Watch Geometry: An Interactive Journey to Mastery

2014
1 Season

Geometry is an essential branch of mathematics that provides a foundation for a wide range of scientific and technical disciplines, including engineering, physics, architecture, and computer graphics. The Great Courses presents a fascinating exploration of this rich subject in their latest offering, Geometry: An Interactive Journey to Mastery.

Hosted by Professor James Tanton, an accomplished mathematician and educator, this course consists of twenty-four lectures that take learners on a fascinating journey through the world of geometry. The course is designed for curious minds of all levels, from high school students to professionals seeking to deepen their understanding of the subject.

The course explores the fundamental building blocks of geometry, including points, lines, and angles, and shows how they combine to create planes, polygons, and other shapes. Professor Tanton takes a hands-on approach, inviting learners to interact with him as he explores the intricacies of geometry. He uses real-world examples and interactive exercises to help learners develop an intuitive understanding of these concepts.

One of the unique features of this course is its highly interactive nature. Throughout the lectures, Professor Tanton poses questions and challenges to the viewer, inviting them to pause the video and work out the answers on their own. Learners can then compare their answers with the professor's, gaining valuable feedback and insights.

The course also introduces learners to the famous geometrical constructions developed by ancient mathematicians, such as Euclid, Archimedes, and Apollonius. These constructions use only a straightedge and a compass to create intricate shapes and patterns, such as regular polygons, tangents to circles, and golden ratios. Professor Tanton demonstrates these constructions in real-time, giving learners a unique insight into the mathematical thinking of these brilliant minds.

The course also explores the connection between geometry and algebra, showing how algebraic formulas can be used to express geometric properties. Learners will discover how to use algebraic techniques to solve geometrical problems, such as finding the area of a triangle, the volume of a cylinder, or the distance between two points.

Another exciting aspect of this course is its emphasis on creativity and discovery. Professor Tanton encourages learners to experiment with geometry, testing out different configurations and exploring what happens when they change the shape or size of a figure. By engaging in this type of experimentation, learners will develop a deeper understanding of geometry and a greater appreciation for its beauty and elegance.

The course also covers some of the more advanced topics in geometry, such as non-Euclidean geometries, topology, and fractals. These topics may be challenging for some learners, but Professor Tanton's clear explanations and engaging presentation style will help make them accessible and understandable.

Overall, Geometry: An Interactive Journey to Mastery is an outstanding course that offers a comprehensive and engaging exploration of one of the most important areas of mathematics. Whether you're a student, a teacher, or a curious adult, this course is guaranteed to provide you with new insights and a deeper appreciation for the power and beauty of geometry. So why not embark on this journey today and discover the wonders of geometry for yourself?

Geometry: An Interactive Journey to Mastery is a series that ran for 1 seasons (36 episodes) between May 1, 2014 and on The Great Courses

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Seasons

36. Bending the Axioms - New Geometries

May 2, 2014

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 geometry is never a closed-book experience.

35. Complex Numbers in Geometry

May 2, 2014

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. Examine how using complex numbers can help solve several tricky geometry problems.

34. The Geometry of Figurate Numbers

May 2, 2014

Ponder another surprising appearance of geometry - the mathematics of numbers and number theory. Look into the properties of square and triangular numbers, and use geometry to do some fancy arithmetic without a calculator.

33. The Geometry of Braids - Curious Applications

May 2, 2014

Wander through the crazy, counterintuitive world of rotations. Use a teacup and string to explore how the mathematics of geometry can describe an interesting result in quantum mechanics.

32. Dido's Problem

May 2, 2014

If you have a fixed-length string, what shape can you create with that string to give you the biggest area? Uncover the answer to this question using the legendary story of Dido and the founding of the city of Carthage.

31. The Mathematics of Fractals

May 2, 2014

Explore the beautiful and mysterious world of fractals. Learn what they are and how to create them. Examine famous examples such as Sierpinski's Triangle and the Koch Snowflake. Then, uncover how fractals appear in nature - from the structure of sea sponges to the walls of our small intestines.

30. The Mathematics of Symmetry

May 2, 2014

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.

29. Folding and Conics

May 2, 2014

Use paper-folding to unveil sets of curves: parabolas, ellipses, and hyperbolas. Study their special properties and see how these curves have applications across physics, astronomy, and mechanical engineering.

28. Tilings, Platonic Solids, and Theorems

May 2, 2014

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 and create your own beautiful patterns - even using more than one type of shape.

27. The Reflection Principle

May 2, 2014

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 ball? Play with these questions and more through an exploration of the reflection principle.

26. Exploring Geometric Constructions

May 2, 2014

Let's say you don't have a marked ruler to measure lengths or a protractor to measure angles. Can you still draw the basic geometric shapes? Explore how the ancient Greeks were able to construct angles and basic geometric shapes using no more than a straight edge for marking lines and a compass for drawing circles.

