Horoball packings related to the 4-dimensional hyperbolic 24 cell honeycomb {3,4,3,4}

In this paper we study the horoball packings related to the hyperbolic 24 cell honeycomb by Coxeter-Schl?fli symbol {3,4,3,4} in the extended hyperbolic 4-spaceH 4 where we allow horoballs in different types centered at the various vertices of the 24 cell. Introducing the notion of the generalized polyhedral density function, we determine the locally densest horoball packing arrangement and its density with respect to the above regular tiling. The maximal density is ? 0:71645 which is equal to the known greatest horoball packing density in hyperbolic 4-space, given in [13].

An upper bound on the growth of Dirichlet tilings of hyperbolic spaces

It is shown that the growth rate [Formula: see text] of any [Formula: see text] faces Dirichlet tiling of [Formula: see text] is at most [Formula: see text], for an [Formula: see text], depending only on [Formula: see text] and [Formula: see text]. We do not know if there is a universal [Formula: see text], such that [Formula: see text] upperbounds the growth rate for any [Formula: see text]-regular tiling, when [Formula: see text]?

Periodic oscillations of the forced Brusselator

We study the organization of stability phases in the control parameter space of a periodically driven Brusselator. Specifically, we report high-resolution stability diagrams classifying periodic phases in terms of the number of spikes per period of their regular oscillations. Such diagrams contain accumulations of periodic oscillations with an apparently unbounded growth in the number of their spikes. In addition to the entrainment horns, we investigate the organization of oscillations in the limit of small frequencies and amplitudes of the drive. We find this limit to be free from chaotic oscillations and to display an extended and regular tiling of periodic phases. The Brusselator contains also several features discovered recently in more complex scenarios like, e.g. in lasers and in biochemical reactions, and exhibits properties which are helpful in the generic classification of entrainment in driven systems. Our stability diagrams reveal snippets of how the full classification of oscillations might look like for a wide class of flows.

Self-Assembly of Block Copolymers - from Periodic to Aperiodic Structures

Block copolymers with incompatible components are known to self-assemble into regular structures with mesoscopic length scales. This paper introduces periodic and aperiodic tiling structures from two kinds of three component terpolymer systems. One includes binary blends of linear poly(isoprene-styrene-2-vinylpyridine)(ISP) triblock terpolymers with cylindrical structures. When average composition of a blend is a little off from symmetry, i.e., composition ratio of three components are 0.24/0.59/0.17 for I/S/P, the system show unusual cylindrical array, where gray P domain has 5 black I domain neighbors, however it has found this structure is a periodic one with hexagonal symmetry. The other system is star-shaped terpolymers from the same polymer species, that is polyisoprene(I), polystyrene(S) and poly(2-vinylpyridine)(P). It was clarified that if composition ratio is in the vicinity of 1/1/1 for I/S/P, terpolymers tend to exhibit cylindrical structures whose cross sections possess the feature of regular two-dimensional tilings, that is, Archimedean tilings. However, when the ratio is varied to some extent, the structure formed deviates from regular tiling, and finally the sample, I1S2.7P2.5, was confirmed to show quasicrystalline tiling with dodecagonal symmetry as also verified by X-ray diffraction experiments.[1]

Chaotic Dynamical Systems Associated with Tilings of RN

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of RN, whose most familiar example is provided by the N-dimensional torus TN. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup.

Polyiamonds and Polyhexes with Minimum Site-Perimeter and Achievement Games

An animal is an edge connected set of finitely many cells of a regular tiling of the plane. The site-perimeter of an animal is the number of empty cells connected to the animal by an edge. The minimum site-perimeter with a given cell size is found for animals on the triangular and hexagonal grid. The formulas are used to show the effectiveness of a simple random strategy in full set animal achievement games.

Colorings of hyperbolic plane crystallographic patterns

In color symmetry the basic problem has always been to classify symmetrically colored symmetrical patterns [13]. An important step in the study of color symmetry in the hyperbolic plane is the determination of a systematic approach in arriving at colored symmetrical hyperbolic patterns. For a given uncolored semi-regular tiling with symmetry group