The Holy Grail is a 3 dimensional semi algebraic surface that represents a cross section of the five dimensional space of real homogeneous quartics:
f(x, y) = ax4 + b x3y + c x2y2 + d xy3 + e y4.
A better sense of the picture can be obtained by looking at an animation that was drawn using the software 3D-Filmstrip of Richard Palais. This is a Quick Time VR animation that requires the Quick Time Plug In plug in for your browser.
(Here I only show a section, the upper and lower bowl extend ad infinitum).The surface of the Holy grail corresponds to the quartics with repeated roots.There is another component (not shown in the picture or movie) consisting of two pairs of"whiskers" that sprout inside the bowls and connect to the swallow tail points of thetetrahedron at the center.
The cross sections by horizontal planes of this figure are well known curves, the most remarkable of which is the astroid, (the second of the following figures):
You can download an animation of the cross sections by horizontal planes, the corresponding plane for the astroid is z=0.
To download this amazing free program 3D-Filmstrip or to know more about mathematical visualization follow the links to Richard Palais's homepage.
I am interested in this surface because it describes the space of Hermitian matrices in GL(4,C). The points in the interior of the tetrahedron correspond to elliptic elements of SU(3,1) or to SU(4). The elements in the bowls are elements of SU(2,2). The loxodromic elements of SU(3,1) are on the exterior of the grail, and the parabolic elements are the points on the surface of the grail. The connection between homogeneous quartics and Hermitian matrices is simply that the characteristic polynomial of a Hermitian matrix is a polynomial of degree four. It is not hard to verify that if the matrix is Hermitian then the polynomial depends on the trace and the second symmetric function that turns out to be real. These are (more or less) the coordinates for the pictures. In a standard 3 dimensional frame the (x,y) coordinates correspond to the trace and the z coordinate to the second symmetric function.
But the real reason is that PU(3,1) (which is SU(3,1) modulo its center) is the group of holomorphic isometries of the complex hyperbolic space of dimension three. Thus, knowing this two numbers for a matrix in SU(3,1) allows me to get information on the induced automorphism in PU(3,1), and I expect to use this information to find interesting discrete groups acting on complex hyperbolic space.
To know more about these topics see:
For homogeneous cubic forms the problem was solved by Ziemann. The animation shows the parameter space of deformations: In this case, a four dimensional cone with a three dimensional base depicted in the final picture of the animation. The base is called the Umbilic Bracelet. Notice the 1/3 turn that the cross section of the Umbilic Bracelet makes as it turns around the circle. For a 3 dimensional object movie click on the animation below.
Up to my home page.