User:Bradley Klee/SeriesInversionIII

Series Inversion III: Monkey Saddles

Introduction

The Monkey Saddle is a humorously-named set of mathematical functions, not only relevant to the circus. The most simple example is a two-dimensional cubic function with Cartesian height equation

$z(x,y)=3yx^{2}-y^{3},\;\;\;\;\;\;(1)$

In cylindrical coordinates,

$h(\rho ,\phi )=\rho ^{3}\sin(3\phi ).\;\;\;\;\;\;(2)$

the function becomes easier to analyze. For example, we invert implicit Eq. 2 to calculate the equipotential contours

$\rho (h,\phi )={\bigg (}{\frac {h}{\sin(3\phi )}}{\bigg )}^{1/3},$

and plot for a few values of $h$.

Fig. 1. Simple Monkey Saddle. Left: Equipotential contours in a regular polygonal region with area $2{\sqrt {3}}$. Each of six function divergences, $\sin(3\phi )=0$, corresponds to one of three seperatrix lines, which divide the hexagonal disk into six congruent quadrilaterals. Right: Monkey saddle over the same domain. Scaling height such that $h(\rho ,\phi )=(27/120)\rho ^{3}\sin(3\phi )$ forces the surface boundary to nearly follow the edges of a cube with volume $2{\sqrt {2}}$.

The set of monkey saddles intersects the set of functions obtained by series expanding triply periodic minimal surfaces around axes of Dihedral symmetry. In particular, the Schwarz P and D surfaces are completely defined by unique monkey saddle expansions combined with a crystallographic symmetry. In the following, we give a 99.99% accurate coordinatization of the Schwarz D "Diamond" surface, and calculate the equipotential contours using a conjectural equation for series inversion with leading cubic term.

Schwarz D "Diamond" Surface

Background

The D Surface admits a precise coordinatization in the Enneper–Weierstrass representation, as discussed in "Exact computation of the triply periodic D (`diamond') minimal surface". However, in some calculations elliptic function formalism may not be necessary. Other approximate and arbitrary-precision methods exist outside of the standard fare. For example, "Nodal surface approximations to the P, G, D and I-WP triply periodic minimal surfaces" discusses a Fourier type approximation. In this article the authors claim to obtain a accurate approximation of the D surface by fitting a model with three free parameters to 8650 numerically computed data points. Although it is difficult to verify the author's claim, we sidestep the issue enitrely by calculating an approximation of greater accuracy using only one free parameter.

Approximate Coordinatization

In the non-parametric problem for minimal surfaces, we determine a height function $h(x,y)=h(\rho \;\cos(\phi ),\rho \;\sin(\phi ))$, which satisfies the condition of zero mean curvature in a bounded region of the plane. Assuming triangular dihedral symmetry, we obtain the following approximate series expansion $h(\rho ,\phi )=c{\bigg (}\rho ^{3}\sin(3\phi ){\bigg )}+c^{3}{\bigg (}{\bigg (}{\frac {27}{20}}{\bigg )}\rho ^{7}\sin(3\phi ){\bigg )}+c^{5}{\bigg (}{\bigg (}{\frac {243}{80}}{\bigg )}\rho ^{11}\sin(3\phi )+{\bigg (}{\frac {-243}{400}}{\bigg )}\rho ^{11}\sin(9\phi ){\bigg )}$

$+c^{7}{\bigg (}{\bigg (}{\frac {15309}{1600}}{\bigg )}\rho ^{15}\sin(3\phi )+{\bigg (}{\frac {-2187}{800}}{\bigg )}\rho ^{15}\sin(9\phi ){\bigg )}$

$+c^{9}{\bigg (}{\bigg (}{\frac {531441}{16000}}{\bigg )}\rho ^{19}\sin(3\phi )+{\bigg (}{\frac {-19683}{1600}}{\bigg )}\rho ^{19}\sin(9\phi )+{\bigg (}{\frac {59049}{68000}}{\bigg )}\rho ^{19}\sin(15\phi ){\bigg )},$

where $c$ is a free parameter set by the boundary conditions. For the D surface, $c=27/120$ suffices.

Surface Area Integral

As minimal surfaces also tend to minimize surface area, the surface area integral provides a natural test of approximation accuracy. The exact result for a surface in a cube of volume $2{\sqrt {2}}$ is already known

$S.A.=3\;K(3/4)/K(1/4)=3.83778...$

in terms of the complete elliptic integral of the first kind $K(m)$ (cf. A249282, A249283, Gergonne Schwartz Surface).

We calculate the surface area by direct integration in cylindrical coordinates

$S.A.=12\;\int _{0}^{\pi /6}d\phi \int _{0}^{\rho _{B}(\phi )}\rho \;d\rho \;\;{\sqrt {1+{\bigg (}{\frac {dh}{d\rho }}{\bigg )}^{2}+{\bigg (}{\frac {1}{\rho }}{\frac {dh}{d\phi }}{\bigg )}^{2}}};\;\;\;\;\;\rho _{B}(\phi )={\sqrt {\frac {1}{1-\sin ^{2}(\phi )}}}.$

$S.A.\approx 3.83804,$

correct to three decimal digits. After Gandy et al., the accuracy statistic is

${\frac {S.A._{fitted}}{S.A._{exact}}}=3.83804/3.83778=1.00007.$

Comparing with Table 5 of "Nodal surface approximations to the P, G, D and I-WP triply periodic minimal surfaces", where others obtain accuracy of 1.000253 using at least two free parameters, shows more convergence toward minimal area, 99.99% as opposed to 99.97%, again using only one free parameter.