Schwarz $\mathrm{P}$ and Schwarz $\mathrm{D}$ as free-boundary minimal disks in a tetragonal disphenoid: exact areas

Abstract

Let $T$ be the tetragonal disphenoid whose four congruent faces are isosceles triangles with sides $2,\sqrt3,\sqrt3$, so that the dihedral angles are $π/2$ along one opposite pair of edges and $π/3$ along the remaining four. By Smyth's theorem $T$ contains three embedded minimal disks meeting $\partial T$ orthogonally; the symmetry group $D_{2d}$ of $T$ fixes one and interchanges the other two. We identify all three completely. The distinguished disk is a patch of $\textbf{Schwarz's $\mathrm{P}$ surface}$ and the congruent pair are patches of $\textbf{Schwarz's $\mathrm{D}$ surface}$, with exact areas \[ \operatorname{Area}(Σ^{*}_{A}) = \frac{K\!\left(\tfrac14\right)} {K\!\left(\tfrac34\right)} = 0.781700961348056\ldots, \] \[ \operatorname{Area}(Σ^{*}_{B}) = \frac{K\!\left(\tfrac34\right)} {2\,K\!\left(\tfrac14\right)} = 0.639630785585503\ldots \] in the normalization $\operatorname{Vol}(T)=2/3$, inradius $r_T=\sqrt2/4$. In particular $2\,\operatorname{Area}(Σ^{*}_{A})\operatorname{Area}(Σ^{*}_{B})=1$, which we show is the Bonnet relation for the adjoint pair $(\mathrm{D},\mathrm{P})$. The proof is a reversal of the usual strategy: rather than uniformizing the Gauss map, we identify Smyth's three Plateau contours as the $\textit{tetrahedral four edges}$ and the $\textit{octahedral four edges}$ --- classically solved by $\mathrm{D}$ and $\mathrm{P}$ respectively --- and then conjugate. Both areas agree with Surface Evolver computations of Ken Brakke to all $15$ reported digits. We also show that all twelve corners of the three disks are midpoints of edges of $T$, each of the six midpoints being used twice, and that $T$ is precisely the alcove of the affine Weyl group $\widetilde{A}_{3}$, confirming a remark of Gandy, Cvijović, et al. (1999).