Grow in accessible directions, like Voronoi diagrams

2020 ◽  
pp. 15-18
Author(s):  
Susan D'Agostino
Keyword(s):  
Voronoi Diagram ◽  
Regional Planning ◽  
Voronoi Diagrams ◽  
Real Life ◽  
Two Dimensional ◽  
Mathematics Students ◽  
Voronoi Tessellations ◽  
Dimensional Plane ◽  
A Cell

“Grow in accessible directions, like Voronoi diagrams” offers an accessible introduction to the mathematics of Voronoi diagrams—a separation of a two-dimensional plane into regions known as “cells” based on “sites.” In a Voronoi diagram, any point inside a cell is closer to the site of its cell than the site of any other cell. The discussion includes numerous real-life examples of Voronoi diagrams—also known as a Voronoi tessellations—in nature and regional planning. The discussion is supplemented with numerous hand-drawn sketches to enhance understanding. Mathematics students and enthusiasts are encouraged to draw inspiration from Voronoi diagrams by growing in accessible directions in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

scholarly journals The Voronoi theory of the normal liver lobular architecture and its applicability in hepatic zonation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
C. Lau ◽  
B. Kalantari ◽  
K. P. Batts ◽  
L. D. Ferrell ◽  
S. L. Nyberg ◽  
...  
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Normal Liver ◽  
Digital Tools ◽  
Liver Pathology ◽  
Two Dimensional ◽  
Lobular Architecture ◽  
Precise Characterization ◽  
Human Livers

AbstractThe precise characterization of the lobular architecture of the liver has been subject of investigation since the earliest historical publications, but an accurate model to describe the hepatic lobular microanatomy is yet to be proposed. Our aim was to evaluate whether Voronoi diagrams can be used to describe the classic liver lobular architecture. We examined the histology of normal porcine and human livers and analyzed the geometric relationships of various microanatomic structures utilizing digital tools. The Voronoi diagram model described the organization of the hepatic classic lobules with overall accuracy nearly 90% based on known histologic landmarks. We have also designed a Voronoi-based algorithm of hepatic zonation, which also showed an overall zonal accuracy of nearly 90%. Therefore, we have presented evidence that Voronoi diagrams represent the basis of the two-dimensional organization of the normal liver and that this concept may have wide applicability in liver pathology and research.

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.

scholarly journals Tunable characteristics of low-frequency bandgaps in two-dimensional multivibrator phononic crystal plates under prestrain

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Hai-Fei Zhu ◽  
Xiao-Wei Sun ◽  
Ting Song ◽  
Xiao-Dong Wen ◽  
Xi-Xuan Liu ◽  
...  
Keyword(s):  
Phononic Crystal ◽  
Real Life ◽  
Low Frequency ◽  
Acoustic Vibration ◽  
Crystal Plate ◽  
Frequency Noise ◽  
Two Dimensional ◽  
The Matrix ◽  
Noise Frequency ◽  
Realtime Control

AbstractIn view of the influence of variability of low-frequency noise frequency on noise prevention in real life, we present a novel two-dimensional tunable phononic crystal plate which is consisted of lead columns deposited in a silicone rubber plate with periodic holes and calculate its bandgap characteristics by finite element method. The low-frequency bandgap mechanism of the designed model is discussed simultaneously. Accordingly, the influence of geometric parameters of the phononic crystal plate on the bandgap characteristics is analyzed and the bandgap adjustability under prestretch strain is further studied. Results show that the new designed phononic crystal plate has lower bandgap starting frequency and wider bandwidth than the traditional single-sided structure, which is due to the coupling between the resonance mode of the scatterer and the long traveling wave in the matrix with the introduction of periodic holes. Applying prestretch strain to the matrix can realize active realtime control of low-frequency bandgap under slight deformation and broaden the low-frequency bandgap, which can be explained as the multiple bands tend to be flattened due to the localization degree of unit cell vibration increases with the rise of prestrain. The presented structure improves the realtime adjustability of sound isolation and vibration reduction frequency for phononic crystal in complex acoustic vibration environments.

scholarly journals Kernel density estimation and its application

2018 ◽  
Vol 23 ◽  
pp. 00037 ◽  
Author(s):  
Stanisław Węglarczyk
Keyword(s):  
Probability Density Function ◽  
Probability Distribution ◽  
Density Estimation ◽  
Kernel Density Estimation ◽  
Real Life ◽  
Kernel Estimation ◽  
Kernel Density ◽  
Two Dimensional ◽  
2D Histogram ◽  
Function Shape

Kernel density estimation is a technique for estimation of probability density function that is a must-have enabling the user to better analyse the studied probability distribution than when using a traditional histogram. Unlike the histogram, the kernel technique produces smooth estimate of the pdf, uses all sample points' locations and more convincingly suggest multimodality. In its two-dimensional applications, kernel estimation is even better as the 2D histogram requires additionally to define the orientation of 2D bins. Two concepts play fundamental role in kernel estimation: kernel function shape and coefficient of smoothness, of which the latter is crucial to the method. Several real-life examples, both for univariate and bivariate applications, are shown.

scholarly journals EVALUATION OF THE FASHION DRAWING (ILLUSTRATION) LESSON TEACHING PROGRAM UNDER THE LIGHT OF LECTURERS’ AND STUDENTS’ OPINIONS

