Voronoi in nature
Voronoi pattern/ tessellation is significant in the skin of different species in nature. Examples are given below:
Giraffe
Has nature created Voronoi Diagrams on these beautiful animals ?
Are they hiding something more ?
Is there any symmetry in shape and size of polygons ?
Is the number of polygons constant on each Giraffe ?
(Source: www.mcs.anl.gov/~csverma/Misc.html)
Turtle exoskeleton ( Source: http://www.desertgold.com/park/pics/turtle.jpg)
Voronoi fractals in plants ( Source: http://www.flickr.com/photos/flight404/538133104/)
"Reticulum Plasmatique." Fig. 321. From On Growth and Form.
D'Arcy Thompson.
Alligator skin_ voronoi tessellation ( Source: tosca.cs.technion.ac.il/book/slides/Stanford0)
Skin of a tiger _ voronoi tessellation (Source: tosca.cs.technion.ac.il/book/slides/Stanford0)
Application of Voronoi diagram in different fields
A. Social & planning aspects
There are a wide variety of applications of Voronoi diagrams. They are more important then what one might come to believe. Some of the applications are as follows:
1. Nearest Neighbour Search: This is the most obvious application of Voronoi Diagrams.
2. Facility Location: The example that is often quoted in this case is the example of choosing where to place a new Antenna in case of cellular mobile systems and similarly deciding the location of a new McDonalds given a number of them already exist in the city.
3. Path Planning: Suppose one models the sites as obstacles, then they can be used to determine the best path (a path that stays at a maximum distance from all obstacles or sites).
There are a number of other applications, such as in Geophysics, Metrology, Computer Graphics, Epidemiology and even pattern recognition. A very good example that illustrates how they can be used was the analysis of the Cholera epidemic in London in 1854, in which physician John Snow determined a very strong correlation of deaths with proximity to a particular infected pump (specific example from Wolfram Mathworld).
Let’s consider the specific example of path planning . Consider a robot placed in one corner of a room with stuff dispersed around.
Image Source:
tosca.cs.technion.ac.il/book/slides/Stanford0
Now the best path from the point where the robot is located to the goal would be the one in which the robot is farthest from the nearest obstacle at any point in time. To find such a path, the Voronoi diagram of the room would be required to be found out. Once it is done, the vertices or the skeleton of the Voronoi Diagram provides the best path. Which path ultimately is to be taken can be found out by comparing the various options (alternative paths) by using search algorithms.
Image Source:
tosca.cs.technion.ac.il/book/slides/Stanford0
B) Polymer physics
The Voronoi diagram is useful in polymer physics. It can be used to represent free volume of the polymer.
C) Wireless network
It is also used in derivations of the capacity of a wireless network.Voronoi graphs are a promising approach to create easy, but realistic movement models for nodes of a mobile network. Artificial scenarios with streets, sidewalks and buildings can thus be calculated and simulated.
Image Source:
www.lkn.ei.tum.de/~hmz/research.html?lang=en
D) Climatology
in climatology, Voronoi diagrams are used to calculate the rainfall of an area, based on a series of point measurements. In this usage, they are generally referred to as Thiessen polygons.
Image Source:
www-personal.umich.edu/.../win01/sarhaus
Bisectors, buffers, and proximity zones (Thiessen polygons)
E) Growth pattern of forest
Voronoi diagrams are used to study the growth patterns of forests and forest canopies, and may also be helpful in developing predictive models for forest fires.
F) Computer graphics
Voronoi diagrams are also used in computer graphics to procedurally generate some kinds of organic looking textures.
Image Source:
tosca.cs.technion.ac.il/book/slides/Stanford0
Golan Levin’s experiments in using Voronoi diagrams to obtain aesthetic forms yielded probably even more pleasant results. The ones below give a very delicate look to their subjects.
image Source:
http://www.flong.com/projects/zoo/
The tilings that are produced by just mild tweaks to the basic definition of a Voronoi Diagram for a 2-D case that I had talked about earlier can give rise to a variety of tilings. Say like the one below:
Image Source:
http://www.cgl.uwaterloo.ca/~csk/papers/kaplan_isama1999.pd
G) Fractals from voronoi
To create a fractal, first create a Voronoi diagram from some points, next add more points and then create the Voronoi diagrams inside individual Voronoi Regions. Some sample progression could be like this:
Image Source:
http://www.righto.com/fractals/vor.html
Repeating the above process recursively on the above would give the following Voronoi fractal.
