Hi

As Steve Smith has been pointing out for years, Lichtenberg figures are not the only indications of electrical shaping of terrain. He has shown that geometric features on Mars are due to high voltage electrical interactions with the planet surface. This electrical carving behavior is especially prominent where the material involved is electrically conductive, having for example a large iron content, as on Mars, or in the Southwestern US. Regular features of electrically carved landscapes have been referred to by Steve in terms of "Rosenwieg instabilities" in ferrofluids, which is a quite striking and correct understanding of the situation.

However, such in situations, there is an electrical basis for the observed behaviors, having to do with the Maxwell equations. It's one thing to see something and give it a name. It is quite another to see something and find out what caused it. Rosenwieg "instabilities" (which are actually islands of stability which are observed against the background of an inherently unstable ferrofluid) are the result of minimal surface solutions of the Maxwell equations.

Bateman’s self-conjugate solutions of the Maxwell equations, which are based on a complex vector with a zero square, define a minimal surface. The norm of the self-conjugate vector solution is given by an expression which is proportional to the energy density of the field. If this term is non-zero, when the equation is solved, it will turn out that the minimal surface will be

regular.

The self-conjugate (minimal surface) condition requires that the E field is orthogonal to the B field, and the electric and magnetic energy densities are equal. The conclusion is reached that propagating electromagnetic waves are to be associated with minimal surfaces. The associated minimal surface is always regular and without singularities for real, non-zero E and B. When the E and B fields are complex, which in a physical sense implies the existence of elliptical polarization, another interpretation is possible.

The association of electromagnetic wave propagation with minimal surface theory was apparently unknown to Bateman. See Osserman’s book "A Survey of Minimal Surfaces". According to Osserman, complex 3-vector representations of minimal surfaces were known to Enneper and Weierstrass. A study of the minimal surfaces generated in E4 is given by Kommerell. The minimal surfaces so generated in E4 by this class of vector fields will have 3-dimensional images that are not always regular.

In general, two dimensional non-regular surfaces may have ”singularities” consisting of ”curves of double points” created by intersections of two local surface patches, or of ”triple” points consisting of intersections of three local surface patches, or of curves of double points which terminate on ”Pinch” points within the interior of the surface. These three types of self-intersection singularities are the only three ”stable” singularities in the sense of Whitney.

Recall that Whitney proved that any N manifold can be embedded in 2N+1 Euclidean space, and immersed in a 2N Euclidean space. The induced surfaces may be orientable or non-orientable. Non-orientable examples are characterized by the Klein-Bottle, or the Projective Plane, and the orientable surfaces, by the Sphere. Each surface may have tubular handles, holes and distortions.

These are explorations regarding minimal surfaces. If the given surface has no singularities, then the surface is said to be regular or embedded. The constraint of regularity implies that the surface normal vector never goes to zero over the surface, or the induced metric on the surface is always invertible.

This implies that are always two linearly independent directions on a regular domain of the surface. If the lines of self-intersection are divergence-free on the domain (meaning that they stop or start only on boundary points), or are closed upon themselves, then the surface is said to be immersed in 3-Dimensions.

The points where the divergence of the lines of intersection is not zero are defined as Pinch points. Such surfaces cannot be immersed in 3-D. Pinch points are the signature of the fact the surface resides in 4-Dimensions (as an immersion), and cannot be immersed in 3-Dimensions.

A flow vector field may have domains where it is irrotational or solenoidal, and these domains may be separated by a surface. If the surface of separation is a minimal surface, then the flow on this surface is harmonic.

The minimal surface need not be regular, and may have lines of self-intersection. These lines of surface

self-intersections (lines of singular double points) are not necessarily solenoidal.

In fact, the Pinch points are points where the lines of self-intersection terminate, not on themselves, and not on a boundary, but in the surface of the interior. The Pinch points may be viewed as the ”sources” of the divergence of lines of self-intersection. Again, the classic example is given by Whitney.