25. Playing with Geometric Probability

May 2, 2014

Unite geometry with the world of probability theory. See how connecting these seemingly unrelated fields offers new ways of solving questions of probability - including figuring out the likelihood of having a short wait for the bus at the bus stop.

24. Introduction to Scale

May 2, 2014

If you double the side-lengths of a shape, what happens to its area? If the shape is three-dimensional, what happens to its volume? In this lecture, you explore the concept of scale. You use this idea to re-derive one of our fundamental assumptions of geometry, the Pythagorean theorem, using the areas of any shape drawn on the edges of the right triangle - not just squares.

23. Three-Dimensional Geometry - Solids

May 2, 2014

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. Finally, you study the properties (like volume) of these shapes.

22. Explorations with Pi

May 2, 2014

We say that pi is 3.14159 ... but what is pi really? Why does it matter? And what does it have to do with the area of a circle? Explore the answer to these questions and more - including how to define pi for shapes other than circles (such as squares).

21. Understanding Area

May 2, 2014

What do we mean when we say "area"? Explore how its definition isn't quite so straightforward. Then, work out the formula for the area of a triangle and see how to use that formula to derive the area of any other polygon.

20. The Equation of a Circle

May 2, 2014

In your study of lines, you used the combination of geometry and algebra to determine all kinds of interesting properties and characteristics. Now, you'll do the same for circles, including deriving the algebraic equation for a circle.

19. The Geometry of a Circle

May 2, 2014

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 inscribed angle theorem and Thales' theorem.

18. What Is the Sine of 1°?

May 2, 2014

So far, you've seen how to calculate the sine, cosine, and tangents of basic angles (0°, 30°, 45°, 60°, and 90°). What about calculating them for other angles - without a calculator? You'll use the Pythagorean theorem to come up with formulas for sums and differences of the trig identities, which then allow you to calculate them for other angles.

17. Trigonometry through Right Triangles

May 2, 2014

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 measure the heights of buildings.

16. Circle-ometry - On Circular Motion

May 2, 2014

How can you figure out the "height" of the sun in the sky without being able to measure it directly with a ruler? Follow the path of ancient Indian scholars to answer this question using "angle of elevation" and a branch of geometry called trigonometry. You learn the basic trig identities (sine, cosine, and tangent) and how physicists use them to describe circular motion.

15. The Classification of Triangles

May 2, 2014

Continue the work of classification with triangles. Find out what mathematicians mean when they use words like scalene, isosceles, equilateral, acute, right, and obtuse. Then, learn how to use the Pythagorean theorem to determine the type of triangle (even if you don't know the measurements of the angles).

14. Exploring Special Quadrilaterals

May 2, 2014

Classify all different types of four-sided polygons (called quadrilaterals) and learn the surprising characteristics about the diagonals and interior angles of rectangles, rhombuses, trapezoids, and more. Also see how real-life objects - like ironing boards - exhibit these geometric characteristics.

13. A Return to Parallelism

May 2, 2014

Continue your study of parallelism by exploring the properties of transversals (lines that intersect two other lines). Prove how corresponding angles are congruent, and see how this fact ties into a particular type of polygon: trapezoids.

12. Equidistance - A Focus on Distance

May 2, 2014

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.

11. Making Use of Linear Equations

May 2, 2014

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 a point and a line.

10. Practical Applications of Similarity

May 2, 2014

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 width of a river using only a baseball cap.

9. Similarity and Congruence

May 2, 2014

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 principle in 600 B.C.E. to impress the Egyptians by calculating the height of the pyramids.

8. Proofs and Proof Writing

May 2, 2014

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.

7. The Nature of Parallelism

May 2, 2014

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, it's possible to compute the circumference of the Earth just by using shadows!

6. Distance, Midpoints, and Folding Ties

May 2, 2014

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.

5. The Pythagorean Theorem

May 2, 2014

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 how it's a critical foundation for the rest of geometry.

4. Understanding Polygons

May 2, 2014

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 them mathematically? What are their properties? Find out the answers to these questions and more.

3. Angles and Pencil-Turning Mysteries

May 2, 2014

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 on the surface of a sphere. With this exercise, Professor Tanton introduces you to the difference between flat and spherical geometry.

2. Beginnings - Jargon and Undefined Terms

May 2, 2014

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 vertical angle theorem.

1. 1. Geometry - Ancient Ropes and Modern Phones

May 1, 2014

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. #Science & Mathematics

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Where to Watch Geometry: An Interactive Journey to Mastery

Geometry: An Interactive Journey to Mastery is available for streaming on the The Great Courses website, both individual episodes and full seasons. You can also watch Geometry: An Interactive Journey to Mastery on demand at Amazon Prime, Amazon and Hoopla.