Similarity Classes in the Eight-Tetrahedron Longest-Edge Partition of a Regular Tetrahedron

Miguel A. Padrón, Ángel Plaza and José Pablo Suárez ()
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Miguel A. Padrón: IUMA Information and Communications System, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
Ángel Plaza: IUMA Information and Communications System, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
José Pablo Suárez: IUMA Information and Communications System, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain

Mathematics, 2023, vol. 11, issue 21, 1-13

Abstract: A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an initial regular tetrahedron R 0 , the iterative eight-tetrahedron longest-edge partition (8T-LE) of R 0 produces an infinity sequence of tetrahedral meshes, τ 0 = { R 0 } , τ 1 = { R i 1 } , τ 2 = { R i 2 } , … , τ n = { R i n } , … . In this paper, it is proven that, in the iterative process just mentioned, only two distinct similarity classes are generated. Therefore, the stability and the non-degeneracy of the generated meshes, as well as the minimum and maximum angle condition straightforwardly follow. Additionally, for a standard-shape tetrahedron quality measure ( η ) and any tetrahedron R i n ∈ τ n , n > 0 , then η R i n ≥ 2 3 η R 0 . The non-degeneracy constant is c = 2 3 in the case of the iterative 8T-LE partition of a regular tetrahedron.

Keywords: regular tetrahedron; similarity classes; 8T-LE partition; normalized sextuple; longest-edge bisection; strong stability; refinement; meshes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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