# James Tanton

### James Tanton >

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Geometry—Ancient Ropes and Modern Phones

Episode 1 of Geometry

Episode 1 of Geometry

Explore the origins of one of the oldest branches of mathematics. See how geometry not only deals with practical concerns such as mapping, navigation, architecture, and engineering, but also offers an intellectual journey in its own right--inviting big, deep questions.

Beginnings—Jargon and Undefined Terms

Episode 2 of Geometry

Episode 2 of Geometry

Lay the basic building blocks of geometry by examining what we mean by the terms point, line, angle, plane, straight, and flat. Then learn the postulates or axioms for how those building blocks interact. Finally, work through your first proof--the…

Understanding Polygons

Episode 4 of Geometry

Episode 4 of Geometry

Shapes with straight lines (called polygons) are all around you, from the pattern on your bathroom floor to the structure of everyday objects. But although we may have an intuitive understanding of what these shapes are, how do we define…

Distance, Midpoints, and Folding Ties

Episode 6 of Geometry

Episode 6 of Geometry

Learn how watching a fly on his ceiling inspired the mathematician Rene Descartes to link geometry and algebra. Find out how this powerful connection allows us to use algebra to calculate distances, midpoints, and more.

The Nature of Parallelism

Episode 7 of Geometry

Episode 7 of Geometry

Examine how our usual definition of parallelism is impossible to check. Use the fundamental assumptions from the previous lectures to follow in Euclid's footsteps and create an alternative way of checking if lines are parallel. See how, using this result,…

The Pythagorean Theorem

Episode 5 of Geometry

Episode 5 of Geometry

We commonly define the Pythagorean theorem using the formula a2 + b2 = c2. But Pythagoras himself would have been confused by that. Explore how this famous theorem can be explained using common geometric shapes (no fancy algebra required), and…

Angles and Pencil-Turning Mysteries

Episode 3 of Geometry

Episode 3 of Geometry

Using nothing more than an ordinary pencil, see how three angles in a triangle can add up to 180 degrees. Then compare how the experience of turning a pencil on a flat triangle differs from walking in a triangular shape…

Complex Numbers in Geometry

Episode 35 of Geometry

Episode 35 of Geometry

In lecture 6, you saw how 17th-century mathematician Rene Descartes united geometry and algebra with the invention of the coordinate plane. Now go a step further and explore the power and surprises that come from using the complex number plane.…

Proofs and Proof Writing

Episode 8 of Geometry

Episode 8 of Geometry

The beauty of geometry is that each result logically builds on the others. Mathematicians demonstrate this chain of deduction using proofs. Learn this step-by-step process of logic and see how to construct your own proofs.

Making Use of Linear Equations

Episode 11 of Geometry

Episode 11 of Geometry

Delve deeper into the connections between algebra and geometry by looking at lines and their equations. Use the three basic assumptions from previous lectures to prove that parallel lines have the same slope and to calculate the shortest distance between…

Bending the Axioms—New Geometries

Episode 36 of Geometry

Episode 36 of Geometry

Wrap up the course by looking at several fun and different ways of reimagining geometry. Explore the counterintuitive behaviors of shapes, angles, and lines in spherical geometry, hyperbolic geometry, finite geometry, and even taxi-cab geometry. See how the world of…

Three-Dimensional Geometry—Solids

Episode 23 of Geometry

Episode 23 of Geometry

So far, you've figured out all kinds of fun properties with two-dimensional shapes. But what if you go up to three dimensions? In this lecture, you classify common 3-D shapes such as cones and cylinders, and learn some surprising definitions.…

The Reflection Principle

Episode 27 of Geometry

Episode 27 of Geometry

If you're playing squash and hit the ball against the wall, at what angle will it bounce back? If you're playing pool and want to play a trick shot against the side edge, how do you need to hit the…

Tilings, Platonic Solids, and Theorems

Episode 28 of Geometry

Episode 28 of Geometry

You've seen geometric tiling patterns on your bathroom floor and in the works of great artists. But what would happen if you made repeating patterns in 3-D space? In this lecture, discover the five platonic solids! Also, become an artist…

Equidistance—A Focus on Distance

Episode 12 of Geometry

Episode 12 of Geometry

You've learned how to find the midpoint between two points. But what if you have three points? Or four points? Explore the concept of equidistance and how it reveals even more about the properties of triangles and other shapes.

The Mathematics of Symmetry

Episode 30 of Geometry

Episode 30 of Geometry

Human aesthetics seem to be drawn to symmetry. Explore this idea mathematically through the study of mappings, translations, dilations, and rotations--and see how symmetry is applied in modern-day examples such as cell phones.

Similarity and Congruence

Episode 9 of Geometry

Episode 9 of Geometry

Define what it means for polygons to be "similar" or "congruent" by thinking about photocopies. Then use that to prove the third key assumption of geometry--the side-angle-side postulate--which lets you verify when triangles are similar. Thales of Ionia used this…

Practical Applications of Similarity

Episode 10 of Geometry

Episode 10 of Geometry

Build on the side-angle-side postulate and derive other ways of testing whether triangles are similar or congruent. Also dive into several practical applications, including a trick botanists use for estimating the heights of trees and a way to measure the…

Trigonometry through Right Triangles

Episode 17 of Geometry

Episode 17 of Geometry

The trig identities you explored in the last lecture go beyond circles. Learn how to define all of them just using triangles (expressed in the famous acronym SOHCAHTOA). Then, uncover how trigonometry is practically applied by architects and engineers to…

The Geometry of a Circle

Episode 19 of Geometry

Episode 19 of Geometry

Explore the world of circles! Learn the definition of a circle as well as what mathematicians mean when they say things like radius, chord, diameter, secant, tangent, and arc. See how these interact, and use that knowledge to prove the…