Hexagrammum Mysticum
Six unordered points on a circle (or on any conic) can be connected as the vertices of a hexagon in 60 different ways, resulting in 60 different instances of Pascal's Theorem and 60 different Pascal lines. This configuration of 60 lines is called the Hexagrammum Mysticum.
This is my attempt at starting to understand the points and lines associated with this configuration, namely, the 60 Kirkman points, the 20 Steiner points, the 20 Cayley lines, the 15 Plücker lines, and the 15 Salmon points.