Explore the projective version of the Quadruple quad formula in this 31-minute mathematics lecture. Delve into the relationship between four projective points and discover its surprising connection to cyclic quadrilateral geometry. Learn about the projective quadruple quad function R(a,b,c,d) and how it compares to the one-dimensional affine Q(a,b,c,d) function. Gain insights into a new approach to metrical geometry that replaces classical trigonometry, applicable to general fields and arbitrary bilinear forms. Cover topics including the Archimedes function, terminology, formula derivation, Quadratic Compatibility Theorem, and practical exercises to reinforce understanding.