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'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.
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.
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.
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.)
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'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.
A raytraced image of Scherk's singly periodic surface in a ball of radius 10.
