Geometry: An Interactive Journey to Mastery

Overview

Course No. 1033

Inscribed over the entrance of Plato’s Academy were the words, “Let no one ignorant of geometry enter my doors.” To ancient scholars, geometry was the gateway to gaining a profound knowledge of the world. Today, geometry’s core skills of logic and reasoning are essential to success in school, work, and many other aspects of life. Like other math fields, geometry teaches us how to think. It leads students to uncover new truths based on already established ideas and facts. It requires us to test and examine the conclusions of others. It teaches us to lay out our thinking clearly, describing each step so that others can follow along and verify our results. This systematic way of thinking is essential in many fields. Drawing conclusions from experimental data is the basis of scientific discovery. Our justice system depends on compelling evidence to render a verdict in a court of law. And we use logical reasoning in everyday conversations to win friends over to our point of view. In fact, the great Greek scholar Euclid demonstrated just how much you can do with logic. He worked out that basically all of geometry stands on just 10 core principles. You can build the rest using straightforward, logical reasoning. In short, geometry is among the great intellectual feats of humankind. However, geometry goes far beyond being just an intellectual exercise. Its real-world applications extend to navigation, architecture, engineering, physics, technology, and even art.

Course Structure

1. Geometry—Ancient Ropes and Modern Phones

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.

Duration: 33 min

2. Beginnings—Jargon and Undefined Terms

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.

Duration: 28 min

3. Angles and Pencil-Turning Mysteries

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.

Duration: 28 min

4. Understanding Polygons

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.

Duration: 31 min

5. The Pythagorean Theorem

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.

Duration: 29 min

6. Distance, Midpoints, and Folding Ties

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.

Duration: 29 min

7. The Nature of Parallelism

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!

Duration: 35 min

8. Proofs and Proof Writing

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.

Duration: 29 min

9. Similarity and Congruence

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.

Duration: 34 min

10. Practical Applications of Similarity

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.

Duration: 31 min

11. Making Use of Linear Equations

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.

Duration: 29 min

12. Equidistance—A Focus on Distance

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.

Duration: 33 min

13. A Return to Parallelism

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.

Duration: 31 min

14. Exploring Special Quadrilaterals

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.

Duration: 30 min

15. The Classification of Triangles

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).

Duration: 30 min

16. Circle-ometry—On Circular Motion

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.

Duration: 32 min

17. Trigonometry through Right Triangles

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.

Duration: 28 min

18. What Is the Sine of 1°?

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.

Duration: 32 min

19. The Geometry of a Circle

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.

Duration: 29 min

20. The Equation of a Circle

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.

Duration: 33 min

21. Understanding Area

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.

Duration: 28 min

22. Explorations with Pi

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).

Duration: 31 min

23. Three-Dimensional Geometry—Solids

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.

Duration: 32 min

24. Introduction to Scale

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.

Duration: 30 min

25. Playing with Geometric Probability

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.

Duration: 30 min

26. Exploring Geometric Constructions

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.

Duration: 29 min

27. The Reflection Principle

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.

Duration: 31 min

28. Tilings, Platonic Solids, and Theorems

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.

Duration: 32 min

29. Folding and Conics

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.

Duration: 29 min

30. The Mathematics of Symmetry

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.

Duration: 28 min

31. The Mathematics of Fractals

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.

Duration: 30 min

32. Dido’s Problem

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.

Duration: 31 min

33. The Geometry of Braids—Curious Applications

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.

Duration: 29 min

34. The Geometry of Figurate Numbers

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.

Duration: 32 min

35. Complex Numbers in 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. Examine how using complex numbers can help solve several tricky geometry problems.

Duration: 32 min

36. Bending the Axioms—New Geometries

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.

Duration: 32 min