text: A packing that can be described as the orbit of a body under the action of a double lattice is called a double lattice packing. In many cases the highest known packing density for a body is achieved by a double lattice. Examples include the regular pentagon, heptagon, and nonagon and the equilateral triangular bipyramid.
  Włodzimierz Kuperberg and Greg Kuperberg showed that all convex planar bodies can pack at a density of at least  by using a double lattice.
  In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that the double lattice packing of the regular pentagon has the optimal density among all packings of regular pentagons in the plane. This packing has been used as a decorative pattern in China since at least 1900, and in this context has been called the "pentagonal ice-ray". , the proof of its optimality has not yet been refereed and published.
  It has been conjectured that, among all convex shapes, the regular heptagon has the lowest packing density for its optimal double lattice packing, but this remains unproven.