Karcher JE Saddle Tower

JE morph thin 001
These doubly periodic embedded minimal surfaces are parametrized by rectangular tori. The shape can be viewed as a doubly periodic version of Riemann's minimal surfaces.
karcher je saddle patch 001
karcher je saddle wire a 001
karcher je saddle
Anaglyph wire frame view

Fundamental Domain

karcher je saddle assf p 001
Karcher JE Saddle Towers, associate family morph of fundamental domain.
The conjugate piece is bounded by four vertical half lines down and two horizontal symmetry curves. The piece can be extended to the full surface by first reflecting in the horizontal plane of the symmetry curves and then repeatedly rotating 180 degrees around the vertical lines. -- The vertical straight lines become horizontal symmetry curves on the fundamental piece of the original surface and the two symmetry curves on the conjugate piece become vertical straight segments on the original piece. Again, repeated reflections and 180 degree rotations eventually make the complete doubly periodic surface.
karcher je saddle assf w 001
anaglyph wire frame version of the associate family above.

Conjugate

karcher je saddle conj m 001
“karcher je saddle conj m 001”
This is a rare case where a family of minimal surfaces and their conjgates are all embedded. The shape of this family was predicted by looking at the Scherk singly periodic surfaces and this prediction led to their discovery. [see Scherk Surface]
karcher je saddle conj ana
JE conjugate (anaglyph)
These conjugate surfaces are also 2-periodic and embedded. The shape can be viewed as an array of Scherk saddle towers, seperated by straight lines (symmetry lines of these surfaces). This picture led to their discovery.
karcher je saddle conj w a
karcher je saddle.conj (anaglyph wireframe)
JE wire anaglyph
JE wire anaglyph

These families are parametrized by 4-punctured rectangular tori; they and their conjugates are embedded.

For the visual appearance of these surfaces it is particularly important that the punctures are centers of polar coordinate lines. Formulas are taken from [K1] or [K2]. The Gauss maps for these surfaces are degree 2 elliptic functions. The cases shown are particularly symmetric, the zeros and poles of the Gauss map are half-period points and the punctures are there. In the case of Karcher JD Saddle tower, the diagonal of the rectangular fundamental domain joins the two zeros, and in the Je-case it joins a zero and a pole of the Gauss map.

Under suitable choices of the modulus of the Torus these surfaces look like a fence of Scherk Saddle Towers — with a vertical straight line ( Karcher JE case), respectively a planar symmetry line ( JD case), separating these towers. The conjugate surfaces look qualitatively the same in the JD-cases and like a checkerboard array of horizontal handles between vertical planes in the JE-cases.

For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see either [KWH], or pages 192--217 of [DHKW].

Karcher_Je_Saddle_Tower.pdf