Category Archives: Media Hunt #2

Sierpinski / Menger Sponge

The Menger Sponge, also known as the Sierpinski Sponge, is an example of a fractal. It is created by dividing a cube into 27 smaller cubes, and removing the center cube and the middle cube on each face. In the image above, we see a plain cube and the first 3 iterations or levels of the fractal.

The Menger Sponge is a three-dimensional extension of the Sierpinski Carpet, which is a 2D shape created by removing the center square from a square divided into 9 parts.

This shape has a fractal dimension [Hausdorff dimension] of approximately 2.726833.

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Media Hunt #2: Fractals and Jackson Pollock

Who’s familiar with Jackson Pollock?  Pollock was a very interesting American abstract artist who created art by essentially splattering paint onto the canvas in a seemingly random pattern.  This somewhat psychedelic video I found starts out rather generally giving information about fractals that we’ve already learned in class, but things get more interesting at about 2:15 when Jackson Pollock himself is addressed.  Many examples of Pollock’s paintings are then shown.  Interestingly, with such great controversy surrounding the legitimacy of his work, only Pollock’s paintings contain fractals, whereas the imitators of such work do not.

link: http://www.youtube.com/watch?v=Rs1GZxmdT6swatch?v=Rs1GZxmdT6s

Burning Ship

 

The burning ship fractal was first described by Michael Michelitsch and Otto Rossler in 1992.  This fractal is created by iterations of the equations:

Xn+1 = Xn^2 – Yn^2 – Cx

Yn+1 = 2 * absvalue( Xn*Yn) – Cy

This video zooms in on a fractal that resembles a burning ship.  As you zoom in closer and closer, you start to see the repeated pattern of the burning ship.  This is much like the Mendelbrot set, since each time you zoomed in you would see a smaller version of the original shape.  I would like to explore the burning ship fractal more to see if there are any other repeated shapes.  For example, this video only zooms in the the area on the right.  I wonder if the “swirly smoke” also has this kind of repeated pattern.

 

For more info:  Burning Ship Fractal

Mandelbulb

The Mandelbulb is a 3-dimensional analog of the Mandelbrot set, created by Daniel White and Paul Nylander. Since the 2d Mandelbrot set exists in the space of complex numbers (a + bi, where i is the square root of 2), the 3d Mandelbrot must occupy a space of “hypercomplex” numbers.

The hypercomplex algebra is complicated, but the important thing is that the Mandelbulb uses the map z -> z^n + c    (different values of n generate different Mandelbulbs). The most well-known Mandelbulb seems to be the one generated by n=8:

Just like the Mandelbrot set, the Mandelbulb displays “pseudo-fractal” morphology. There don’t appear to be any exact repetitions, but there are bulbs upon bulbs upon bulbs……. Additionally, we cannot discern our level of zoom just by looking at the level of detail.

Here is a video of a zoom into the Mandelbulb:

For more information, see http://en.wikipedia.org/wiki/Mandelbulb and http://www.skytopia.com/project/fractal/mandelbulb.html

Fractal Carrots

Many plants in nature produce fractal growth patterns, and a plant called Queen Anne’s Lace is one of them.  Queen Anne’s Lace is the special name given to wild carrots, and its floral arrangement displays certain characteristics of a fractal.  From the main stem, each blossom produces smaller iterative blooms, so that each blossom looks just like a copy of the main stem.  In this way, Queen Anne’s Lace shows self-similarity, at least up to a point.  As a biological/physical object, the flower’s fractal characteristics are not infinite, only showing a handful of iterations.  However, its self-budding still gives Queen Anne’s Lace its characteristic, large-scale fractal qualities.

Top View

Bottom View

Broccoli

Broccoli is not only my favorite vegetable but also a fractal.  Broccoli is an example of one of the naturally occurring fractals in the environment.  Broccoli meets the characteristics of being a fractal because it has a small-scale structure, it is split into parts that are smaller than the origin and it can not be described in regular geometric terms because of its irregular structure.    A piece of broccoli would be difficult to measure because of its small structure and infinite complexity.

Media Hunt #2: Fractal Diamonds

    Here is a picture of a diamond fractal (self-named by the author; taken from deviantart.com drawn by JASONQUANTUM1). It is a self- similar pattern, one of the characteristics of fractals. Mathematically, you could find a way to draw this pattern.  What I would like to discuss in class is if real diamonds could be considered fractals? They have very sharp lines inside that cross planes; however, there are no examples in our readings of them. Given that most people have had interaction with diamonds, they probably should be a go-to example. And if we determine that diamonds are not in fact fractals, what makes them different from crystals?

