Explore the Folium of Descartes in this 32-minute lecture from the MathFoundations series. Delve into an extended exercise in algebraic calculus, examining the curve both as a cubic in the plane and a surface in three-dimensional space. Learn to parametrize the curve, calculate Taylor polynumbers and expansions, and determine tangents. Interpret these concepts as tangent planes, lines, quadrics, and conics. Discover a fascinating duality satisfied by the tangent conics. Gain insights into the logical foundations of modern pure mathematics through this in-depth exploration of a classic geometric object.