Skip to content
DraconiticResearch
Search Sources API
182,469 studies31 sourcesUpdated 31 min ago
PreprintCC BY 4.0arXiv

Largest-dihedral-angle bisection algorithm does not preserve mesh regularity for tetrahedral partitions

  • Sergey Korotov
  • Jérôme Michaud
Open at sourcePDFDOICite

Abstract

We present a fixed nondegenerate tetrahedron with a nested largest-dihedral-anglebisection branch that collapses onto a unit edge. The selected dihedral angles to bisect are uniquely largest ones at every bisection step. Some face angles of the branch elements tend to zero, the corresponding diameter-to-inradius ratio diverges, and the diameters of the elements produced do not tend to zero, although every dihedral angle stays uniformly away from both 0 and π and all face and dihedral angles satisfy a uniform maximum-angle bound. The proof is based on an exact invariant-range argument. This example shows that the largest-dihedral-angle bisection algorithm can produce degenerating tetrahedral partitions.

Related

  1. Degenerating orbits of the Longest Edge Bisection process
    Karim Adiprasito, Daniel Kalmanovich, Yaar Solomon
    HALPreprintCC BY-NC-ND 4.0Full textCombinatorics & discrete mathematicsDynamical systemsHALarXiv
  2. Closure complexity of longest-edge bisection for triangular meshes
    Yuwen Li, Zhiyuan Yang
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  3. Maximum cut and maximum bisection in random regular graphs
    P. M. Aronow, Patrick Lopatto
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsProbabilityarXiv
  4. Multiform Longest Edge Bisection of Tetrahedra via Sextuple Permutations
    Agustin Trujillo, Jose Pablo Suarez, Tania Moreno-García
    arXivPreprintCC BY-NC-ND 4.0Full textNumerical analysis & scientific computingarXiv
  5. Classification of skew left braces with additive group isomorphic to the infinite dihedral group
    Akihide Hanaki, Yuto Sakata, Hiroki Yoshino
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{korotov2026largestdihedralangle,
  title = {{Largest-dihedral-angle bisection algorithm does not preserve mesh regularity for tetrahedral partitions}},
  author = {Sergey Korotov and Jérôme Michaud},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18788},
  eprint = {2609.18788},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18788},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Largest-dihedral-angle bisection algorithm does not preserve mesh regularity for tetrahedral partitions
AU  - Sergey Korotov
AU  - Jérôme Michaud
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18788
UR  - https://arxiv.org/abs/2609.18788
AB  - We present a fixed nondegenerate tetrahedron with a nested largest-dihedral-anglebisection branch that collapses onto a unit edge. The selected dihedral angles to bisect are uniquely largest ones at every bisection step. Some face angles of the branch elements tend to zero, the corresponding diameter-to-inradius ratio diverges, and the diameters of the elements produced do not tend to zero, although every dihedral angle stays uniformly away from both 0 and π and all face and dihedral angles satisfy a uniform maximum-angle bound. The proof is based on an exact invariant-range argument. This example shows that the largest-dihedral-angle bisection algorithm can produce degenerating tetrahedral partitions.
ER  -