Onionesque Reality

Probably the Most Tragic Images of the Year

Posted in Famous Photographs, Nature, Random, tagged Amu Darya, Aral Sea, Aral Sea Death, Geology, Kazakhstan, Soviet Union, Syr Darya, Uzbekistan, Water on July 16, 2009| 4 Comments »

I picked up these images at Wired two days ago and just could not fit in the time to put them up earlier.

There is a remarkable quote by Einstein –

“Two things are infinite: the universe and human stupidity; and I’m not sure about the the universe.”

It isn’t definite to me if I liked this quotation earlier. But I am NOW wholly convinced that I love it. And this new found unequivocalness for it is due to the following:

As a kid, I used to have a large collection of encyclopedias. I remember reading about the Aral Sea in the picture atlas, and that it mentioned that there was increasing salination of the sea water and that it would disappear in some decades.

Over the years that time, we were fed with doomsday scenarios all the while. Like all the coastal cities would be soon under sea due to rising ocean levels, and that the Himalayas would soon be ice free etc etc. Over a period of time you get fed up with such idle talk and since you don’t see anyone giving convincing answers, you tend to believe that nothing like that is true. Secondly, the eternal optimist that I am, I just probably wished that what that encyclopedia said about the Aral was some “minor” problem.

I last read about the problem many years ago and after that never came across anything on it. And just a couple of days back was shocked by these images. I have only one word for them : Tragic!

The images are from 1973, 1987, 1999, 2006 and 2009. The two recent images were released by the European Space Agency, the earlier ones were taken by the United States Geological Survey.

Aral Sea - 1973

Aral Sea - 1987

Aral Sea - 1999

Aral Sea - 2006

Aral Sea - 2009

[Image(s) Source : Wired Science]

The South Aral Sea, the remnant of the original lake that you can see to your left on the above image is also expected to vanish by 2020, thankfully the North Aral sea (the part on the right) has been saved due to a world bank funded dam project.

The Aral sea, once the world’s fourth largest lake at roughly around 68,000 sq kms is now just about one-tenth that size. The trouble started when it was decided by the Soviets in 1918 that the two rivers that drained into the Aral – The Amu Darya and Syr Darya would be largely diverted to the deserts to develop them into cotton growing lands. The Soviet plan worked and cotton became one of the most important exports from that area. By the 1960s massive amount of water was being diverted and the sea began to shrink steadily. And how that happened is spoken out loud by the pictures.

The death of the Aral is extremely sad. It’s death has left it’s once thriving fishing industry destroyed, the diverting of the rivers has mostly reduced the two rivers to a shadow of their former selves. The Aral served as a climate moderator in the largely arid lands there, it’s death might herald major environmental catastrophe in the region.

This is a prime example of what human stupidity could lead to and leaves me short of words to describe my anguish at the same.

It has a number of things to say:

Ignoring warnings which have clear proof is just plain stupidity. There is ample proof for example of climate change and its bad impact. For example, I have been visiting the Himalayas once every few years since 1991. And the change there is apparent, as compared to the 80s the glaciers that make up the Ganges have shrunk by several kilometers. I don’t know what the solutions are, nor am I comparing the Aral problem with it. I understand that the Aral was a different kind of a problem. Different because it was known to the Soviets that the lake would dry up from the start. Climate change can not be compared to it as we do not yet fully understand a number of things about it, so how effective the correctives would be is debatable. It would be for our good if that debate is settled soon with good and incisive scientific evidence.

It also is a comment on how totalitarian regimes can be dangerous. In such regimes, since a decision taken can not be opposed, such a decision could either lead to major dividends/progress as it would be implemented very rapidly or major catastrophe as was in the above case. Soviet officials were aware that the Aral would sooner or later evaporate. In 1964 Aleksandr Asarin noted that :

“It was part of the five-year plans, approved by the council of ministers and the Politburo. Nobody on a lower level would dare to say a word contradicting those plans, even if it was the fate of the Aral Sea.”

Ofcourse he was right, there is rarely any way to convince or reason with or oppose supercilious totalitarian regimes even if their decisions are clearly suicidal. I am tempted to make a political comment on two present day countries (one a totalitarian state and one a liberal democracy) here, but would avoid the temptation.

Anyhow, the images above disturbed me enough to lose sleep.

