepts, and astonishing power of geometry.

In 24 richly illustrated lectures, you learn that geometry is everywhere. It is the key to scientific disciplines from cosmology to chemistry. It is central to art and architecture. It provides deep insights into algebra, calculus, and other mathematical fields. And it is stunning to contemplate in its beauty.

Consider these intriguing applications of geometry:

  • Conic sections: Euclid and other ancient mathematicians investigated conic sections—the shapes produced by the intersection of a plane and a cone. Two thousand years later, Galileo, Kepler, and Newton discovered that these shapes describe the paths followed by free-falling objects in a gravitational field.
  • Non-Euclidean geometry: Euclidean geometry is simple and intuitive, and it appears to govern the world around us. But a nagging problem with Euclid's concept of parallel lines led to the discovery of new geometries in the 1800s. These non-Euclidean geometries accurately reflect phenomena in physics and other disciplines.
  • Topology: Under what conditions can a coffee cup and a doughnut be considered the same? When they are analyzed in topology—the branch of mathematics that deals with shapes that retain their identity after twisting and stretching. Topology captures fundamental geometric properties of objects, giving us a new perspective on reality.

Intellect and Eye

From the simplicity of the golden rectangle to the infinitely complex realm of fractals, no other area of mathematics is so richly endowed with interesting examples as geometry, which appeals to both the intellect and the eye. All of geometry's many applications make use of the bedrock concepts of axioms, theorems, and proofs. In Mathematics from the Visual World, you discover that these traditional techniques are not ends in themselves but tools for gaining new insights such as these:

  • In exploring the surprisingly diverse ways of defining the center of a triangle, you learn that one type of center, and the associated circle that inscribes the triangle with that center, led to a breakthrough in skin-grafting techniques for surgeons.
  • The unusual shape of art galleries, with many nooks and crannies, raises the question of how many security cameras suffice to protect the room. You learn creative strategies for attacking this problem and reaching a solution.
  • The shape of the universe itself is subject to simple geometric analysis. The observations themselves may be tricky, but Dr. Starbird shows that distinguishing among three possible geometries is relatively straightforward once we have the data.

On a more everyday level, you may be interested to know that the age-old problem of how to cut a square cake so that each piece has the same quantity of icing is easily solved.

Famous Problems

Geometry is also richly endowed with famous problems, some with life-or-death implications. Take the Delian Problem: Legend has it that in ancient Athens the citizens consulted the oracle at Delos for advice on how to stop a deadly plague. The oracle replied that the plague would end if the Athenians doubled the size of their cube-shaped altar to the god Apollo. So the Athenians doubled each side. But the plague continued unabated. The oracle had meant that they should double the altar's volume, not its linear dimensions.

Doubling the cube in this way is a classic problem from antiquity, which Professor Starbird proves is impossible to solve with the traditional tools of a straightedge and compass. However, in the 17th century Isaac Newton showed that the construction can be done if one is allowed to make two marks on the straightedge. Dr. Starbird explains how this clever trick works.

Here are some other famous problems that you investigate in Mathematics from the Visual World:

  • How large is the Earth? The problem of measuring the Earth was solved around 200 B.C. by the Greek mathematician Eratosthenes. All he needed were observations of the shadow cast by the sun at two particular locations on a special date—plus a bit of geometry.
  • Why is it dark at night? A geometrical argument by 19th-century German astronomer Heinrich Wilhelm Olbers proved that the universe cannot be infinite in size, infinitely old, and compositionally the same in all directions. Otherwise, the night sky would be ablaze with light—which it isn't.
  • Königsberg bridges: Walkers in 18th-century Königsberg in Prussia amused themselves by seeing if they could cross all seven bridges in the central city without passing over the same bridge twice. Mathematician Leonhard Euler showed there is no solution, laying the foundation for the field of graph theory.

A Delightful, Enlightening, and Invigorating Journey

A specialist in geometry and topology, Dr. Starbird is not only Professor of Mathematics at The University of Texas at Austin but also University Distinguished Teaching Professor. He has won an impressive array of teaching awards, including most of the major teaching awards at UT, a prestigious statewide teaching award, and the national teaching award from the Mathematical Association of America.

Professor Starbird believes that there is no excuse for a dull course on mathematics, a philosophy he pursues throughout Mathematics from the Visual World. In Lecture 1 he says, "To me, the satisfying aspect of a great proof occurs when the proof reveals some underlying, often surprising connection or relationship from which we see some truth that we previously could not fathom. When we see such a proof, we might say, 'Aha, that's why it's true.'" Although they don't always come easily, you have many such "aha" moments in this course.

An old story recounts that King Ptolemy of Egypt asked Euclid, the father of geometry, whether there was a simpler way to understand the axioms, theorems, and proofs of the subject. Euclid's famous answer was, "There is no royal road to geometry." However, now there is Professor Starbird's road, which is a delightful, enlightening, and invigorating journey through one of the most glorious inventions of the human mind.

