The 20 Cayley lines each contain one of the 20 Steiner points and three of the 60 Kirkman points arising from the 60 Pascal lines of a hexagon inscribed in a conic. The Pascal lines intersect three at a time at the Steiner points and also three at a time at the Kirkman points. The 20 Cayley lines pass four at a time through 15 points known as Salmon points (Wells 1991). There is a dual relationship between the 20 Cayley lines and the 20 Steiner points.
The figure above is the Levi graph of the 15 Salmon points and the 20 Cayley lines: a point vertex is joined to a line vertex exactly when the point lies on the line. It records the incidence structure rather than a projective realization of the lines themselves.