_____

http://www.wired.com/wiredscience/2009/07/aralsea/

Voronoi Art

Posted in Art, Computer Science, Mathematics, Nature, tagged Art, Computational Geometry, Computer Graphics, Digital Art, Fractals, Mathematics, Mosaics, Path Picking, Robotics, Space Partitioning, Tessellations, Voronoi Diagrams on December 13, 2008| 6 Comments »

While searching for some methods for face representation in connection with my recent project, I lost the way clicking on some stray links and landed up on some beautiful art work involving Voronoi diagrams. I was aware of art work based on Voronoi diagrams (it kind of follows naturally that Voronoi diagrams can lead to very elegant designs, isn’t it?) but a couple of images on them were enough to re-ignite interest. It was also interesting to see an alternate solution to my problem based on Voronoi diagrams as well. However I intend to share some of the art work I came across.

———-

Before I get to the actual art-work I suppose it would be handy to give a very basic introduction to Voronoi Diagrams with a couple of handy applications.

Introduction: Voronoi diagrams are named after Georgy Voronoy (1868-1908), an eminent Russian/Ukrainian mathematician. A number of mathematicians before Voronoy such as Descartes and Dirichlet have been known to have used them, Voronoy extended the idea to $\mathcal{N}$ dimensions. The Voronoi diagram is a tessellation, or a tiling. A tiling of a plane is simply a collection of plane figures that fills the plane with no overlaps and no gaps in between. This idea can ofcourse be extended to $\mathcal{N}$ dimensions, but for simplicity let us stick with 2 dimensions.

[A pavement Tessellation/Tiling]

Definition: A Voronoi diagram is a special kind of a decomposition of a metric space which is determined by a discrete set of points.

Generally speaking for a 2-D case:

>> Let us designate a set of $n$ distinct points that we call sites as $\mathcal{P}$ . i.e $\mathcal{P}=\{P_1, P_2\ldots, P_n\}$

>> We may then define the Voronoi Diagram of P as a collection $\mathcal{V}=\{V_1,V_2,\ldots, V_n\}$ of subsets of the plane. These subsets are called as Voronoi Regions. Each point in $V_i$ is such that it is closer to $\mathcal{P}_i$ than any other point in $\mathcal{P}$ .

To be more precise:

A point $Q$ lies in the Voronoi Region corresponding to a site $P_i \in \mathcal{P}$ if and only if –

$Euclidean\_Distance(Q,P_i) < Euclidean\_Distance(Q,P_j)$ for each $P_i \neq P_j$

However it might be the case that there are some points in the plane that might have more than one site that is the closest to it. These points do not lie in either Voronoi Region, but simply lie on the boundary of two adjacent regions. All such points form a skeleton of lines that is called the Voronoi Skeleton of $\mathcal{P}$ .

We can say that $\mathcal{V(P)}$ is the Voronoi Transform, that transforms a set of discrete points (sites) into a Voronoi Diagram.

———-

For more clarity on the above, consider a sample Voronoi Diagram below:

The dots are called the Sites. The Voronoi Regions are simply areas around a site but enclosed by the lines around them. The network of lines is simply the Voronoi Skeleton.

———-

Some Extensions to the above Definition: The above definition which is ofcourse extremely simple can be extended very easily^[1] for getting some fun art forms.

Some of these extensions could be as follows^[1]:

1. Since each region corresponds to a site, each site can be associated with a color. Hence the Voronoi diagram can be colored accordingly.

2. In the definition the sites were considered to be simply points, we can obtain a variety of figures by allowing the “sites” to be subsets of the plane than just points. We see, that if the sites are defined as simply points, the Voronoi skeleton would always be composed of straight lines. With this change there could be interesting skeletal figures emerging.

3. We could also modify the distance metric from the Euclidean distance to some other to get some very interesting figures.

This just shows the kind of variety of figures that can be generated by just a small change in one aspect of the basic definition.

———-

Constructing a Voronoi Diagram:

Let us forget the extensions that we spoke of for a moment and come back to the basic definition. Looking at the definition it seems constructing Voronoi diagrams is a simple process. And it is not difficult at all. The steps are as follows:

1. Consider a random set of points.

2. Connect ALL of these points by straight lines.

3. Draw a perpendicular bisector to EACH of these connecting lines.

4. Now select pieces that are formed, such that each site (point) is encapsulated.

Voronoi Diagrams can be very easily made by direct commands in both MATLAB and MATHEMATICA.

