Modeling crystal growth: Polyhedra with faces parallel to the planes x1 ± x2 ± x3 = 0

  • Jerzy Grzybowski Faculty of Mathematics and Computer Science, Adam Mickiewicz University
  • Ryszard Urbański Faculty of Mathematics and Computer Science, Adam Mickiewicz University
Keywords: crystal growth, abstract cone of convex polyhedra, modified Minkowski addition

Abstract

A growing crystal can be understood as a series of polyhedra with faces parallel to planes from a fixed finite set. Such series of polyhedra forms a curve, often broken line, in a vector space of virtual polyhedra. In this paper we apply a modified Minkowski addition to study geometry of a cone of all polyhedra with faces parallel the faces of regular octahedron.

References

J. Grzybowski and R. Urba´nski, Crystal growth in terms of Minkowski–R°adstr¨om– H¨ormander space, Bull. Soc. Sci. Lettres, L´od´z S´er. Rech. D´eform. 59, no. 1 (2009), 91–101. J. Grzybowski and R. Urbański, Modeling crystal growth: Polyhedra with faces parallel to planes from a fixed finite set, Bull. Soc. Sci. Lettres, L´od´z Ser. R´ech. D´eform. 65, no. 1 (2015), 21–36.

D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, Fractional Arithmetic with Convex Sets, Mathematics and Its Applications, Kluwer Academic. Publisher, Dortrecht–Boston–London, 2002.

A. Reinhold and H. Briesen, Convex geometry for the morphological modeling and characterization of crystal shapes, Part. Part. Syst. Charact. 28 (2011), 37–56.

R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2014.

T. Stroiński, Classes of convex polyhedra closed under Minkowski addition, Bull. Soc. Sci. Lettres, L´od´z S´er. Rech. D´eform. 67, no. 2 (2017), 43–60.

Published
2019-04-26
Section
Articles