2,324 research outputs found
Tessellations and Pattern Formation in Plant Growth and Development
The shoot apical meristem (SAM) is a dome-shaped collection of cells at the
apex of growing plants from which all above-ground tissue ultimately derives.
In Arabidopsis thaliana (thale cress), a small flowering weed of the
Brassicaceae family (related to mustard and cabbage), the SAM typically
contains some three to five hundred cells that range from five to ten microns
in diameter. These cells are organized into several distinct zones that
maintain their topological and functional relationships throughout the life of
the plant. As the plant grows, organs (primordia) form on its surface flanks in
a phyllotactic pattern that develop into new shoots, leaves, and flowers.
Cross-sections through the meristem reveal a pattern of polygonal tessellation
that is suggestive of Voronoi diagrams derived from the centroids of cellular
nuclei. In this chapter we explore some of the properties of these patterns
within the meristem and explore the applicability of simple, standard
mathematical models of their geometry.Comment: Originally presented at: "The World is a Jigsaw: Tessellations in the
Sciences," Lorentz Center, Leiden, The Netherlands, March 200
Fitting Voronoi Diagrams to Planar Tesselations
Given a tesselation of the plane, defined by a planar straight-line graph
, we want to find a minimal set of points in the plane, such that the
Voronoi diagram associated with "fits" \ . This is the Generalized
Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered
recently in \cite{Baner12}. Here we give an algorithm that solves this problem
with a number of points that is linear in the size of , assuming that the
smallest angle in is constant.Comment: 14 pages, 8 figures, 1 table. Presented at IWOCA 2013 (Int. Workshop
on Combinatorial Algorithms), Rouen, France, July 201
Subset Warping: Rubber Sheeting with Cuts
Image warping, often referred to as "rubber sheeting" represents the deformation of a domain image space into a range image space. In this paper, a technique is described which extends the definition of a rubber-sheet transformation to allow a polygonal region to be warped into one or more subsets of itself, where the subsets may be multiply connected. To do this, it constructs a set of "slits" in the domain image, which correspond to discontinuities in the range image, using a technique based on generalized Voronoi diagrams. The concept of medial axis is extended to describe inner and outer medial contours of a polygon. Polygonal regions are decomposed into annular subregions, and path homotopies are introduced to describe the annular subregions. These constructions motivate the definition of a ladder, which guides the construction of grid point pairs necessary to effect the warp itself
Pomelo, a tool for computing Generic Set Voronoi Diagrams of Aspherical Particles of Arbitrary Shape
We describe the development of a new software tool, called "Pomelo", for the
calculation of Set Voronoi diagrams. Voronoi diagrams are a spatial partition
of the space around the particles into separate Voronoi cells, e.g. applicable
to granular materials. A generalization of the conventional Voronoi diagram for
points or monodisperse spheres is the Set Voronoi diagram, also known as
navigational map or tessellation by zone of influence. In this construction, a
Set Voronoi cell contains the volume that is closer to the surface of one
particle than to the surface of any other particle. This is required for
aspherical or polydisperse systems.
Pomelo is designed to be easy to use and as generic as possible. It directly
supports common particle shapes and offers a generic mode, which allows to deal
with any type of particles that can be described mathematically. Pomelo can
create output in different standard formats, which allows direct visualization
and further processing. Finally, we describe three applications of the Set
Voronoi code in granular and soft matter physics, namely the problem of
packings of ellipsoidal particles with varying degrees of particle-particle
friction, mechanical stable packings of tetrahedra and a model for liquid
crystal systems of particles with shapes reminiscent of pearsComment: 4 pages, 9 figures, Submitted to Powders and Grains 201
The oscillating behavior of the pair correlation function in galaxies
The pair correlation function (PCF) for galaxies presents typical
oscillations in the range 20-200 Mpc/h which are named baryon acoustic
oscillation (BAO). We first review and test the oscillations of the PCF when
the 2D/3D vertexes of the Poissonian Voronoi Tessellation (PVT) are considered.
We then model the behavior of the PCF at a small scale in the presence of an
auto gravitating medium having a line/plane of symmetry in 2D/3D. The analysis
of the PCF in an astrophysical context was split into two, adopting a
non-Poissonian Voronoi Tessellation (NPVT). We first analyzed the case of a 2D
cut which covers few voids and a 2D cut which covers approximately 50 voids.
The obtained PCF in the case of many voids was then discussed in comparison to
the bootstrap predictions for a PVT process and the observed PCF for an
astronomical catalog. An approximated formula which connects the averaged
radius of the cosmic voids to the first minimum of the PCF is given.Comment: 19 pages 14 figure
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