Related papers: Freeform Islamic Geometric Patterns
We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct sum of free groups, then the complements of the arrangements are homotopy equivalent. For…
The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is…
We propose a framework to model elastic properties of polycrystals by coupling crystal orientational degrees of freedom with elastic strains. Our model encodes crystal symmetries and takes into account explicitly the strain compatibility…
The use of pyramid elements is crucial to the construction of efficient hex-dominant meshes. For conforming nodal finite element methods with mixed element types, it is advantageous for nodal distributions on the faces of the pyramid to…
One well studied way to construct quasicrystalline tilings is via inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We present a…
We are introducing a general framework for the construction of polyhedra and simplicial comlexes that are {\em bifoldable}, i.e. foldable into two two different planes. This vastly generalizes Origami folds known as the Miura pattern, the…
The reader is informed about a method for the objective identification of the plane symmetry group of a "noisy" crystal pattern. Without giving numerical details, this information theory based method is applied to two beautiful pieces of…
We construct models, by three-dimensional arrays of ccc posets, where many classical cardinal characteristics of the continuum are pairwise different.
We show that, for an arrangement of subspaces in a complex vector space with geometric intersection lattice, the complement of the arrangement is formal. We prove that the Morgan rational model for such an arrangement complement is formal…
We develop an optimization-based method to model smocking, a surface embroidery technique that provides decorative geometric texturing while maintaining stretch properties of the fabric. During smocking, multiple pairs of points on the…
We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex…
The shape of semiflexible polymer rings is studied over their whole range of flexibility. Investigating the joint distribution of asphericity and nature of asphericity as well as their respective averages we find two distinct shape regimes…
In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties.…
Traditional origami structures can be continuously deformed back to a flat sheet of paper, while traditional kirigami requires glue or seams in order to maintain its rigidity. In the former, non-trivial geometry can be created through…
We construct multi-field generalisations of the Cosserat continuum model on the basis of the square lattice model that takes into account rotational degree of freedom of microstructural elements. This approach allows us to model not only…
This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the {\it generalized corner-vector method}. First we establish a uniform upper bound for the degree…
Optical vortices carrying orbital angular momentum offer additional degrees of freedom. According to the orientation of orbital angular momentum, optical vortices can be classified into spatial optical vortex beam carrying…
We study homogeneity aspects of metric spaces in which all triples of distinct points admit pairwise different distances; such spaces are called isosceles-free. In particular, we characterize all homogeneous isosceles-free spaces up to…
This work presents a recursive construction for simple $t$-designs using resolutions of the ingredient designs. The result extends a construction of $t$-designs in our recent paper [39]. Essentially, the method in [39] describes the blocks…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…