## The lines on the Cayley surface

### This page is dedicated in memory of my friend Sevín Recillas, who got me interested in this surface.

 Scroll down   to the   discussion   of the main features of the Cayley surface. Viewing hints:   This is a large figure and it simply may load slowly. So, the first thing to try is just to be patient for a little while.   If nothing loads except the light blue background, it may help to (i) click on your browser's "back" button, and then (ii) click on the "forward" button to get back to this page.   If the picture absolutely won't load, please click on the Static figure link.
 Cayley surface links: Duality Equation View without  the lines Static figure   (Visit this link if the figure won't load.) Steiner surface links: Other links: Introduction:      The figure shows part of the Cayley surface. It is a surface of degree 3, with 4 singular points. These singular points are often called nodes. Near each singular point, the surface is closely approximated by a quadric cone.     There are 9 lines on the Cayley surface. All of these lines are shown in this figure. Six of the lines join pairs of nodes. These lines are traced in black on the figure. Thus, the nodes are the vertices of a tetrahedron, and these six lines are the edges of the tetrahedron. The other three lines lie in the tritangent plane, which is discussed on the duality page. These lines are traced in white on the surface.     The Cayley surface is the dual variety of the Steiner surface.  This means that the points of the Cayley surface correspond bijectively to the tangent planes of the Steiner surface (except that the correspondence is not bijective along finitely many subvarieties).     For a discussion of specific features of the duality correspondence, please click on the Duality link at the left.