Minimal surfaces are objects defined as critical points to the area function for a given perimeter, that is they are the mathematic equivalent to soap-films that span a rigid wire loop. Minimal surfaces came out as a result of Lagranges calculus of variations in his original paper on the subject. He also observed that the plane is a trivial case. An important key to the understanding of surfaces is their curvatures, plural because there are several.

Consider a point, p, on a surface, S, and a surface normal, n, perpendicular to the tangent plane, T, in the point. A (normal) plane, N, spanned by n and a vector v in T will cut S along a planar curve G_v. The normal curvature of S at p along v is the inverse of the radius of the circle that best approximates G_v at p. The normal curvature at p is clearly a cyclic function of the direction v in T, and so it will assume a maximum and a minimum value. These values, k₁ and k₂ are the principal curvatures of S at p.

The Gaussian curvature, K, of S at p is the product k₁ k₂ while the arithmetic mean, 1/2( k₁+ k₂), is called the mean curvature, H. K is related to the local metric of the surface, and in order for two surfaces to be isometric, i.e. Minimal surfaces have the unique property that the mean curvature vanishes at every point, H=0.

Minimal surfaces may be generated using Weierstrass equations, a triplet of integral equations that transform any analytic function, R(w), to the Cartesian coordinates of a minimal surface.

In this setting the Gaussian curvature is given by

K=-4 |R(w)|^-2(1+w²)^-4

Since K depends only on the absolute value of R(w) the expression e^iqR(w) will yield a one parameter family of isometric surfaces. From the expression for K, it is clear that R(w) must be of the order w^-4 if K is to be bounded and non-zero as w approaches infinity. Such a choice is always allowed by a simple reorientation of the surface, and we may generate a catalogue of surfaces simply by distributing the zeros of such functions in the complex plane C. Sine the normal (Gauss) map of a minimal surface is conformal, we may consider C as a stereographic projection of the Gauss map sphere, and commence by considering highly symmetric distributions on the sphere.

The simplest choice is

R(w) = w^-4

^{which corresponds to a forth order zero at the origin of C, or the south pole of the Gauss map sphere. This is in fact equivalent to}

R(w) = 1

where the same high-order flat point occurs at infinity in C, or the north pole of the Gauss map sphere. The minimal surface generated is the classical Ennepers surface.

Placing second order flat points at both poles yields

R(w) = w^-2

and the integrals yield the expression for the catenoid and the helicoid and the continuous isometric family the constitute the endpoints of.

Four flat points in a circle around the equator of the Gauss map sphere (or on the unit circle in C) generates

R(w) = (w⁴- 1)^-1

and the image is the Scherk surface family. Going beyond four flat points introduces zeros of order less than one, and this means that the simple Gauss map sphere is replaced by a more complex Riemann surface. Eight equivalent half order flat points give rise to a two sheeted Riemann surface, and the most symmetric expression, that generated by the distribution of the flat points on the vertices of a regular cube

R(w) = (w⁸-14w⁴+1)^-1/2

is the expression for the isometric family of surfaces containing the periodic P,D and G surfaces.