Diamond vs. Crystal

(Diamond picture from jewishjournal.com article; crystal from USDA file found on wikimedia.com)

Falling Sand Game

When looking at the many branch patterns we discussed in class, I couldn’t help but remember a game I played a few years ago. This one was called Falling Sand. It is a complete game of emergence, as there is no progression. All one does is add different types of particles and see how they interact. The interaction that relates to out subject matter is that between the Water and Plant. Water is blue and falls down until it leaves the screen. Plant is green and stays in the same place on the screen. When a blue particle touches a green particle, it stops and turns green, making the plant “grow”. The branching pattern occurs when you place a spout (light blue) above a plant. The spout stays still, but produces water at a seemingly random timing.

The water falls and hits the plant, making it grow. The spots that get more water grow faster and expand, cutting off water to lower parts. This can be furthered by segmenting the spout (as seen in the image) or can be minimized by starting all points at the same time. Here’s the link to try it out yourself:

http://fallingsandgame.com/sand/pyro.html

Media Hunt #2: Peacock Feathers

While it may not seem evident at first, the feathers of a peacock, or any bird feather, show some amount of a fractal pattern.  A feather’s fractal nature does not repeat infinitely, only a few times.  The feather has a main vein off which many barbs extend.  Off of these barbs, extend barbules, and then off the barbules are tiny hooklets.  It is these hooklets which hold the barbules, and therefore the barbs, together giving the feather the shape that we see.  Fractals can be found everywhere in nature, and here we see them found in birds.  The peacock is an especially beautiful bird where the feathers themselves are not only fractals, but also create an exquisite pattern with the other feathers.

Media Hunt 2: Mandelbrot TED Talk

The following is a video of Mandelbrot himself explaining the concept of roughness and how he found regularity within roughness. In the beginning he goes through how a cauliflower, for example, has a repetitive morphology; the more you cut it, the smaller yet same cauliflowers you have. What’s really amazing is how he is able to estimate, just by looking, the fractal dimension of a certain picture. He also talks about how he came up with certain assumptions that were correct, though he had no proofs for them (his friends later found the proofs).

In the video, Mandelbrot  describes himself as “constructing a geometry, a geometry of things which had no geometry” which I thought was pretty interesting.

He also talks about the use of his findings in CGI images in movies, the debate over how large the area of a lung is, and other interesting facts about his career and the Mandelbrot set. He is a really funny guy and the video was very entertaining to watch.

Stress Fractures in Metal

Seen in the video is an example of a stress fracture occurring in aluminum. Stress fractures are common and are caused by metal fatigue through excess use or strain.

Stress Fracture

When observing fractures on a microscopic scale, self-similarity becomes apparent. In the above image, there appears to be stress fractures inside of the larger crack.

When viewed from a mathematical standpoint, cracks propagate in a nearly linear fashion. They can be predicted to a certain extent within bounds, as cracks tend to stay along the same path. These properties show that they can be thought of as fractals.

For more information on fatigue, view the notes at the link below from a Materials Science course:
http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/anal/kelly/fatigue.html

The Human Genome – A Fractal

By breaking the human genome into millions of pieces and reverse-engineering their arrangement, researchers have produced the highest-resolution picture ever of the genome’s three-dimensional structure. The picture is one of mind-blowing fractal glory, and the technique could help scientists investigate how the very shape of the genome, and not just its DNA content, affects human development and disease.

To read more about this please go the following link.

Media Hunt #2: Nautilus Shell

A nautilus shell is an example of something that demonstrates fractal-like behavior. When looking at the cross-section of one of these shells, it is easy to see the presence of a fractal. The inside of a nautilus shell is made up of numerous chambers which are created by the nautilus as it grows throughout its life time. The chambers spiral inwards towards a focal point with each chamber being much like the previous one except for the smaller size. While this real world fractal is finite in size, one can easily imagine the pattern continuing outwards with larger sized chambers. Another interesting thing to point out about the nautilus shell is that the spiral created by the shell itself is very similar to a Fibonacci spiral. While the shell does not create an exact Fibonacci spiral, the shape is very close and one can easily see the similarities between the two.

Lightning in Slow Motion, Lichtenberg Figures Videos

Lichtenberg figures are fractal formations caused by electrical discharge. They can occur in rocks, grass, wood, or even people (as shown by Abby)! In Abby’s post, I think it is interesting to see the interaction of one fractal (electricity) with another (capillary and blood flow in the human body).