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24 lectures
 |  Average 30 minutes each
  • 1
    Seeing with Our Eyes, Seeing with Our Minds
    Shapes, patterns, and forms have intrigued humans for millennia. You start your exploration of the world of geometry by examining the contributions of the ancient Greek mathematician Euclid, who wrote the most famous textbook in any subject for all time: the Elements. x
  • 2
    Congruence, Similarity, and Pythagoras
    What geometrical objects qualify as being the same? This lecture explores the concepts of congruence and similarity, which Professor Starbird uses to give two proofs of the Pythagorean theorem, including one discovered by Leonardo da Vinci. x
  • 3
    The Circle
    You investigate basic features of the circle, including its radius, diameter, circumference, and the famous constant pi. On the practical side, you learn that a belt that is snuggly encircling the Earth can be comfortably loosened by adding just a few feet to the circumference, and that manhole covers need not be circular. x
  • 4
    Centers of Triangles
    Delving into the hidden complexity of triangles, you discover the many ways of defining the center. There are the incenter, circumcenter, and orthocenter, to name just a few. Every triangle has circles naturally associated with it, which recently inspired an innovative technique for grafting skin. x
  • 5
    Surprising Complexity of Simple Triangles
    This lecture looks at three theorems about triangles that illustrate different strategies of proofs. The nine-point circle proof takes simple geometric properties and extends them to explain an amazing relationship. Napoleon's theorem can be proved with a process called tessellation. And the proof of Morley's Miracle proceeds backward! x
  • 6
    Clever Constructions
    Every student of Euclidean geometry learns how to construct basic geometric figures using a straightedge and a compass. You see how these methods reveal a connection between the construction of the golden rectangle and the regular pentagon. A surprisingly deep question is, Which of the other regular polygons can also be constructed? x
  • 7
    Impossible Geometry—Squaring the Circle
    You investigate three famous construction problems that were posed in antiquity and remained unsolved until the 1800s. Using a straightedge and a compass, is it possible to (1) double a cube, (2) trisect every angle, or (3) construct a square with the same area as a given circle? x
  • 8
    Classic Conics
    A plane passing through a right circular cone produces one of four classic shapes depending on the angle at which it intersects the cone. These "conic sections" are a circle, ellipse, parabola, or hyperbola. They arise frequently in physics; for example, the orbits of the planets are ellipses. x
  • 9
    Amazing Areas
    Professor Starbird starts with formulas for simple polygons such as a rectangle, a parallelogram, and a triangle. Then he shows how to deduce the area formulas for a circle and an ellipse. Finally, he demonstrates ingenious methods developed recently to compute the areas of various curved figures. x
  • 10
    Guarding Art Galleries
    How many security cameras are needed in an art gallery that has many nooks and crannies? You examine a clever proof that illustrates two effective strategies for analyzing the problem: divide and conquer, and seek essential ideas. The proof delivers an "aha" moment when the pieces fall into place. x
  • 11
    Illusive Perspective
    The challenge of depicting three dimensions on a two-dimensional plane leads you to an exploration of map projections, in which various strategies are used to render a globe on a flat surface. Artistic perspective is another technique for dealing with three dimensions on two. x
  • 12
    Planes in Space
    You investigate the method devised by the ancient Greek mathematician Archimedes for determining the volume of a sphere. Then you explore some surprising features of the two-dimensional plane that are revealed by projecting shapes into a third dimension. x
  • 13
    Cooling Towers and Hyperboloids
    Challenging you to imagine what a cube that is spinning on two opposite corners looks like, Professor Starbird uses this exercise to introduce a proof of Brianchon's theorem, in which you discover the fascinating properties of the architectural shape common to nuclear reactor cooling towers. x
  • 14
    A Non-Euclidean Spherical World
    The most controversial of Euclid's axioms was his parallel postulate, which mathematicians sought in vain to prove from Euclid's other axioms. Two millennia later, this problem led to the breakthrough of non-Euclidean geometries. One of these is spherical geometry, which you study in this lecture. x
  • 15
    Hyperbolic Geometry
    You explore hyperbolic non-Euclidean geometry, which has the property that for any point not on a given line there are infinitely many lines through the point that are all parallel to the line. A model for hyperbolic geometry called the Poincaré disk was the source for artistic work by x
  • 16
    The Dark Night Sky Paradox
    The dark night sky is proof that the universe is not infinitely expansive, infinitely old, and isotropic. You see how geometry is used to prove this and other features of the universe, including the size of the Earth and the nature of planetary orbits. x
  • 17
    The Shape of the Universe
    Is the universe best described as having spherical, hyperbolic, or Euclidean geometry? Another way of asking this question is, Does the universe have positive, negative, or zero curvature? You examine the possible observations that would help determine the true shape of the universe. x
  • 18
    The Fourth Dimension
    Higher-dimensional geometry is a rich domain with truly surprising insights. This lecture uses thought experiments in the first, second, and third dimensions to help you reason by analogy into the fourth dimension. Once you have this skill, there's no obstacle to going to even higher dimensions. x
  • 19
    Patterns of Patterns
    One of the most fundamental features of decorative designs is symmetry, seen in the repeated patterns on floor tiles, carpets, wall coverings, building ornamentation, screensavers, and paintings. You learn that different patterns have different ways of repeating. x
  • 20
    Aperiodic Tilings and Chaotic Order
    This lecture investigates Penrose and pinwheel tilings as illustrations of symmetry that is, paradoxically, at once orderly and chaotic. Such examples of aperiodic geometry have an uncanny ability to describe the real physical world and also lead to a new aesthetic sense. x
  • 21
    The Mandelbrot and Julia Sets
    Fractals have caught the popular imagination due to their beautiful complexity, and apparent symmetry and self-similarity. But how are they made? In this lecture, you see how infinitely intricate images arise naturally from repeating a simple process infinitely many times. Examples include Mandelbrot and Julia sets. x
  • 22
    Pathways to Graphs
    You focus on three famous geometric problems that relate to graph theory: the Königsberg bridge problem, the traveling salesman problem, and the four-color problem. Although easy to state, each leads into a fascinating thicket of mathematical ideas that can be explored with graphs. x
  • 23
    A Rubber-Sheet World
    Topology deals with shapes that retain their identity after twisting and stretching. For example, a coffee cup and a doughnut are topologically equivalent because each can be continuously deformed to produce the other. You look at surprising transformations that can occur in the topological realm. x
  • 24
    The Shape of Geometry
    Professor Starbird concludes by stepping back to survey the big picture of the geometrical questions explored during these lectures. From Euclid to fractals, the evolution of geometrical ideas over thousands of years is a model for how concepts spring from one another in marvelous profusion and grow in unexpected directions. x

