Series Approximation for the Schwarz D Surface

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The Schwarz D "diamond" surface is a simple example of a triply periodic minimal surface with local dihedral symmetry [1, 2]. The semi-numerical series approximation here nearly satisfies all desired conditions: zero mean curvature, dihedral symmetry and matching along the boundary. In polar coordinates, the height function takes the form of a Fourier series along the angular dimension and a power series along the radial dimension. A patch of the surface is plotted, boundary convergence is analyzed, and the surface area is calculated by direct integration of the Fourier or power series. Comparing the numerical result with solutions in terms of elliptic functions shows agreement to four significant figures [3]. It is difficult to imagine a soap film experiment that could measure surface area more accurately than this benchmark.

Contributed by: Brad Klee (April 2017)
Open content licensed under CC BY-NC-SA


Snapshots


Details

A minimal surface obeys everywhere the condition of zero mean curvature. In the nonparametric problem [4], the curvature constraint applies directly to a height function ,

, (1)

where are first and second partial derivatives with respect to the Cartesian coordinates. A formal solution can be represented by a series:

, (2)

which should converge on a limited region around . Constraint equation (1) introduces dependencies into the set of , but does not entirely determine the shape of the surface. The remaining degrees of freedom depend uniquely on a boundary condition,

,

with a -periodic parameter around the boundary , a closed curve in .

Equations (1)–(3) locally determine a surface of zero mean curvature. A minimal surface in all exists whenever the boundary admits a valid tiling. Valid tilings obey a few additional constraints: every segment of must coincide with a segment of exactly one adjacent , the interiors of and must be mutually disjoint, and the surface must obey equation (1) across all adjacencies.

The Schwarz D surface is an simple example in which the boundary extends along the six edges of a cube with vertices :

,

without visiting or . Taking the surface normal along the axis , the projection becomes a regular hexagon, while the height alternates with a period of . To solve for this boundary, we rewrite equation (2) as

, (3)

with polynomials

,

.

After satisfying equation (1), we obtain the height function displayed above. The remaining degrees of freedom determine the shape along the boundary. Settling on a numerical method, we insert decimal values and apply variations until the approximate boundary most nearly matches the exact boundary. The procedure determines one- and two-parameter approximations:

,

,

and

,

,

.

The "boundary convergence" plot shows the difference between approximate and exact boundaries. To make the tiny difference more visible, we magnify by a factor of 10 and by a factor of 100. In principle, the series method extends to arbitrary precision, where the approximate boundary continues to converge to the exact boundary.

As a test of approximation validity, we directly integrate the surface area

,

where the definite integral goes over one hexagonal domain. The integral equals

(1 parameter),

(2 parameters).

The approximate calculation reaches excellent precision, differing by less than percent from the exact value from [3],

,

where is the complete elliptic integral of the first kind.

References

[1] E. W. Weisstein. "Minimal Surface" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/MinimalSurface.html (Wolfram MathWorld).

[2] Wikipedia. "Schwarz Minimal Surface." (Apr 17, 2017) en.wikipedia.org/wiki/Schwarz_minimal_surface.

[3] P. J. F. Gandy, D. Cvijović, A. L. Mackay and J. Klinowski, "Exact Computation of the Triply Periodic D ('Diamond') Minimal Surface," Chemical Physics Letters, 314(5–6), 1999 pp. 543–551. doi:10.1016/S0009-2614(99)01000-3.

[4] J. C. C. Nitsche, Lectures on Minimal Surfaces, Vol. 1. (J. M. Feinberg, trans., A. Schmidt, ed.), New York: Cambridge University Press, 1989.



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