Multiform Longest Edge Bisection of Tetrahedra via Sextuple Permutations
Abstract
We introduce a new formulation of the Longest Edge Bisection (LEB) of tetrahedra entirely in sextuple space R6, where tetrahedra are represented by the squares of their edge lengths. This representation renders the LEB refinement equations fully linear and eliminates the need for coordinate-based data structures. A central difficulty in three-dimensional LEB arises when a tetrahedron possesses multiple longest edges, making the refinement rule intrinsically multivalued. We formalize this phenomenon through the notion of Multiform Longest Edge Bisection (MLEB), which systematically explores all admissible longest-edge choices. To encode this multivalued behavior, we introduce the concept of bisection patterns, defined as sequences of sextuple permutations governing the refinement process. We prove that the set of sextuples sharing a common LEB pattern forms a convex region in R6. For structurally significant families of tetrahedra, including the R1+ family and the Liu-Joe family, we show that the infinite refinement tree collapses into a finite directed graph with eight states. Remarkably, both families are governed by the same graph, differing only in their initial state. This directed-graph formulation provides a unified combinatorial description of the refinement process and offers an efficient computational framework for deep iterative LEB analysis.
Related
- Degenerating orbits of the Longest Edge Bisection process
- Closure complexity of longest-edge bisection for triangular meshes
- On degeneration of tetrahedra under longest-edge n-section refinement
- The longest increasing subsequences of Luce permutations and permutons
- Largest-dihedral-angle bisection algorithm does not preserve mesh regularity for tetrahedral partitions
BibTeX
@misc{trujillo2026multiform,
title = {{Multiform Longest Edge Bisection of Tetrahedra via Sextuple Permutations}},
author = {Agustin Trujillo and Jose Pablo Suarez and Tania Moreno-García},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26522},
eprint = {2609.26522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26522},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Multiform Longest Edge Bisection of Tetrahedra via Sextuple Permutations
AU - Agustin Trujillo
AU - Jose Pablo Suarez
AU - Tania Moreno-García
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26522
UR - https://arxiv.org/abs/2609.26522
AB - We introduce a new formulation of the Longest Edge Bisection (LEB) of tetrahedra entirely in sextuple space R6, where tetrahedra are represented by the squares of their edge lengths. This representation renders the LEB refinement equations fully linear and eliminates the need for coordinate-based data structures. A central difficulty in three-dimensional LEB arises when a tetrahedron possesses multiple longest edges, making the refinement rule intrinsically multivalued. We formalize this phenomenon through the notion of Multiform Longest Edge Bisection (MLEB), which systematically explores all admissible longest-edge choices. To encode this multivalued behavior, we introduce the concept of bisection patterns, defined as sequences of sextuple permutations governing the refinement process. We prove that the set of sextuples sharing a common LEB pattern forms a convex region in R6. For structurally significant families of tetrahedra, including the R1+ family and the Liu-Joe family, we show that the infinite refinement tree collapses into a finite directed graph with eight states. Remarkably, both families are governed by the same graph, differing only in their initial state. This directed-graph formulation provides a unified combinatorial description of the refinement process and offers an efficient computational framework for deep iterative LEB analysis.
ER -