Schwarz PD Family Minimal Surfaces

schwarz PD family morph 011
Schwarz PD Family Minimal Surfaces. This is a 2-parameter family of triply periodic genus 3 minimal surfaces.

Imagine these surfaces built from pieces which sit in a “brick” and meet the brick's 6 faces in convex curves. Reflection in the planes of these curves extends the surface to a larger one. Eventually it fills R^3. It has translation symmetries in 3 directions, hence it is called “triply periodic”.

schwarz PD family morph 001
schwarz PD family morph
Schwarz PD Fundament
This piece is one 8th of the piece in the brick. It is also bounded by planar symmetry lines. This piece is computed from the “Weierstrass representation” and the surface is built from it by reflections.
schwarz PD ConjFund
The conjugate of the fundamental piece has 6 straight boundary segments which meet under 90 degree angles. This conjugate piece can be numerically computed from its boundary hexagon. It is the surface of smallest area with this boundary.
schwarz conj PD morph 001
schwarz conj PD morph. This animation shows the conjugate pieces of part of the first animation. The surfaces of these two animations are both “embedded”, they do not intersect themselves. It is rare that a conjugate pair of minimal surfaces is embedded.
schwarz pd family st
schwarz pd family st
schwarz pd family sw
schwarz pd family sw

About the Schwarz PD Family

This is a 2-parameter family of triply periodic genus 3 surfaces. In each case the original surface and the conjugate surface are embedded. The most symmetric example (with a cubic lattice) which is obtained when cc = 0, dd = 0, was already constructed by H. A. Schwarz. When Alan Schoen found more triply periodic surfaces around 1970 he named the two surfaces which Schwarz found the P-surface (P for cubic primitive) and the D-surface (D for diamond). He also found a third embedded(!) surface in the associate family of these, the Gyroid (associate parameter 0.577 which is approx. 52 degrees). If dd=0 then a fundamental cell for the lattice is a prisma with square base, in the morphing cc changes the height of the prisma.

K. Grosse-Brauckmann, M. Wohlgemuth: The Gyroid is embedded and has constant mean curvature companions. To appear Calc. Var. 1996

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].