scholarly journals Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation

2019 ◽  
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel
Keyword(s):  
Voronoi Tessellation ◽  
Voronoi Diagrams ◽  
Random Patterns ◽  
Seed Points ◽  
Symmetric Patterns

Voronoi entropy for the random patterns and patterns demonstrating various elements of symmetry are calculated. The symmetric patterns are characterized by the values of the Voronoi entropy very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points, constituting the pattern. The extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables revealing of the elements of symmetry of the pattern.

scholarly journals Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 452 ◽  
Author(s):  
Bormashenko ◽  
Legchenkova ◽  
Frenkel
Keyword(s):  
Voronoi Tessellation ◽  
Voronoi Diagrams ◽  
Random Patterns ◽  
Seed Points ◽  
Symmetric Patterns

The Voronoi entropy for random patterns and patterns demonstrating various elements of symmetry was calculated. The symmetric patterns were characterized by the values of the Voronoi entropy being very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points constituting the pattern. Extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables the elements of symmetry of the patterns to be revealed.

scholarly journals Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1659
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel ◽  
Nir Shvalb ◽  
Shraga Shoval
Keyword(s):  
Symmetry Group ◽  
Voronoi Diagrams ◽  
Continuous Measure ◽  
Penrose Tiling ◽  
Continuous Symmetry ◽  
Measure Of Symmetry ◽  
Symmetry Measure ◽  
Seed Points

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

Violations of Symmetry and Repetition in Visual Patterns

Perception ◽  
1975 ◽  
Vol 4 (3) ◽  
pp. 239-249 ◽  
Author(s):  
Vicky G Bruce ◽  
Michael J Morgan
Keyword(s):  
Decision Time ◽  
Considerable Extent ◽  
Visual Patterns ◽  
Repeated Pattern ◽  
Random Patterns ◽  
Line Patterns ◽  
Symmetric Patterns

Observers were asked to detect small violations in otherwise left-right symmetric or repeated line patterns. The decision time for symmetric patterns tended to be shorter than for repeated patterns, confirming an intuitive impression that symmetry is a very easily extracted feature, even of otherwise random patterns. The importance of the pattern midline in determining the salience of symmetry was investigated by systematically controlling the position of the violations in the patterns. Small departures from symmetry near the midline were much easier to detect than the same violations in a repeated pattern, but there were no differences where the violation was at the edge of the pattern. The salience of symmetry seems to a considerable extent to depend upon the ease of comparing spatially-contiguous elements near the midline of the pattern. However, both symmetry and departures from it are easier to detect than repetition, even when the nature of the violation is random.

scholarly journals The transition from order to disorder in Voronoi Diagrams with applications

2018 ◽  
Vol 10 (6) ◽  
pp. 82
Author(s):  
L. Zaninetti
Keyword(s):  
Local Structure ◽  
Geometrical Parameter ◽  
Voronoi Tessellation ◽  
Voronoi Diagrams ◽  
Disordered Structures ◽  
Cosmic Voids ◽  
Supercritical Ammonia

The transition from ordered to disordered structures in Voronoi tessellation is obtained by perturbing the seeds that were originally identified with two types of lattice in 2D and one type in 3D. The area in 2D and the volume in 3D are modeled with the Kiang function. A new relationship that models the scaling of the Kiang function with a geometrical parameter is introduced. A first application models the local structure of sub- and supercritical ammonia as function of the temperature and a second application models the volumes of cosmic voids.

scholarly journals Numerical Investigation of Effective Thermal Conductivity of Strut-Based Cellular Structures Designed by Spatial Voronoi Tessellation

Materials ◽  
2020 ◽  
Vol 14 (1) ◽  
pp. 138
Author(s):  
Minghao Zhang ◽  
Junteng Shang ◽  
Shiyue Guo ◽  
Boyoung Hur ◽  
Xuezheng Yue
Keyword(s):  
Thermal Conductivity ◽  
Effective Thermal Conductivity ◽  
Dimensional Space ◽  
Voronoi Tessellation ◽  
Matrix Material ◽  
Three Dimensional ◽  
Filling Material ◽  
The Matrix ◽  
Seed Points ◽  
Insulation Performance

Porous materials possess light weight and excellent thermal insulation performance. For disordered porous structures, the number of seed points is an important design parameter which is closely related to the morphology and mean pore size of the structure. Based on the arrangement of points in three-dimensional space, seven kinds of structures were designed by spatial Voronoi tessellation in this paper. The effect of the number of seed points on effective thermal conductivity for Voronoi was studied. Numerical simulation was conducted to research the effects of structural porosity, filling material and structural orientation on the effective thermal conductivity and heat transfer characteristics. The results showed that the effective thermal conductivity is closely related to the porosity and the matrix material. Different number and arrangement of seed points make the structure have different anisotropic performance due to different thermal paths. In addition, required the least number of seed points was obtained for the designation of isotropic random Voronoi.

