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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. [ABSTRACT FROM AUTHOR]

In this paper we present growing crystal as a point in finite dimentional space moving inside a cone SG ⊂ ℝm corresponding to the family BG of polyhedra with faces parallel to planes from a fixed finite set G. For this purpose we modify Minkowski addition of convex sets and explain in depth a representation of a growth of certain crystal of NaCl. We also provide a number of illustrations of a respective cone S'G of symmetric polyhedra and the trajectory of our growing crystal which is a broken line. [ABSTRACT FROM AUTHOR]