Regular Packings of PG (3,q)

Regular Sphere Packings

A collection of non-overlapping spheres in the space is called a packing. Two spheres are said to be neighbours if they have a boundary point in common. A packing is called k-regular if each sphere has exactly k neighbours. We are concerned with the following question. What is the minimum number of not necessarily congruent spheres which may form a k-regular packing? In general, for which natur...

Regular Circle Packings

Dense Crystalline Dimer Packings of Regular Tetrahedra

We present the densest known packing of regular tetrahedra with density φ = 4000 4671 = 0.856347 . . . . Like the recently discovered packings of Kallus et al. and Torquato–Jiao, our packing is crystalline with a unit cell of four tetrahedra forming two triangular dipyramids (dimer clusters). We show that our packing has maximal density within a three-parameter family of dimer packings. Numeric...

Dense regular packings of irregular nonconvex particles.

We present a new numerical scheme to study systems of nonconvex, irregular, and punctured particles in an efficient manner. We employ this method to analyze regular packings of odd-shaped bodies, both from a nanoparticle and from a computational geometry perspective. Besides determining close-packed structures for 17 irregular shapes, we confirm several conjectures for the packings of a large s...

On regular semiovals in PG(2, q)

In this paper we prove that a point set in PG(2, q) meeting every line in 0, 1 or r points and having a unique tangent at each of its points is either an oval or a unital. This answers a question of Blokhuis and Szőnyi [1].