Episode 31 of Geometry

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.

Running Time

31 mins

Year

2014

Kanopy ID

1338449

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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.

Equidistanceâ€”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.

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).

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.

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…

Beginningsâ€”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.

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.

Divide and Conquer

Turn now to the last fundamental operation of arithmetic: division. Explore a variety of shortcuts for dividing by one- and two-digit numbers; learn how to convert fractions such as 1/7 and 3/16 into decimals; and discover methods for determining when a large number is divisible by numbers such as 3,…

Mathematics and Chess

Chess is more like doing real mathematics than almost any other game out there. You'll get a quick overview of how it's is played; learn how to see connections between math and chess; explore some classic chess puzzles and problems; tap into strategies and tactics for the opening, middle, and…

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…

The Joy of Pascal's Triangle

A geometric arrangement of binomial coefficients called Pascal's triangle is a treasure trove of beautiful number patterns. It even provides an answer to the song "The Twelve Days of Christmas": Exactly how many gifts did my true love give to me?

The Joy of Integral Calculus

Geometry and trigonometry are used to determine the areas of simple figures such as triangles and circles. But how are more complex shapes measured? Calculus comes to the rescue with a technique called integration, which adds the simple areas of many tiny quantities.

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