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PreprintCC BY-NC-ND 4.0HAL

Degenerating orbits of the Longest Edge Bisection process

  • Karim Adiprasito
  • Daniel Kalmanovich
  • Yaar Solomon
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Abstract

We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: • There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. • There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. • If parametrizing the space of d-dimensional simplices by independent standard Gaussian vectors, then as d increases, a random simplex degenerates asymptotically almost surely. This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.

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Cite

BibTeXRIS
BibTeX
@misc{adiprasito2026degenerating,
  title = {{Degenerating orbits of the Longest Edge Bisection process}},
  author = {Karim Adiprasito and Daniel Kalmanovich and Yaar Solomon},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08846},
  eprint = {2609.08846},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05742991v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Degenerating orbits of the Longest Edge Bisection process
AU  - Karim Adiprasito
AU  - Daniel Kalmanovich
AU  - Yaar Solomon
PY  - 2026
DA  - 2026/09/08
PB  - HAL
DO  - 10.48550/arxiv.2609.08846
UR  - https://hal.science/hal-05742991v1
AB  - We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: • There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. • There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. • If parametrizing the space of d-dimensional simplices by independent standard Gaussian vectors, then as d increases, a random simplex degenerates asymptotically almost surely. This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.
ER  -