scholarly journals On the Structural Properties of Voronoi Diagrams

KoG ◽  
2021 ◽  
pp. 72-77
Author(s):  
Leonard Weydemann ◽  
Christian Clemenz ◽  
Clemens Preisinger
Keyword(s):  
Physical Properties ◽  
Structural Properties ◽  
Voronoi Diagram ◽  
Structural Engineering ◽  
Voronoi Tessellation ◽  
Voronoi Diagrams ◽  
Biological Processes ◽  
Voronoi Tessellations ◽  
Cell Tissue

A Voronoi diagram is a tessellation technique, which subdivides space into regions in proximity to a given set of objects called seeds. Patterns emerging naturally in biological processes (for example, in cell tissue) can be modelled in a biomimicry process via Voronoi diagrams. As they originate in nature, we investigate the physical properties of such patterns to determine whether they are optimal given the constraints imposed by surrounding geometry and natural forces. This paper describes under what circumstances the Voronoi tessellation has optimal (structural) properties by surveying recent studies that apply this tessellation technique across different scales. To investigate the properties of random and optimized Voronoi tessellations in comparison to a regular tessellation method, we additionally run and evaluate a simulation in Karamba3D, a parametric structural engineering tool for Rhinoceros3D. The novelty of this research lies in presenting a simple and straightforward simulation of Voronoi diagrams and highlighting how and where their advantages over regular tessellations can be exploited by surveying more advanced approaches as found in literature.

Anticipatory memory for regular and random patterns.

2020 ◽  
Vol 46 (9) ◽  
pp. 965-978 ◽  
Author(s):  
Irem Yildirim ◽  
Helene Intraub
Keyword(s):  
Random Patterns

scholarly journals An optical observational cluster mass function at z ∼ 1 with the ORELSE survey

2021 ◽  
Vol 502 (3) ◽  
pp. 3942-3954
Author(s):  
D Hung ◽  
B C Lemaux ◽  
R R Gal ◽  
A R Tomczak ◽  
L M Lubin ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Large Scale ◽  
Near Infrared ◽  
Function Analysis ◽  
Voronoi Tessellation ◽  
Cosmological Parameters ◽  
Mass Range ◽  
Mass Function ◽  
Cluster Mass ◽  
Theoretical Mass

ABSTRACT We present a new mass function of galaxy clusters and groups using optical/near-infrared (NIR) wavelength spectroscopic and photometric data from the Observations of Redshift Evolution in Large-Scale Environments (ORELSE) survey. At z ∼ 1, cluster mass function studies are rare regardless of wavelength and have never been attempted from an optical/NIR perspective. This work serves as a proof of concept that z ∼ 1 cluster mass functions are achievable without supplemental X-ray or Sunyaev-Zel’dovich data. Measurements of the cluster mass function provide important contraints on cosmological parameters and are complementary to other probes. With ORELSE, a new cluster finding technique based on Voronoi tessellation Monte Carlo (VMC) mapping, and rigorous purity and completeness testing, we have obtained ∼240 galaxy overdensity candidates in the redshift range 0.55 < z < 1.37 at a mass range of 13.6 < log (M/M⊙) < 14.8. This mass range is comparable to existing optical cluster mass function studies for the local universe. Our candidate numbers vary based on the choice of multiple input parameters related to detection and characterization in our cluster finding algorithm, which we incorporated into the mass function analysis through a Monte Carlo scheme. We find cosmological constraints on the matter density, Ωm, and the amplitude of fluctuations, σ8, of $\Omega _{m} = 0.250^{+0.104}_{-0.099}$ and $\sigma _{8} = 1.150^{+0.260}_{-0.163}$. While our Ωm value is close to concordance, our σ8 value is ∼2σ higher because of the inflated observed number densities compared to theoretical mass function models owing to how our survey targeted overdense regions. With Euclid and several other large, unbiased optical surveys on the horizon, VMC mapping will enable optical/NIR cluster cosmology at redshifts much higher than what has been possible before.

Polygon visibility ordering via Voronoi diagrams

2007 ◽  
Vol 23 (7) ◽  
pp. 503-511 ◽  
Author(s):  
Shinichi Fukushige ◽  
Hiromasa Suzuki
Keyword(s):  
Voronoi Diagrams ◽  
Visibility Ordering
Sign in / Sign up

Export Citation Format

Share Document