Scherk Surface

It was a sensation when Heinrich Scherk discovered in 1834 new minimal surfaces since during the previous 50 years only the Catenoid and the Helicoid where known. (The plane is a minimal surface, but not counted as interesting.)

Two of his discoveries were without singularities, namely a doubly periodic one which Scherk described as the graph of a function and a singly periodic one which Scherk presented by an implicit equation. (doubly periodic = translational symmetries in two directions, singly periodic = translational symmetries in one direction)

These two surfaces turned out to be fundamental examples when in the 1980s a wealth of unexpected minimal surfaces were found. These modern surfaces are all made with the Weierstrass representation - which leaves Scherk's discoveries as a singular success, since no methods were found to write down functions whose graphs are minimal surfaces nor functions which have one minimal level.

Scherk's Doubly Periodic

scherk 001
Scherk's doubly periodic minimal surface in the historic graph representation. 5 fundamental pieces for the translational symmetries are shown. The surface extends to infinitely many such fundamental pieces over the black squares of an infinite checkerboard. The surface contains vertical lines over the vertices of this checkerboard, these lines are missing because a graph representation cannot contain vertical lines.
scherk horizontal
To see the checkerboard symmetries we had to use a downward view of the surface. This downward perspective distorts the vertical dimension of the surface. Our horizontally placed fundamental piece shows its true shape.
scherk rotate 001
This cross-eyed stereo view is intended to show that the surface is only rotated and definitely not stretched while the fundamental piece turns from vertical to horizontal.
scherk rvature circles
The two principal curvature circles at each point of a minimal surface have the same size but lie on different sides of the surface. Another way of saying this is: “The mean curvature of a minimal surface is zero.” The image shows the principal circles at two points. (The surface normals at these two points are away from the viewer.)
scherk doubly raytrace10
The graph of a function f(x,y) is also the solution of the implicit equation F(x,y,z) := f(x,y) - z = 0 . We can therefore compute raytraced images of Scherk's doubly periodic surface, here the part in a ball of radius 10.

Scherk's Singly Periodic

scherk implicit 1 periodic
Scherk's singly periodic minimal surface is conjugate to Scherk's doubly periodic surface. This image shows the portion in a sphere of radius 4.5. It can be described by the implicit equation sin(z) - sinh(x)*sinh(y) = 0 see Saddle Tower for generalizations and renderings made with the Weierstrass representation. We are not aware of images from the 19th century. Our rendering raytraces the solution of the implicit equation.
scherk singly3
A raytraced image of Scherk's singly periodic surface in a ball of radius 10.

See also: Saddle Tower

Scherk Minimal Surface Parametric Equations

  x = u
  y = v
  z = (ln(cos(aa * v) / cos(aa * u))) / aa
scherk 07444
Illustration from 1988. 〔Karcher, H.: Embedded Minimal Surfaces Drived From Scherk's Examples. Manuscripta Math. 62, 83 - 114 (1988)〕
scherk 59454
Illustration from 1988. 〔Karcher, H.: Embedded Minimal Surfaces Drived From Scherk's Examples. Manuscripta Math. 62, 83 - 114 (1988)〕

Related Surface

  1. Catalan Surface
  2. Henneberg Surface
  3. Scherk Surface
  4. Saddle Tower

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