## 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: Introduction 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 them join pairs of nodes. Thus, we can view the nodes as vertices of a tetrahedron, and these six lines are the edges of the tetrahedron. The other three lines lie in the trigangent plane, which is discussed on the duality page.     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.

The Java files used in this page were downloaded from the Geometry Center webpage.
I generated the geometric data for this figure in March 2009.
Updates completed on August 16, 2010.

Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA

Office: 531 Vincent Hall
Phone: (612) 625-9135
Dept. FAX: (612) 626-2017
e-mail: roberts@math.umn.edu
http://www.math.umn.edu/~roberts