Related papers: Constructing Interlocking Assemblies with Crystall…
We develop further Cannon's method of cone types for finding the growth function of a group, which can also be used to find the coordination sequences of certain infinite graphs. We then apply this method to compute the growth functions and…
We argue for Brackets Consistency to be a `Pillar of Geometry', i.e. a foundational approach, other Pillars being 1) Euclid's constructive approach, 2) the algebraic approach, 3) the projective approach, and 4) the geometrical automorphism…
A panoptic view of architectured planar lattices based on star-polygon tilings was developed. Four star-polygon-based lattice sub-families, formed of systematically arranged triangles, squares, or hexagons, were investigated numerically and…
The Pattern self-Assembly Tile set Synthesis (PATS) problem is to determine a set of coloured tiles that self-assemble to implement a given rectangular colour pattern. We give an exhaustive branch-and-bound algorithm to find tile sets of…
In this paper, we introduce a natural dynamical analogue of crystalline order, which we call choreographic order. In an ordinary (static) crystal, a high degree of symmetry may be achieved through a careful arrangement of the fundamental…
Creating materials with structure that is independently controllable at a range of scales requires breaking naturally occurring hierarchies. Breaking these hierarchies can be achieved via the decoupling of building block attributes from…
Ring polymers are characterized by topology-specific entanglements called threadings. In the limit of large rings, it is conjectured that a "topological glass" should emerge due to the proliferation of threadings. In this study, we used…
Anisotropy at the level of the inter-particle interaction provides the particles with specific instructions for the self-assembly of target structures. The ability to synthesize non-spherical colloids, together with the possibility of…
The design of irregular planar phased arrays (PAs) characterized by a highly-modular architecture is addressed. By exploiting the property of self-replicating tile shapes, also known as rep-tiles, the arising array layouts consist of tiles…
Entangled structures such as textiles and architected materials are often doubly periodic. Due to this property and their finite transverse thickness, the symmetries of these materials are described by the crystallographic layer groups.…
Mutually repelling particles form spontaneously ordered clusters when forced into confinement. The clusters may adopt similar spatial arrangements even if the underlying particle interactions are contrastingly different. Here we demonstrate…
We consider a class of cut-and-project sets $\Lambda = \Lambda_F \times \zahl$ in the plane. Let $L=\Lambda+w\real$, $w\in\real^2$, be a countable union of parallel lines. Then either (1) $L$ is a discrete family of lines, (2) $L$ is a…
The ability to control forces between sub-micron-scale building blocks offers considerable potential for designing new materials through self-assembly. A typical paradigm is to first identify a particular (crystal) structure that has some…
A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which…
If particles interact according to isotropic pair potentials that favor multiple length scales, in principle a large variety of different complex structures can be achieved by self-assembly. We present, motivate, and discuss a conjecture…
Many remarkably robust, rapid and spontaneous self-assembly phenomena in nature can be modeled geometrically starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly…
This paper explores the problem of clustering ensemble, which aims to combine multiple base clusterings to produce better performance than that of the individual one. The existing clustering ensemble methods generally construct a…
This paper presents a complete Pascal interpolation scheme for use in the plane geometry mapping applied in association with numerical methods. The geometry of a domain element is approximated by a complete Pascal polynomial. The…
In this paper we consider a random partition of the plane into cells, the partition being based on the nodes and links of a {\it random planar geometric graph}. The resulting structure generalises the \emph{random \tes}\ hitherto studied in…
We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a…