———-

Voronoi Diagrams in Nature: It is interesting to see how often Voronoi diagrams occur in nature. Just consider two examples:

[Left: Reticulum Plasmatique (Image Source) Right: Polygons on Giraffes]

———-

Uses of Voronoi Diagrams: 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^[2]. Consider a robot placed in one corner of a room with stuff dispersed around.

[Illustration Source]

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.

[Illustration Source]

Now finally after the background on Voronoi Diagrams let’s look at some cool artwork that i came across. ;-)

———-

Fractals from Voronoi Diagrams:

This I came across on the page of Kim Sherriff^[3]. The idea is straightforward to say the least.

It is: 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:

Repeating the above process recursively on the above would give the following Voronoi fractal.

Interestingly, this fractal looks like the structure of a leaf.

The above was repeated in color by Frederik Vanhoutte^[4] to get some spectacular results. Also I would highly recommend his blog!

$voronoi-fractal$

[Voronoi Fractal – Image Source]

I am really going to try this myself, it seems a few hours of work at first sight. Ofcourse I won’t use the code that the author has provided. ;-)

———-

Mosaic Images Using Voronoi Diagrams^[5] :

I have not had the time to read this paper. However, I am always attracted by mosaics, and these ones (the ones in the paper) as created using Voronoi Diagrams have an increased coolness quotient for me. Sample this image:

Golan Levin’s^[6] 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.

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:

[5] A Method for creating Mosiac Images using Voronoi Diagrams.

Also, today I came across a nice Voronoi Diagram on The Reference Frame:

The diagram is a representation of 17,168 weather stations around the world. Dr Motl illustrates how handy MATHEMATICA is for such things.

Click to Enlarge

———-

References:

[1] Craig Kaplan, “Voronoi Diagrams and Ornamental Design“.

[2] Introduction to Voronoi Diagrams – Example.

[3] Kim Sherriff, “Fractals from Voronoi Diagrams“.

[4] Frederik Vanhoutte, “Voronoi Fractal“.

[6] Golan Levin, “Segmentation and Symptom“

———-

Applets:

1. Voronoi Diagram Applet.

2. Bubble Harp.

———-

Additional Links:

1. Image Stained Glass using Voronoi Diagrams.

2. Interactive Design of Authentic Looking Mosaics using Voronoi Diagrams.

3. Voronoi Diagrams of Music.

———-

Lenfirewood’s Ingenious Digital Artwork

Posted in Art, Image Processing, Random, tagged Art, Digital Art, Digital Photography, Image Processing, Image Processing Tools, Lenfirewood, Random on November 18, 2008| 2 Comments »

I got linked to by a blog today morning and I wanted to see what it was about. It turned out to be a web-log on the author’s digital photography, compositions of multiple images taken by him and images post processing. Some of the processed images are beautiful. Consider a sample:

[The Spirit of Autumn]

The above image christened The Spirit of Autumn is actually a composition of 3 different images taken on a digital camera. The resultant image was then processed further in GIMP (GNU based image manipulation tools) to get a wonderful output.

[Through a Glass Wetly]

I am strongly attracted to painted abstract art, and very rarely to digital art (though I like ingenious fractals and mathematical figures) however this work is truly deserving of a hearty applause. Interestingly, the author pursues photography as a hobby. You can check out some of his other art work at his web-log here.

Image processing is one of my most favorite subjects and I have been involved in projects concerning some Image Processing too, however I think I should move beyond looking at MRI scans or X-rays once in a while and try my hand at post processing photographs taken by me (using tools ofcourse, not algorithms as such) to TRY and get as spectacular results as obtained by DJ Lenfirewood as above. LOL.

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Links:

1. Download GIMP

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3. Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters.

The Camera Lucida

Posted in Art, Physics, Random, Vintage, tagged Art, Devices, Optics, Vintage on October 8, 2008| 5 Comments »

While reading something by Leonel Moura a few months ago which described in some sense the evolution of art. I came across the Camera Lucida, I had not heard or read about it before and reading about it was fun. Not only is it elegant but is something that can be used with good effect even today and is available quite easily with art suppliers.

The Camera Lucida is a very elegant device, it works to the effect as if the the object we have to draw is reflected on the paper or the canvas we are drawing on. So one would only have to trace the object without having to worry about the perspective. Camera Lucida is Latin for light chamber which is the exact opposite of Camera Obscura which means dark chamber. That ancestral thread lead to the modern photography and as an illustration of that fact we still call our photographic devices as cameras. However there is no optical similarity between these two devices.