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What's Included

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Video DVD
DVD Includes:
  • 24 lectures on 4 DVDs
  • 120-page printed course guidebook

What Does The Course Guidebook Include?

Video DVD
Course Guidebook Details:
  • 120-page printed course guidebook
  • Suggested readings
  • Questions to consider
  • Timeline

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Your professor

Michael Starbird

About Your Professor

Michael Starbird, Ph.D.
The University of Texas at Austin
Dr. Michael Starbird is Professor of Mathematics and University Distinguished Teaching Professor at The University of Texas at Austin, where he has been teaching since 1974. He received his B.A. from Pomona College in 1970 and his Ph.D. in Mathematics from the University of Wisconsin-Madison in 1974. Professor Starbird's textbook, The Heart of Mathematics: An Invitation to Effective Thinking, coauthored with Edward B. Burger,...
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Mathematics from the Visual World is rated 4.3 out of 5 by 32.
Rated 4 out of 5 by from This is not a review, but a request. I've watched several of your courses. A common experience with all of them is a desire, at one point or another, to be able to ask the teacher a question, or to expand on some point. Is it possible to institute a mechanism whereby a viewer can send the teacher a message and get a response to his question? If the course were "live" I would have, say, raised my hand when the question occurred to me, but since that's not possible, some sort of e-mail exchange might serve the purpose. I had to mark the rating down a notch for the lack of that capability, otherwise I was quite satisfied with the course.,. Thank you.
Date published: 2020-08-04
Rated 2 out of 5 by from Few Visuals for a Visual Course I was very disappointed in this course. It made very little use of actual visual images and most of the lectures featured Prof Starbird standing and talking. It was a course from the 1960s (or perhaps 1900) with no effort to take advantage of the marvelous programs, such as Matlab, which allow one to SHOW mathematics. I have purchased over 140 courses and this was the most boring course of all of them.
Date published: 2018-11-25
Rated 4 out of 5 by from Geometry Class on DVD is Great. I like "Mathematics from the Visual World". I bought the DVD version so I can watch it as many times as I want. Professor Starbird is very good. He uses a lot of visual aids and explains everything very well.
Date published: 2018-07-17
Rated 5 out of 5 by from Challenged me with new ideas In the interest of full disclosure, I have taken several of the Professor's courses and enjoyed them all. And I found this course to be up to his standards. I especially enjoyed the series beginning with the non-Euclidean geometry lecture and the ones that followed. I don't say the lectures before were not professional, informative and wonderfully presented; its just that my background includes experience in these areas. And Professor Starbird always has a few "zingers" of new information to arouse any student whose mind is wandering. One example in this course is the revelation of when it was understood - using geometry - that the world was round and not flat. Professor Starbird's love of and enthusiasm with mathematics is contagious and inspiring.
Date published: 2018-03-26
Rated 4 out of 5 by from This course is instructive even though it is less demanding than courses such as Prof. Wolfson "Physics in Your Universe" or Prof. Grossman "Thermodynamics". Less demanding meaning it is not requiring taking as much notes and replaying several sections. The importance of similar triangles and practical uses of it are exposed. On the practical side, Prof. Starbird mentions that, contrarily to a square, a triangle cannot change its shape without stretching or shortening of its sides and there is an illustration of the triangle forming the basic cell of a bridge's structure. Having studied geometry and calculus, I was astonished to discover how a simple shape such as a triangle can be linked to a level of complexity I had never imagined. The example of applying this complexity to skin grafts is worth noting. The exposition of non-planar geometry is also worthy of note. A practical application is given showing the shortest path an airplane can follow. I especially liked the last lecture which shows how geometry can be used to understand the convergence of a series such 1/4 + (1/4)^2 + (1/4)^3 + ... This one is nothing short of amazing. For someone who has never been exposed to geometry (in Greek, short for "Earth" and "measure" - it started out with the need to measure the surface of fields), chapter 4 provides an excellent illustration of applying a logical process of simple steps to prove a conclusion. This is where taking notes will be most likely be required. This provides an excellent introduction to logical thinking. Even if you find it difficult to follow, it is worth the effort if you are in any way curious. Note that this is definitely not a roadblock, you can skip it and move on to the next lectures. On the slightly minus side, Prof. Starbird could have eased the understanding of complex numbers by showing that multiplying a vector in the plane by i corresponds to a 90 degree rotation (counter-clockwise). Multiplying again by i means another rotation by 90 degrees, with the net result of multiplying a vector in the plane by -1 produces a rotation of 180 degrees - making it less difficult to swallow something like the square root of -1 which will be barfed out by your calculator if you try it. In short, another way of obtaining two orthogonal axes. Prof. Starbird also missed an opportunity to further expose the complexity of the Mandelbrot set by not showing that a Mandelbrot set can even exhibit perfect tri-axial symmetry, the so-called "Bad Mandelbrot set" instead of being merely symmetric with respect to the x-axis. No big deal, just a missed opportunity. In short, a good exposition of the various members of the geometry domain.
Date published: 2016-08-21
Rated 4 out of 5 by from Insight Laced with Obfuscation and Brilliance This is a course which is very worthwhile overall, but which varies widely in quality. Reading the reviews will give you a good idea of how various students have responded to this. Unlike some others, I appreciated the early, basic lectures which cover topics from high school geometry. Aside from being a helpful review, they often present the essentials in, for me, new and memorable ways. One example would be the discussion of the several concepts of the "center" of a triangle, and the surprising ways in which they are related. (Also unlike some others, btw, I do feel the course description gives a fair overview and summary.) Certainly the latter two thirds or so of the course covers more complex and engrossing material. The topics discussed are many, diverse, and of high intrinsic interest, even fascination. From the apparent simplicity of the art gallery guard problem to the intricacies of non-Euclidean geometry, all are worthy objects of study for anyone with an interest in mathematics, and a number of times I felt I had been granted an insight into ideas of sheer brilliance and beauty. The variable quality of the course, for me, rests primarily not with the topics but with the professor. Professor Starbird is undeniably a deeply knowledgeable and enthusiastic teacher who has devoted much of his career to sharing his love for mathematics with others. It is apparent that the mathematics is crystal-clear to him, and it is often explained clearly to us, with the help of excellent illustrations. The occasional problem is its translation into English. It sometimes seems as though our good professor's native language is mathematics, and that he has not quite mastered English as a second language when it is used to communicate mathematical concepts. And complexities are sometimes given rapid and less than lucid treatment, while simple concepts may be repeated past the point of helpfulness. Please understand - these concerns are not true of all, or even most, of the course. But this happens sufficiently often to be frustrating and to detract somewhat from the overall quality. So - I do recommend the course for all with an interest in mathematics, and particularly in "mathematics from the visual world", a.k.a. geometry and topology. Regardless of the caveats I have mentioned, I felt my time taking it was well spent.
Date published: 2016-06-13
Rated 5 out of 5 by from Mathematics Video. An enlightening and entertaining course superbly taught and presented. The animations graphics and video clips were outstanding. Very Highly Recommended!!
Date published: 2016-01-25
Rated 5 out of 5 by from Help for an H.S. student I sat with a high school student and together we went through this course. It was the first time she really enjoyed a high school math class. The presentations were excellent. She followed easily. After the class, she commented that she wished all of her H.S. math classes were this interesting. My comment was that I wished I'd had Prof. Starbird for my Topology class in Grad school. Thanks Prof. Starbird.
Date published: 2015-03-14
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