Schwarz P and Schwarz 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,√3,√3, 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 ∂ T orthogonally; the symmetry group D2d of T fixes one and interchanges the other two. We identify all three completely. The distinguished disk is a patch of $\textbf{Schwarz's P surface}$ and the congruent pair are patches of $\textbf{Schwarz's D surface}$, with exact areas Area(Σ*A) = (K (1/4))/(K (3/4)) = 0.781700961348056…, Area(Σ*B) = (K (3/4))/(2 K (1/4)) = 0.639630785585503… in the normalization Vol(T)=2/3, inradius rT=√2/4. In particular 2 Area(Σ*A)Area(Σ*B)=1, which we show is the Bonnet relation for the adjoint pair (D,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 tetrahedral four edges and the octahedral four edges --- classically solved by D and 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 Ã3, confirming a remark of Gandy, Cvijović, et al. (1999).
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BibTeX
@misc{finch2026schwarz,
title = {{Schwarz $\mathrmP$ and Schwarz $\mathrmD$ as free-boundary minimal disks in a tetragonal disphenoid: exact areas}},
author = {Steven Finch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22013},
eprint = {2610.22013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22013},
}
RIS
TY - UNPB
TI - Schwarz P and Schwarz D as free-boundary minimal disks in a tetragonal disphenoid: exact areas
AU - Steven Finch
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22013
UR - https://arxiv.org/abs/2610.22013
AB - Let T be the tetragonal disphenoid whose four congruent faces are isosceles triangles with sides 2,√3,√3, 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 ∂ 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 $\textbf{Schwarz's P surface}$ and the congruent pair are patches of $\textbf{Schwarz's D surface}$, with exact areas Area(Σ^*_A) = (K (1/4))/(K (3/4)) = 0.781700961348056…, Area(Σ^*_B) = (K (3/4))/(2 K (1/4)) = 0.639630785585503… in the normalization Vol(T)=2/3, inradius r_T=√2/4. In particular 2 Area(Σ^*_A)Area(Σ^*_B)=1, which we show is the Bonnet relation for the adjoint pair (D,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 tetrahedral four edges and the octahedral four edges --- classically solved by D and 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 Ã_3, confirming a remark of Gandy, Cvijović, et al. (1999).
ER -