Geometry: An Interactive Journey to Mastery

Exploring the origins of one of the oldest branches of mathematics; how geometry not only deals with practical concerns like mapping and navigation, but also offers an intellectual journey in its own right; the big and deep questions it invites.

Laying the basic building blocks of geometry by examining the terms point, line, angle, plane, straight and flat; learning the postulates or axioms for how those building blocks interact; working through the vertical angle theorem.

How three angles in a triangle can add up to 180 degrees; how the experience of turning a pencil on a flat triangle differs from walking in a triangular shape on the surface of a sphere; the difference between flat and spherical geometry.

The ubiquity of shapes with straight lines; how to define polygons mathematically despite understanding them intuitively; analyzing their properties; finding out the answers to these questions and more.

The common definition of the Pythagorean theorem; Pythagoras himself would be confused by that; how this famous theorem can be explained using common geometric shapes; how it is a critical foundation for the rest of geometry.

How watching a fly on his ceiling inspires mathematician Rene Descartes to link geometry and algebra; how this powerful connection allows people to use algebra to calculate distances, midpoints and more.

How the usual definition of parallelism is impossible to check; using fundamental assumptions to create an alternative way of checking if lines are parallel; how it is possible to compute the circumference of the Earth.

The beauty of geometry is that each result logically builds on the others; mathematicians demonstrate this chain of deduction using proofs, learning this step-by-step process of logic and how to construct one's own proofs.

What it means for polygons to be similar or congruent by thinking about photocopies; proving the third key assumption of geometry, which is the side-angle-side postulate; Thales of Ionia uses this principle in 600 B.C. to impress the Egyptians.

Building on the side-angle-side postulate; deriving other ways of testing whether triangles are similar or congruent; diving into several practical applications, which include estimating the heights of trees and measuring the width of a river.

Delving into the connections between algebra and geometry by looking at lines and their equations; using the three basic assumptions to prove that parallel lines have the same slope; calculating the shortest distance between a point and a line.

Knowing how to find the midpoint between two points; what if one has three or four points; exploring the concept of equidistance; how it reveals even more about the properties of triangles and other shapes.

Continuing the study of parallelism by exploring the properties of transversals, which are lines that intersect two other lines; proving how corresponding angles are congruent; how this fact ties into the trapezoid, which is a type of polygon.

Classifying the different types of four-sided polygons called quadrilaterals; the surprising characteristics of the diagonals and interior angles of various shapes; how real-life objects like ironing boards exhibit these geometric characteristics.

Continuing the work of classification with triangles; what mathematicians mean when they use words like scalene, isosceles, equilateral, acute, right and obtuse; how to use the Pythagorean theorem to determine the type of triangle.

How one can figure out the "height" of the sun in the sky; answering this question using angle of elevation and a branch of geometry called trigonometry; the basic trigonometric identities; how physicists use them to describe circular motion.

Trigonometric identities go beyond circles; how to define all of them using just triangles, which is expressed in a famous acronym; uncovering how trigonometry is practically applied by architects and engineers to measure the heights of buildings.

How to calculate the sine, cosine and tangent of basic angles; calculating them for other angles without a calculator; using the Pythagorean theorem to come up with formulas for sums and differences of the trigonometric identities.

Exploring the world of circles; the definition of a circle; what mathematicians mean when they say things like radius, chord, diameter and secant; how these interact; using that knowledge to prove the inscribed angle theorem and Thales' theorem.

Using a combination of geometry and algebra to determine all kinds of interesting properties and characteristics; doing the same for circles, including deriving the algebraic equation for a circle.

The meaning of area; exploring how its definition is not quite so straightforward; working out the formula for the area of a triangle; how to use that formula to derive the area of any other polygon.

What pi really is; why it matters; what it has to do with the area of a circle; exploring the answers to these questions and more; how to define pi for shapes other than circles, such as squares.

Figuring out all kinds of fun properties with two-dimensional shapes and moving to three dimensions; classifying common 3-D shapes such as cones and cylinders; learning surprising definitions; studying the properties of these shapes.

Learning what happens to a shape's area when its side-lengths are doubled; exploring the concepts of scale; applying this idea to derive the Pythagorean theorem; using the areas of any shape drawn on the edges of the right triangle.

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

Drawing basic geometric shapes without having a marked ruler to measure lengths or a protractor to measure angles; exploring how the ancient Greeks construct angles and basic geometric shapes using no more than a straight edge for marking lines.

If one plays squash and hits the ball against the wall, knowing at what angle the ball will bounce back; if one is playing pool and wants to play a trick shot against the side edge, how one needs to hit the ball; exploring the reflection principle.

Seeing geometric tiling patterns on the bathroom floor and in the works of great artists; knowing what happens when one makes repeating patterns in 3-D space; discovering the five Platonic solids; creating beautiful patterns.

Using paper-folding to unveil sets of curves including parabolas, ellipses and hyperbolas; studying their special properties and seeing how these curves have applications across physics, astronomy and mechanical engineering.

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

Exploring the mysterious world of fractals; learning how to create them; examining examples such as Sierpinski's Triangle and the Koch Snowflake; uncovering how fractals appear in nature; the structure of sea sponges and walls of small intestines.

If one has a fixed-length string, knowing what shape one can create with that string to give the biggest area; uncovering the answer to this question using the legendary story of Dido and the founding of the city of Carthage.

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

Pondering another surprising appearance of geometry, the mathematics of numbers and number theory; looking into the properties of square and triangular numbers; using geometry to do some fancy arithmetic without a calculator.

Rene Descartes unites geometry and algebra with the invention of the coordinate plane; exploring the power and surprises that come from using the complex number plane; examining how using complex numbers can help solve tricky geometry problems.

Exploring the counterintuitive behaviors of shapes, angles and lines in spherical geometry, hyperbolic geometry, finite geometry and even taxi-cab geometry; seeing how the world of geometry is never a closed-book experience.