It was used as a handy drawing and painting tool by artists and even microbiologists till a few decades ago and was invented by William Hyde Wollaston in 1807. There is some evidence that the device was first described by Johannes Kepler but over time his contribution it seems was forgotten and now the invention is largely attributed to Wollaston. Wollaston’s original design is given below

[Image Source: Wikipedia Commons]

In such an arrangement an artist looks down at the fabric or paper (labeled as P) through a half silvered mirror which is placed at 45 degrees. The mirror is adjusted so that the source or the object to be drawn (Label S) is in the field of view. Given the arrangement, a virtual image of the source is formed on the paper, this superimposition appears as if the object or person of who you are making a painting of is reflected on the sheet and thus the job is reduced to simply making the outline and coloring it aptly.

Note: The light coming from S is totally internally reflected at the surfaces of the four sided glass prism allowing all of the light from the source to the eye.

[Images Courtesy of The Camera Lucida Company]

The image on the left is simply of an artist using a camera lucida to paint a subject on paper, and the one on the right is simply a photograph taken with a camera lens in place of the artist’s eye and it shows how the image of the subject appears on the paper with the hand of the artist. The Camera Lucida as I mentioned earlier was for obvious reasons used by microbiologists as till recently photomicrographs were expensive!

A sample sketch of a Camera Lucida used for this purpose is shown below:

Click to Enlarge

[A Camera Lucida: Image Courtesy, The Botanic Gardens Trust, Sydney ]

Controversy

According to a controversial art history theory called the Hockney-Falco hypothesis advanced by a British-American artist David Hockney, quite a few of the great artists of the past whose works lead to advances in Realism were using optical aids and that their creations were not entirely due to their skill as is held. The evidence for this proposal is based solely on the characteristics of the paintings. Hockney’s collaborator Charles Falco who is a condensed matter physicist and an expert on optics calculated the amount of distortions that would result with the use of certain optical aids. Such distortions have been found in the works of quite a few artists such as Ingres, Carvaggio etc. Their controversial idea is summarized in Hockney’s book: Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters.

[Image Source: Amazon]

The Camera Lucida is also very easy to make and I going to make one of these soon!

PS: Also check out the David Hockney page at Artsy.

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Quick Links:

1. Buy a Camera Lucida

2. The Camera Obscura

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Swarm Paintings: Non-Human Art

Posted in Art, Artificial Intelligence, Artificial Life, Biology, Complexity, Robotics, Swarm Intelligence, tagged A-Life, Ants, Art, Artificial Intelligence, Artificial Life, Biology, Complexity, Emergence, Non Human Art, Swarm Intelligence on May 13, 2008| 10 Comments »

General Background: Since childhood i have enjoyed sketching and painting, and very much at that! Sometimes i found myself copying an existing image or painting, making small changes here and there in it. Yes, the paintings came out beautiful (or so i think!), but one thing always made me unhappy, i thought that the creativity needed to make original stuff was missing at times (not always). It was not there all the time. It came in bursts and went away.

I agree with Leonel Moura (from his article) that creativity is basically produced due to different experiences and interactions. Absence or lack of which could make art lose novelty.

Talking of novelty, how about looking at art in nature? Richard Dawkins states that the difference between human art or design and the amazingly “ingenious” forms that we encounter in nature, is due tho the fact that Human art originates in the mind , while the natural designs result from natural selection. Which is very true. However it is another matter that natural selection and cultural selection, that will ultimately decide on the “popularity” of an art don’t function in the same way. Anyhow How can we remove the cultural bias or the human bias that we have in our art forms?

Answers in Artificial Life: Artificial life may be defined as “A field of study devoted to understanding life by attempting to derive general theories underlying biological phenomena, and recreating these dynamics in other physical media – such as computers – making them accessible to new kinds of experimental manipulation and testing. This scientific research links biology and computer science.”

Most of the A-Life simulations today can not be considered truly alive, as they still can not show some properties of truly alive systems and also that they have considerable human bias in design. However there are two views that have existed on the whole idea of Artificial Life and the extent it can go.

Weak A-Life is the idea that the “living process” can not be achieved beyond a chemical domain. Weak A-life researchers concentrate on simulating life processes with an underlying aim to understand the biological processes.

