  Special types of quartic surfaces were much studied objects during the 1860s.
Quartics were thus very much in the air when Sophus Lie and Felix Klein first
met in Berlin in 1869. As this study shows, such surfaces played a major role
in their subsequent work, much of which centered on linear and quadratic line
complexes. This mutual interest led them to a number of new results on the
quartic surfaces of Steiner, Pl\"ucker, and Kummer, as well as various types of
ruled quartics studied earlier by Cremona. This paper, which draws on
unpublished archival sources as well as published work from the period
1869-1872, underscores the importance of this aspect of the early geometrical
work of these two famous figures. A highlight was Lie's line-to-sphere
transformation, which led to surprising new findings on properties of
asymptotic curves on Kummer surfaces.
