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.