The Holy Grail

In "The cross-ratio foliation of binary quartic forms", Geom. Dedicata 27 (1988), no 3, 263-280, MR 89i:58013), Tim Poston and Ian Stewart study the geometry of the classification of real binary quartics

f(x, y) = ax⁴ + bx³y + cx²y² + dxy³ + ey⁴

under linear coordinate changes of the $($ x, y) variables. They show that the key to the classification is a certain "surface with whiskers" that they have named the "Holy Grail":

(Click on the window if your browswer can launch Geomview; you will get an image that you can manipulate interactively.) Roughly speaking, the surface consists of two bowls--one opening upward, the other downward--joined in the center by a tetrahedron with curved faces and edges. There is a pair of (red) whiskers inside each bowl. The image above is an animated GIF created by Pau Atela. See acknowledgment at the end.

The classification

The classification depends on two observations:

By the Fundamental Theorem of Algebra, any quartic form can be decomposed into four linear factors p_ix + q_iy $.
Of course, the coefficients$ p_i, q_i $may be complex (and if they are, they occur in complex conjugate pairs
because the form is real). Under linear coordinate changes, the
character of these factors cannot be changed. In particular, if$ f has a repeated factor, that will persist under coordinate changes.
Each f(x, y) has a critical point at the origin; its type is either a minimum, a saddle, or a maximum. Under linear cordinate changes, the type of that critical point cannot change.

Observation 1 suggests that we focus on the discriminant set, which consists of those forms that have two (or more) equal factors. If we identify the form

f(x, y) above with the point

(

a, b, c, d, e) in

R⁵, then Poston and Stewart have shown that, in a certain three-dimensional section of

R⁵, the discriminant set is the "Holy Grail" shown above. Moreover, the Grail separates

R⁵ into four open regions, depending on the nature of the critical point of f at the origin (observation 2) and the number of real factors it has:

inside the upper bowl: minimum with no real factors (e.g. x⁴ + y⁴ )
inside the lower bowl: maximum with no real factors (e.g. $-$ x⁴ - y⁴ )
inside the central tetrahedron: saddle with four real factors (e.g. xy(x² - y² ))
outside the whole Grail: saddle with two real factors (e.g. xy(x² + y² ))

The red whiskers are also part of the discriminant set; they consist of forms that have two pairs of equal complex factors. On the whiskers inside the upper bowl, f is like

(

x² + y² )²; inside the lower bowl, like

-(

x² + y² )².

If you look carefully (better, if you handle the interactive Geomview image), you'll notice that the upper red whiskers continue through the Grail surface along the seam between the upper bowl and the central tetrahedron. Along this seam (actually, a curve of double points of the surface), f has a pair of repeated factors, but the factors are real instead of complex; f is like x²y². The same is true for the lower red whiskers; along the lower seam, f is like $-$ x²y².

Finally, at each the four points where the red whiskers join up with the Grail surface, the surface has a swallowtail point. The quartic form f associated with this point has a quadruple factor and is therefore like x⁴ or $-$ x⁴.

Question: What is f like on the different parts of the Grail surface itself? You should be able to figure this out using the fact that f has a double factor and is a transitional form between the types that lie in the open regions on either side.

The surface

The Holy Grail has a number of interesting features. First of all, it is a ruled surface (i.e., generated by a family of straight lines). Some of the generating lines are faintly visible in the image above.

In fact, the Grail is actually a developable surface; that is, the tangent planes to the surface at all points along a given line are the same). You can see this best in the Geomview image: Turn the surface until part of it is particularly shiny. Now the light comes from a point just behind you, so a piece of the surface is shiny if its tangent plane is perpendicular to your line of sight. Thus all shiny parts must have the same orientation. Finally, notice that, as you turn the surface, the shiniest parts of the surface always lie along a single generating line.

Next, consider the central tetrahedron. Its vertices are the four swallowtail points. Two of its six edges are the "seams" (the curves of double points) that join it to the upper and the lower bowls. The other four edges are cuspidal (that is, they have a cusp-shaped cross section) and they go from the top bowl to the bottom. These four edges make up a single curve with singular points at the vertices. The straight lines that generate the Grail are tangent to this boundary curve. (This is most evident in the Geomview image if you select the Inspect option, open the Appearances window, and choose to Show Edges.) Thus the Grail is the tangent developable of this curve, giving a second proof that the Grail is developable.

To understand the shape of the Grail, it helps to see that the surface "morphs" (or evolves) from a much simpler and more familiar surface--an ordinary cone. If you click on the image below, you'll see a Geomview animation that shows the cone on the left "morphing" into the Grail on the right. (Note: the animation file is large--1.8 MB.)

You can think of the metamorphosis this way. Start with a circle and its normals in a plane. The normals all meet in the center of the circle. If you lift the normals up 45° out of that plane, they generate a cone and meet at the vertex of the cone. Now perturb the circle into an oval, keeping the normals perpendicular to the oval and lifted up 45° out of the plane. The normals then generate the Grail.

The Holy Grail also appears in the work of Alberto Castro on the space of Hermitian matrices in $Gl(4,$ C).

Acknowledgment

I thank Pau Atela for his Geomview tutorial and for extra help in setting up the animations and other images, especially the large animated GIF at the top of the page.