**Models with tetrahedral Symmetry**

In 1862, Kummer himself was involved in the building of several models of quartic surfaces. This family of models is obtained by varying the real parameters $\lambda$ and $\mu$ in the equation $$(\phi_{\mu}(x, y, z))^2 - \lambda p(x, y, z) q(x, y, z) r(x, y, z) s(x, y, z) = 0$$where $\phi_{\mu} = x^2 + y^2 + z^2 - \mu$ and $p, q, r, s$ are degree one polynomials. Let us denote the surface defined by this equation by $K$. Clearly $\phi_{\mu} = 0$ defines a sphere of radius $\sqrt{\mu}$, which we call $S_{\mu}$. The equations $p=0, q=0, r=0, s=0$ define four planes which, in this series of models, are chosen to form a regular tetrahedron concentric with $S_\mu$; we call it $T$. $K$ is a quartic surface: each of $p, q, r, s$ have degree one, and multiply to become degree four, while $\phi_{\mu}$ is degree two, squaring to become degree four as well. The parameter $\lambda$ determines how the sphere $S_{\mu}$ and tetrahedron $T$ interact, and determines the "shape" of the surface $K$.

More precisely, $$\begin{array}{c} p = z - 1 + \sqrt{2} x \\ q = z - 1 - \sqrt{2} x \\ r = -z - 1 + \sqrt{2} y \\ s = -z - 1 - \sqrt{2} y \end{array}$$

**Model IX 1 **shows $K$ for values $\mu=1$ and $\lambda= -1/8$.

Sadly our copy of model IX 2, for which $\mu=1$ and $\lambda=9/10$, is badly damaged, but see here for an intact model in Göttingen.

As can be seen from the variety in appearance of these models, varying $\mu$ and $\lambda$ can have significant effects. Let us first explore the effect on the sphere and tetrahedron of varying $\mu$, before examining the specific values in each of our models. At points where $S_{\mu}$ and the edges of $T$ intersect, $K$ is singular. We will see that $\mu$ determines the type and location of such singularities.

If $\mu <1$ then $S_{\mu}$ is entirely contained within the frame formed by $T$'s edges. Hence the edges do not intersect the sphere (at real points - recall our discussion of real and complex surfaces in Singularities), which is a somewhat degenerate case.

If $\mu = 1$ then $S_{\mu}$ meets $T$'s six edges tangentially. This gives six double points. This case is exhibited by Model IX 1, for $\lambda < 0 $. Similarly, IX 2 shows such a quartic, but this time the four tangent planes at the nodes are complex.

For $\lambda = 0$ we have the degenerate case of a double sphere. Changing the sign of $\lambda$ alters the shape of $K$ significantly. This can be seen by comparing IX 1 and IX 2. For the model IX 2, $\lambda$ is positive, and for IX 1 it is negative. This pair of surfaces was chosen by Kummer to show the amazing property that the space occupied by IX 1 becomes empty when we change to IX 2, and vice versa.

In terms of modern classification of singularities, we have six $A_{3}$ double points in each model.

The wire rings on IX 2 show where the four tetrahedral planes (as discussed above) meet the surface: each plane meets in a circle. These are double-planes, as discussed in Kummer Surfaces. In this case, each double plane meets 3 double points, and each double point is contained in 3 double planes. Likewise for IX 1.

Note that models IX 1 and IX 2 have the same symmetries as a tetrahedron, as does Steiner's surface. This means (informally) that they can be rotated and reflected in the same ways as a tetrahedron, and still look exactly the same.