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## Sisters and Book

Painting is just another way of keeping a diary.

– Pablo Picasso.

One of the things that I have done right from my puerility, discontinuously sadly, is painting and sketching. One of my oldest hobbies and something that gives me immense peace.

This post finds it way here as :

1. It reminds me that I should spend less time wasting on communities on the internet  when I need to “pass” time as a diversion or something (whatever little I spend anyway) and use it to paint instead whenever the usual workload is a little slack.

2. I come across scores of extremely wonderful things on the internet every week . Why share this then? Oh because it (the painting “Sisters and Book “) moves me in a way that I’d rather not try and describe on a blog or for what reasons it does so. Especially a blog that’s not a personal one and is rather shifting focus towards Machine Learning gradually. :)

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[Sisters and Book :  By Iman Maleki]

Another beautiful painting, which almost looks like a photograph to me is this!

[Omens of Hafez – Iman Maleki]

Though I must admit I do not get attracted to realism much, I find Imam Maleki’s work extremely beautiful! Especially the way he paints ladies.

And that’s why this finds its way here. And point noted again to give more time to my beloved hobby.

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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.

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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.

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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.

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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.

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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.

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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]

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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.

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.

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

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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 – 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. ;-)

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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:

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

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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

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Applets:

2. Bubble Harp.

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## Lenfirewood’s Ingenious Digital Artwork

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 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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## The Camera Lucida

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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## Swarm Paintings: Non-Human Art

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?
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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.
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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.
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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)
(Source: Here)
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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>>

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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.
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2. The “life” of Artificial Life shows characteristics like natural life itself namely Morphogenesis, ability to adapt to changing environments, evolution etc.
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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!
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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
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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.
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3. A Strange Metamorphosis [From Kafka to Red Ant], Vitorino Ramos
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Follow the following links to follow on more exciting papers and paintings.

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## A Fractal with Zero Area and an Infinite Perimeter?

Following a discussion on Reasonable Deviations. I was prompted to write on this.

The above is an illustration on the generation of the Sierpinski Gasket. For simplicity we assume that the area of the initial triangle is 1. We split this triangle into four triangles by joining the mid-points of the sides of the triangle. These smaller triangles as shown in figure 2 have equal areas. We then remove the middle triangle. We adopt the convention that we will only remove the middle triangle and not a triangle at the edge.

In each of the three remaining triangles we repeat the process and remove the middle triangles. This is where self-symmetry comes in. If we pretend to see only one of the three small triangles then we are actually doing the same thing as we did to the original triangle. Albeit on a smaller scale. The above figure shows the third and fourth iterates of the original triangle. This process repeated ad infinitum gives rise to the Sierpinski gasket.

The L-System representation of the process is:

variables : A B

constants : + −

start : A

rules : (A → B−A−B),(B → A+B+A)

angle : 60°

Anyway, now what is interesting about this figure is its area and the perimeter.

Continuing with our earlier assumption, suppose the initial triangle had an area

A0= 1;

Now in the first iteration we remove one of the four equal areas in that triangle and keeping the other three that remain.

Therefore the total area of the first iterate will be equal to

A1 = (3/4) x 1;

Similarly in the second iteration, we repeat the process as I have already noted above. Thus,

A2 = (3/4)(3/4) x 1;

For n iterations the area will be given by:

An = (3/4)n x 1;

Now if n is arbitrarily large then the area, it would follow will be ZERO.

Now finding out the perimeter is a similar exercise. The length of the boundary of the nth iterate of the original triangle is the total length of the boundaries of all the shaded small triangles in the nth iterate. It can be shown that this gets arbitrarily large as n gets arbitrarily large. Therefore we conclude that a Sierpinski gasket has infinite perimeter!

And that it has zero area inside and infinite perimeter.

This is against the Koch Snowflake, which is a figure having a finite area inside an infinite perimeter. This goes against the way we think and according to geometric intuition that we have. But it actually is characteristic of many shpaes in nature. For example (i really like this example :D ) if we take all the arteries, veins and capillaries in the human body, then they occupy a relative small fraction of the body. Yet if we were able to lay them out end to end, we would find that the total length would come to over 60000 kilometres. This example i am quoting from an article i had copied years ago.

Related Article:

Lindenmayer systems and Fractals

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## Lindenmayer Systems and Fractals

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.

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.

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.

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:

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.

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