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

Author

Listed:
  • Miguel A. Padrón

    (IUMA Information and Communications System, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
    These authors contributed equally to this work.)

  • Ángel Plaza

    (IUMA Information and Communications System, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
    These authors contributed equally to this work.)

  • José Pablo Suárez

    (IUMA Information and Communications System, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Miguel A. Padrón & Ángel Plaza & José Pablo Suárez, 2023. "Similarity Classes in the Eight-Tetrahedron Longest-Edge Partition of a Regular Tetrahedron," Mathematics, MDPI, vol. 11(21), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4456-:d:1268904
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    References listed on IDEAS

    as
    1. Padrón, Miguel A. & Plaza, Ángel, 2020. "The 8T-LE partition applied to the obtuse triangulations of the 3D-cube," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 176(C), pages 254-265.
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