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Mathematics from the Visual World

Mathematics from the Visual World

Professor Michael Starbird Ph.D.
The University of Texas at Austin

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Mathematics from the Visual World

Mathematics from the Visual World

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

About This Course

24 lectures  |  30 minutes per lecture
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.
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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
  • 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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Michael Starbird
Ph.D. Michael Starbird
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, won a 2001 Robert W. Hamilton Book Award. Professors Starbird and Burger also collaborated on Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, published in 2005. Professor Starbird has won many teaching awards, including the Mathematical Association of America's 2007 Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics, which is the association's most prestigious teaching award. It is awarded nationally to 3 people from its membership of 27,000. Professor Starbird is interested in bringing authentic understanding of significant ideas in mathematics to people who are not necessarily mathematically oriented. He has developed and taught an acclaimed class that presents higher-level mathematics to liberal arts students.

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Reviews

Rated 4.3 out of 5 by 23 reviewers.
Rated 5 out of 5 by Visually Enticing My first thought to share is that this would be a very hard course to teach! Proofs of visual/geometric concepts are not easy. The course is very wide ranging, extending from Euclidian geometry to symmetric patterns, Julia sets and topology. The course only requires some high school level geometry and a minor bit of trigonometry. One lesson uses rudimentary complex variables, but that is one of the few places that the mathematically (un)inclined might be stumped. The professor's use of well-designed visual props is very good. The professor does more with triangular geometry than anything I have ever seen! Interesting stuff. August 5, 2014
Rated 4 out of 5 by Entertaining Math In the early part of the course, it was fun to work out the proofs myself. As the course progressed, the math came to be beyond my scope for solving, but interesting as new information. Professor Starbird has a good way of holding your attention. August 22, 2013
Rated 5 out of 5 by Wakes me up! This lecturer sneaks up on your brain, and presents complex ideas in a very accessible way. For example, he had someone (his wife?) make a patchwork representing a hyperbolic surface with straight lines. It looks too ordinary to be 'maths', but there it is! January 29, 2013
Rated 5 out of 5 by Fun, Interesting, and Valuable This is a great course not just for review of rusty math skills from years ago, but for getting a clear and visual approach that high school and college did not provide. There were also some new topics and perspectives that I really appreciated. Professor Starbird is just wonderful with his enthusiasm and use of supportive visuals. I would buy another of his courses in a moment. I did find the background fake looking window, red brick and plant distracting and irritating. I think this would be improved by having a background of chalkboard, whiteboard, or any other means to display supporting material that the professor would like to bring in. That irritation was not enough to bring down my rating though - I loved the course and professor - 5 stars all the way. August 5, 2011
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