Mathematics from the Visual World

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*.

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.

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.

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.

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!

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?

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?

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.