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

Let $T$ be the tetragonal disphenoid whose four congruent faces are isosceles triangles with sides $2,\sqrt{3},\sqrt{3}$, so that the dihedral angles are $\pi \mathrm{/}2$ along one opposite pair of edges and $\pi \mathrm{/}3$ along the remaining four. By Smyth’s theorem $T$ contains three embedded minimal disks meeting $\mathrm{\partial }T$ 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 $$\mathrm{Area}({\mathrm{\Sigma }}_A^{\ast })=\frac{K\text{\hspace{0.17em}⁣}\left(\frac{1}{4}\right)}{K\text{\hspace{0.17em}⁣}\left(\frac{3}{4}\right)}=0.781700961348056\dots ,$$ $$\mathrm{Area}({\mathrm{\Sigma }}_B^{\ast })=\frac{K\text{\hspace{0.17em}⁣}\left(\frac{3}{4}\right)}{2K\text{\hspace{0.17em}⁣}\left(\frac{1}{4}\right)}=0.639630785585503\dots$$ in the normalization $\mathrm{Vol}(T)=2\mathrm{/}3$, inradius $r_T=\sqrt{2}\mathrm{/}4$. In particular $2\mathrm{Area}({\mathrm{\Sigma }}_A^{\ast })\mathrm{Area}({\mathrm{\Sigma }}_B^{\ast })=1$, 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 $\mathit{\text{tetrahedral}}\mathit{\text{four}}\mathit{\text{edges}}$ and the $\mathit{\text{octahedral}}\mathit{\text{four}}\mathit{\text{edges}}$ — 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 ${\tilde{A}}_3$, confirming a remark of Gandy, Cvijović, et al. (1999).