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A Plasticity Principle of Closed Hexahedra in the Three-Dimensional Euclidean Space

A Plasticity Principle of Closed Hexahedra in the Three-Dimensional Euclidean Space We prove a plasticity principle of closed hexahedra in the three dimensional Euclidean space which states that: Suppose that the closed hexahedron A 1 A 2⋯A 5 has an interior weighted Fermat-Torricelli point A 0 with respects to the weights B i and let α i0j =∠A i A 0 A j . Then these 10 angles are determined completely by 7 of them and considering these five prescribed rays which meet at the weighted Fermat-Torricelli point A 0, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. By applying the plasticity principle of closed hexahedra to the n-inverse weighted Fermat-Torricelli problem, we derive some new evolutionary structures of closed polyhedra for n>5. Finally, we derive some evolutionary structures of pentagons in the two dimensional Euclidean space from the plasticity of weighted hexahedra as a limiting case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

A Plasticity Principle of Closed Hexahedra in the Three-Dimensional Euclidean Space

Acta Applicandae Mathematicae , Volume 125 (1) – Aug 25, 2012

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References (11)

Publisher
Springer Journals
Copyright
Copyright © 2012 by Springer Science+Business Media B.V.
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Statistical Physics, Dynamical Systems and Complexity; Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-012-9778-z
Publisher site
See Article on Publisher Site

Abstract

We prove a plasticity principle of closed hexahedra in the three dimensional Euclidean space which states that: Suppose that the closed hexahedron A 1 A 2⋯A 5 has an interior weighted Fermat-Torricelli point A 0 with respects to the weights B i and let α i0j =∠A i A 0 A j . Then these 10 angles are determined completely by 7 of them and considering these five prescribed rays which meet at the weighted Fermat-Torricelli point A 0, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. By applying the plasticity principle of closed hexahedra to the n-inverse weighted Fermat-Torricelli problem, we derive some new evolutionary structures of closed polyhedra for n>5. Finally, we derive some evolutionary structures of pentagons in the two dimensional Euclidean space from the plasticity of weighted hexahedra as a limiting case.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Aug 25, 2012

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