Strong A-Life is exactly the reverse. John Von Neumann once remarked “life is a process which can be abstracted away from any particular medium“. In recent times Ecologist Tom Ray declared that his computer simulation Tierra was not a simulation of life but a synthesis of life. In Tierra, computer programmes compete for CPU time and access to the main memory. These programs are also evolvable, can replicate, mutate and recombine.

Relating A-Life to Art: While researching on these ideas and the fact that these could be used to generate the art forms that i talked about in the first paragraph i came across a few papers by Swarm Intelligence Guru Vitorino Ramos and a couple of articles by Leonel Moura who had worked in collaboration with Dr Ramos on precisly this theme.

Swarm Paintings: Thus the idea as i had mentioned in my very first paragraph is to create an organism ideally with minimum pre-commitment to any representational art scheme or human style or taste. Sounds simple but is not so simple to implement!

There are a number of projects that have dealt with creating art, but these mostly have been evolutionary algorithms that learn from human behavior, and learn about human mannerisms and try to create art according to that. The idea here is to create art with a minimum of human intervention.

I came across a project by Dr Vitorino Ramos to which i had pointed out implicitly in the last paragraph. This project called ARTSBOT (ARTistic Swarm roBOTs) project. This project tries to address this issue of minimizing the human intervention in aesthetics , ethnicity, taste,style etc. In short their idea was to remove or to minimize the anthropocentric bias that pervades all our art forms. Obviously all this can have massive implications in our understanding of the biological processes also, however here we’ll talk of only art.

Two of the first paintings that emerged were:

(Source: Here)

These paintings were among the first swarm paintings by Leonel Moura and Vitorino Ramos. Now we see that these seem detached from a functional human pre-commitment. They don’t seem to represent any emotion or style or taste. However they still look very pleasant!

However the point to be understood and to be noted is that these are NOT random pictures created either by a programme or by a swarm of robots moving “randomly”. These pictures were generated by a horde of artificial ants and also by robots. They are not random, but they EMERGE from a process of pheromone deposition and evaporation as was simulated in this system from ants. Thus the result that we have above is a Colony Cognitive Map. The colony cognitive map is analogous to a cognitive map in the brain. I will cover the idea of a colony cognitive map in the next post.

A couple of more beautiful paintings can be seen below!

(Source for both images : Here>>)

Though i have already mentioned how these art forms emerge, i would still like to quote a paragraph from here:

The painting robots are artificial ‘organisms’ able to create their own art forms. They are equipped with environmental awareness and a small brain that runs algorithms based on simple rules. The resulting paintings are not predetermined, emerging rather from the combined effects of randomness and stigmergy, that is, indirect communication trough the environment.
Although the robots are autonomous they depend on a symbiotic relationship with human partners Not only in terms of starting and ending the procedure, but also and more deeply in the fact that the final configuration of each painting is the result of a certain gestalt fired in the brain of the human viewer. Therefore what we can consider ‘art’ here, is the result of multiple agents, some human, some artificial, immerged in a chaotic process where no one is in control and whose output is impossible to determine.
Hence, a ‘new kind of art’ represents the introduction of the complexity paradigm in the cultural and artistic realm.’

A Painting bot is something like in the picture shown below:

A swarm of robots at work:

The final art generated by the swarm of these robots is beautiful!

(Photo Credit for the three pictures above: Here>>

Conclusions:

The work of Dr Ramos and Leonel Maura can be summed up as:

1. The human is only the “art-architect”, the “swarm” is the artist.

2. The “life” of Artificial Life shows characteristics like natural life itself namely Morphogenesis, ability to adapt to changing environments, evolution etc.

Leonel Moura’s wonderful article states that the final aim is to create an “Artificial Autopoietic System”, intriguing indeed and eagerly awaited!!
Such simulations could change the way we understand the biological processes and life.

Also i am now thinking how could music be produced based on the same or similar ideas. I wonder if Swarm music could be available. It would be most interesting and i can’t wait to listen to it already!

Have a look at this video by Leonel Moura, having some time lapse footage of robots painting.

References:

1. Ant- Swarm Morphogenese By Leonel Moura

2. On the Implicit and on the Artificial – Morphogenesis and Emergent Aesthetics in Autonomous Collective Systems, in ARCHITOPIA Book, Art, Architecture and Science, INSTITUT D’ART CONTEMPORAIN, J.L. Maubant et al. (Eds.), pp. 25-57, Chapter 2, Vitorino Ramos.