Image Source:
http://www.righto.com/fractals/vor.html
Interestingly, this fractal looks like the structure of a leaf. The above was repeated in color by Frederik Vanhoutte to get some spectacular results.
Image Source:
onionesquereality.wordpress.com/.../voronoi-art/
H) Robot navigation
In autonomous robot navigation, Voronoi diagrams are used to find clear routes. If the points are obstacles, then the edges of the graph will be the routes furthest from obstacles (and theoretically any collisions).
I) Computational chemistry
In computational chemistry, Voronoi cells defined by the positions of the nuclei in a molecule are used to compute atomic charges. This is done using the Voronoi deformation density method.
voronoi in architecture
a) Voronoi bookshelf by Marc Newson
The Voronoi Bookshelf by Marc Newson is made from a single slab of white carrara marble. The organic shape gives the eye a welcome break from the monotony of rectangular shelving. Owning a unit like this would force you to look at your personal objects in a whole new way as you hunted for the right alcove in which to put them. This is a fantastic piece of design that will no doubt have lasting appeal. I have no idea how this would make it into the average home, but perhaps Newson liked the irony of using a totally impractical material for such a banal piece of household furniture. A shelving unit turned into glorious sculpture! I would not be surprised if it is reproduced in a more practical material in the future.
b) Melbourne Federation Square - Lab Architecture
Type:
Public Square and cultural and commercial buildings.
Construction years:
Winner announced:1999
Date started: 2000
Date opened: Saturday 26 October 2002
Location:
Flinders Street, Melbourne, Victoria, Australia
Top view
Facade/ fractal/ voronoi system
Source:
www.pushpullbar.com/forums/australia/7203-mel
Inside
c) Study by Object e-architecture
Brief: This is a first attempt to use the voronoi diagrams for a specific design. The initial set of points is defined by the program requirements. The edges of the voronoi cells become the structure, while a first idea for enclosed space was to use again the voronoi cells (in a 'smoothed' version) as clusters of space in a configuration that resembles the relation of bones to organs. This project though was not developed further...
For more information about computing convex hulls, voronoi diagrams, and other triangulations, check out the qhull website. Qhull is used in Matlab and many other computational geometry applications.
Source: object-e.blogspot.com
d) Voronoi by D.FILTER 2.0
Warsaw Museum of Modern Art [competition entry]
e) Analysis by MatSys (evolution of “Material Systems”)
Established in 2004 by Andrew Kudless, Matsys is an architectural design studio that explores the emergent relationships between architecture, engineering, biology, and computation. Based on the idea that architecture can be understood as a material body with its own intrinsic and extrinsic forces relating to form, growth, and behavior, the studio investigates methodologies of performative integration through geometric and material differentiation. The studio’s work ranges from speculative and built projects to the crafting of new tools which facilitate an interdisciplinary approach to the design and fabrication of architecture.
2005-2006
Voronoi Morphologies is the latest development in an ongoing area of research into cellular aggregate structures. The voronoi algorithm is used in a wide range of fields including satellite navigation, animal habitat mapping, and urban planning as it can easily adapt to local contingent conditions. Within our research, it is used as a tool to facilitate the translation and materialization of data from particle-simulations and other point-based data into volumetric form. Through this process, it becomes much easier to produce highly differentiated structures that are responsive to local performance criteria.
The project was developed though both 2D and 3D voronoi cellular structures. In both cases, a field of points is used to determine regions of space, or cells, that are closer to a certain point than any other point. As the cells are not constrained by a fixed geometric topology, the cells properties can be tuned in much more specific ways than a tradition rectangular or hexagonal cell arrangement. A custom-designed script was written to connect Rhino with Qhull which did the actual voronoi calculations. The script also digitally unfolds, labels, and prepares the geometry for CNC fabrication.
This technique was developed in collaboration with Jelle Feringa of EZCT Architecture and Design Research in Paris.
Source: www.materialsystems.org
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