Packing of R3 by crosses

The existence of tilings of R^n by crosses, a cluster of unit cubes comprising a central one and 2n arms, has been studied by several authors. We have completely solved the problem for R^2, characterizing the crosses which lattice tile R^2, as well as determining the maximum packing density for the crosses which do not lattice tile the plane. In this paper we motivate a similar approach to study lattice packings of R^3 by crosses.The existence of tilings of Rn by crosses, a cluster of unit cubes comprising a central one and 2n arms, has been studied by several authors. We have completely solved the problem for R2 characterizing the crosses which lattice tile R2 as well as determining the maximum packing density for the crosses which do not lattice tile the plane. In this paper we motivate a similar approach to study lattice packings of R3 by crosses.publishe

Image sets of folding surfaces

Isometric foldings are a special class of length-preserving maps of Riemannian manifolds and were initially studied by S. Robertson. For an explanation of their topological and combinatorial properties, see the related works of Ana Breda, Altino Santos, M. El-Ghoul, and E. M. Elkholy. Here, we explore some properties of the singular set and describe the image set of planar, spherical, and hyperbolic foldings