3. A Strange Metamorphosis [From Kafka to Red Ant], Vitorino Ramos

Links:

Follow the following links to follow on more exciting papers and paintings.

1. Dr Vitorino Ramos

2. Leonel Moura

$cantordustfractal_700.gif$

Lindenmayer Systems and Fractals

Posted in Art, Computer Science, Mathematics, Nature Inspired Computing, tagged Art, Fractal, Fractals, Generative Art., Geometry, Grammer, Mathematics, Recursion, Self-Similar Structures on February 5, 2008| 6 Comments »

L-systems are very simple and elegant grammars (a set of rules and symbols) originally developed by biologist Aristid Lindenmayer to describe the growth of plants and trees.

$tsr-fractals.jpg$

These as we can see can lead to very beautiful figures as above!

Photo Credit.

The Lindenmayer system is recursive in nature which leads to self-similarity and hence fractal like structures. Plant models and natural-looking organic forms are similarly easy to define, as by increasing the recursion level the form slowly ‘grows’ and becomes more complex.

The components of a L-system are:

A) The alphabet, which is a finite set V of formal symbols or characters, generally they are taken to be letters A,B,C etc. These are the variables as they can be replaced. A set of symbols which will remain fixed are constants. These may be represented as another finite set.

B) The start, axiom or initiator ω, it basically is a string of symbols from V. If suppose V = {a,b,c} then ω can be aabc, abcc, abcab etc. There are many possibilities. This defines the initial state of the system.

C) Production rules, P that define how a variable can be replaced by a combination of variables and constants.

The rules of the L-system grammar are applied iteratively starting from the initial state. As many rules as possible are applied simultaneously, per iteration. This is the distinguishing feature of a L-system from a language.

There are many examples of the same that lead to very beautiful structures!

Some are:

A) Cantor Dust:

variables : A B

constants : none

start : A {starting character string}

rules : (A → ABA), (B → BBB)

Let A mean “draw forward” and B mean “move forward”.

Put in a different manner. We could say that a cantor dust fractal may be reconstructed using string rewriting using an initial cell {1} and then iterating the below rules.

{0->[0 0 0; 0 0 0; 0 0 0],1->[1 0 1; 0 0 0; 1 0 1]}.

Thus we would get the fractal as :

B) Fractal Tree:

variables : X F

constants : + −

start : X

rules : (X → F-[[X]+X]+F[+FX]-X),(F → FF)

angle : 25°

Here, F means “draw forward”, – means “turn left 25º”, and + means “turn right 25º”. X does not correspond to any drawing action and is used to control the evolution of the curve. For n=6 the following figure may be generated.

$453px-fractal-plantsvg.png$

C) The Sierpinski Triangle:
variables : A B

constants : + −

start : A

rules : (A → B−A−B),(B → A+B+A)

angle : 60°

Here, A and B mean both “draw forward”, + means “turn left by angle”, and − means “turn right by angle”. The angle changes sign at each iteration so that the base of the triangular shapes are always in the bottom (they would be in the top and bottom, alternatively, otherwise).

Evolution for n = 2, n = 4, n = 6, n = 9

Other curves include the famous Koch Curve, Dragon Curve, Penrose Tilings etc.

Good places to do initial research on L-Systems:

1. Wikipedia

2. Fractinct L-system True Fractals, A tutorial by William Mcworter.

3. Fractals and Cellular Automata. by Daniel Shiffman.

4. Fractals and Recursion in the Nature of Code.

I would try to follow these initial posts on fractals with applications in RFID and in other smart antenna applications.

Also check out this voronoi fractal on this post. Credits to the image given in the original post.

$voronoi-fractal$

The Road Ahead

Posted in Art, Famous Photographs, tagged Ansel Adams, Art, Famous Photographs, Innovators and Artists on January 11, 2008| 2 Comments »

Here is a wonderful photograph by Ansel Adams, one of the most influential photographers of the 20th century. He is my personal favorite as well. I greatly venerate Adams.

This single frame speaks volumes about the American west. Like other photographs by Adams, this one also carries a message, and talks about The Road Ahead. One can interpret it differently.

Like it can be considered as a pictorial representation of the phrase “Life Goes On”.

This can be taken as an inspirational photo as well. I for some reason just love this photo and could not resist posting it on the blog!

This wonderful photograph was obtained from the photographymuseum.com, check this website for other wonderful pieces of art!