2021 ◽  
Vol 9 (60) ◽  
Keyword(s):  
Art Education ◽  
Design Education ◽  
Formal Education ◽  
Fashion Design ◽  
Two Dimensional ◽  
Examination System ◽  
Art And Design ◽  
Face To Face ◽  
Dimensional Plane ◽  
Figure Drawing

While clothing is produced and consumed rapidly, it strives to create appropriate forms and to create appropriate identities by giving correct messages; therefore, it is one of the most important fields requiring art and design education. The fashion design student, who forms the infrastructure with basic art education and figure drawing, not only transfers the imagination to the two-dimensional plane with the fashion drawing lesson but also transfers it to the two-dimensional plane; at the same time, he takes art and design education, which is aesthetic education, and feeds his visual vision and mind. In this research, the Fashion Drawing Lesson has been examined in detail in terms of determining the place, position, importance, processing and content of the Fashion Drawing Lesson in the official fashion design curriculum and a comprehensive literature search has been conducted; then the matter has been tried to be examined thanks to a link has been established between education, fashion, art and design disciplines. The course syllabus in Turkey and some of fashion design course programs in the world were examined; the experts providing fashion formal education and the students who received fashion formal education were interviewed face-to-face to obtain information first-hand. First of all, the study started by determining the fashion design undergraduate programs in the OSYM (Assessment Selection and Placement Center) placement guide and examining the course hours, course times and course contents of the fashion formal courses in these programs. The researcher visited the designated universities for the purpose of on-site observation and interview, and conducted semi-structured interview questions and face-to-face interviews with experts and students and recorded them. The transcribed interviews were analyzed using content analysis method and the findings were interpreted within the framework of the research questions. The findings were interpreted and discussed, and after the results were written, suggestions about the study were included. As a result, it has been determined that the admission of students to the fashion design program should be made with the combination of central examination system and talent exam suitable for fashion design education. It was seen that the time granted for the program was not sufficient in mean of implementation of the fashion drawing lessons properly in the fashion design education programs. Keywords: Fashion drawing, fashion illustration, design education, fashion design education, art education

Self-Replicating Patterns in 2D Linear Cellular Automata

2014 ◽  
Vol 24 (01) ◽  
pp. 1430002 ◽  
Author(s):  
Selman Uguz ◽  
Uḡur Sahin ◽  
Hasan Akin ◽  
Irfan Siap
Keyword(s):  
Cellular Automata ◽  
Self Assembly ◽  
Statistical Physics ◽  
Spatial Scales ◽  
Real Life ◽  
Continuous System ◽  
Random Number Generation ◽  
Two Dimensional ◽  
Textile Design ◽  
Genetics Research

This paper studies the theoretical aspects of two-dimensional cellular automata (CAs), it classifies this family into subfamilies with respect to their visual behavior and presents an application to pseudo random number generation by hybridization of these subfamilies. Even though the basic construction of a cellular automaton is a discrete model, its macroscopic behavior at large evolution times and on large spatial scales can be a close approximation to a continuous system. Beyond some statistical properties, we consider geometrical and visual aspects of patterns generated by CA evolution. The present work focuses on the theory of two-dimensional CA with respect to uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) conditions. In total, there are 512 linear rules over the binary field ℤ2for each boundary condition and the effects of these CA are studied on applications of image processing for self-replicating patterns. After establishing the representation matrices of 2D CA, these linear CA rules are classified into groups of nine and eight types according to their boundary conditions and the number of neighboring cells influencing the cells under consideration. All linear rules have been found to be rendering multiple self-replicating copies of a given image depending on these types. Multiple copies of any arbitrary image corresponding to CA find innumerable applications in real life situation, e.g. textile design, DNA genetics research, statistical physics, molecular self-assembly and artificial life, etc. We conclude by presenting a successful application for generating pseudo numbers to be used in cryptography by hybridization of these 2D CA subfamilies.

On the existence of metric differential geometries based on the notion of area

1950 ◽  
Vol 46 (1) ◽  
pp. 67-72 ◽  
Author(s):  
F. Brickell
Keyword(s):  
Differential Geometry ◽  
Fourth Order ◽  
Homogeneous Function ◽  
Two Dimensional ◽  
Fundamental Function ◽  
Metric Tensor ◽  
Partial Derivatives ◽  
Dimensional Plane ◽  
Plane Element

The problem of constructing an n-dimensional metric differential geometry based on the idea of a two-dimensional area has given rise to several publications, notably by A. Kawaguchi and S. Hokari (1), E. T. Davies (2), and R. Debever (3). In this geometry the area of a two-dimensional plane element is defined by a fundamental function L(xi, uhk), where the xi are point coordinates and the uhk are the coordinates of the simple bivector representing the plane element. L is supposed to be a positive homogeneous function of the first degree with respect to the variables uij, and to possess continuous partial derivatives up to and including those of the fourth order. With these assumptions the problem of the construction of the metric differential geometry splits into two problems; the first of these is the problem of constructing a metric tensor gij(xr, uhk), and the second is the problem of constructing an affine connexion. We deal with the first problem only in this paper.

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