Related papers: An algebraic approach to circle packing
Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list of geometric primitives, e.g., splines, triangles, and hexahedra,…
Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials.…
We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite…
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which…
We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a…
Random packing of unoriented regular polygons and star polygons on a two-dimensional flat, continuous surface is studied numerically using random sequential adsorption algorithm. Obtained results are analyzed to determine saturated random…
This paper proves a bottom-left placement theorem for the rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given rectangles into a rectangular container without overlapping, then we can achieve a…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing…
Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.
Packing problems have been of great interest in many diverse contexts for many centuries. The optimal packing of identical objects has been often invoked to understand the nature of low temperature phases of matter. In celebrated work,…
A Bl-packing is a (branched) circle packing that ``properly covers'' the unit disc. We establish some fundamental properties of such packings. We give necessary and sufficient conditions for their existence, prove their uniqueness, and show…
In this paper we present a new algorithm for a layout optimization problem: this concerns the placement of weighted polygons inside a circular container, the two objectives being to minimize imbalance of mass and to minimize the radius of…
Haag, Kertzer, Rickards, and Stange disprove the Local-Global Conjecture for Apollonian circle packings. We extend their disproof to four more types of integral circle packing: the octahedral, cubic, square, and triangular packings. In each…
The problem of packing unequal circles into a circular container stands as a classic and challenging optimization problem in computational geometry. This study introduces a suite of innovative and efficient methods to tackle this problem.…
Let $S$ be a set of $n$ points in the unit square $[0,1]^2$, one of which is the origin. We construct $n$ pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in $S$, and the…
In this work we propose a heuristic algorithm for the layout optimization for disks installed in a rotating circular container. This is a unequal circle packing problem with additional balance constraints. It proved to be an NP-hard…
Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought to be solved, as for each pair $(p,q)$ a new parabola has to be drawn.…
This paper deals with the problem of circle packing, in which the largest radii circle is to be fit in a confined space filled with arbitrary circles of different radii and centers. A circle packing problem is one of a variety of cutting…
We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…