This video shows the creation of many Lichtenberg figures in glass. I couldn’t figure out how to embed the video, but if you copy and paste the link it should work. This video displays sensitivity to initial conditions – look at how different the fractals are from slight differences in how they are created:

http://link.brightcove.com/services/player/bcpid1417302205?bctid=1419758473

However, the most “striking” video that I found was this one, showing a lightning bolt in slow motion. Amazing! Makes me wonder if this could be used as a model for evolution, showing chaos, sensitivity to initial conditions, extinction, and selection:

Romanesco Broccoli

No one is quite sure what exactly Romanesco “broccoli” should be classified as, in different languages it is called cabbage, broccoli, and pyramid cauliflower.  This is certainly in part due to the unique fractal appearance of it.  The fractal is naturally occuring in the plant and is surprisingly complete and has more depth to it then I expected to find in something natural.  I have attached three pictures, one shows the entire broccoli, the next in a zoomed in image that shows the continuing fractal, and the last is where you can see the fractal beginning to break down.  The fractal continues for a while but eventually as you zoom in close enough the shape begins to break down and become more smooth.

Media Hunt #2: The snowflake

A snowflake is a fractal.  It branches out six directions that look fairly similar no matter which direction you go.  For all intents and purposes, I am going to assume that the snowflake in this picture is flat, like a snowflake that you created in kindergarten made out of paper.   Because the snowflake has no height, it can not have a dimension greater than two.  The snowflake is certainly not in the first dimension because it has both length and width.  The snowflake is a two dimensional object with spaces in it (like our balled up piece of paper in class, just one dimension lower).  This would imply that the snowflake exists somewhere in between the first and second dimensions.  Now for the interesting part.  When a snowflake melts, it turns into water.  Water/ice is one of those strange things in nature that expands when it freezes and shrinks when it melts.  So when a snowflake melts, it gets smaller.  Now consider that drop of water.  A drop of water exists in the third dimension (just think of water in a glass; it has height, length, and width, but no holes).  This means that when a snowflake melts and shrinks, it goes from the 1.something dimension to the third dimension: a greater dimension.  Normally when we think of something going up in dimensions, we think of it expanding.  Strangely, a snowflake works in the opposite direction.

Computer Generated Tree

As we discussed in class, tree structures can be made by following a defined set of rules for a certain number of iterations. However, the tree structures in the pictures of the “Tree and Leaf” chapter of Branches did not seem to closely resemble what a natural tree would look like. This video shows a computer program called L-konna, which is used to produce a very realistic looking tree that not only has branches that bend downward but also has colorful leaves. L-konna is a Java-based application that can create L-systems. An L-system, or Lindenmayer system, is commonly used recursive system that can generate self-similar fractals and model the growth of plants. The L-system was discovered by a Hungarian theoretical biologist and botanist named Astrid Lindenmayer, who first used it to model the growth of yeast, fungi, and algae. It was later developed to describe higher plants and complex branching structures.

http://en.wikipedia.org/wiki/L-system

Fractal Solids D < 3 < D

I will comment on it later but here it is,

 

EPIC

 

You can click the image to go to the rest of the site

 

We have gone over on how when we take away and add matter to get objects that exhibit these partial dimensions.

The fractal natural of the Sierpinski Triangle could also be applied in reverse by adding material (similar to Koch) This was shown with the Tapered cube, regular cube (hexahedron), and the octahedron

The website shows it applied to several seed shapes. This was brought up in class on whether or not the shape used could be altered to obtain these interesting patterns.
What was interesting with these fractal solids was how the ones where material is removed could be zoomed in on and the knowledge on what scaling was done is gone. However with the shapes where it is added you can tell that you are zoomed in just not how much. I feel that this is a difference between this. I also think I know why we see that.

When we remove you are bringing the shape toward a 2D shape. We being 3 dimensional can see this easily and imagine it easily too.
When we add we are bringing the shape toward a 4D shape. Yeah try drawing (sculpting) a 4D cube right quick, you cannot because we are limited to the 3D we exist in thus when a near 4D object exists in our 3D we see very interesting properties (such as knowledge that we have zoomed in)

Just thought I would note that.

The fractal natural of the Sierpinski Triangle could also be applied in reverse by adding material (similar to Koch) This was shown with the Tapered cube, regular cube (hexahedron), and the octahedron

Lightening Strikes Create Fractals

These images show the effects of the current of a lightening strike.  When a human is struck there appears a Lichtenburg pattern which is a fractal resulting from electric discharge.  It will show up on the surface of insulating materials such as our skin.  The figure with a rod of the Lichtenburg pattern was created by exposing the lucite cylinder to an electron beam.  This pattern is exactly what happens with the lightening strike on our skin.  Some scientists even believe that this pattern continues to smaller and smaller levels all the way to the molecular level which we cannot see.

Media Hunt #2: Fractals in African Villages

The below video is of a presentation given by a mathematician/ anthropologist named Ron Eglash.  (He is considered an “ethnomathematician” because he studies mathematics as it relates to culture.)  Several years ago he was looking at aerial photographs of African villages and saw what he considers to be fractal patterns.  He later found that these villages were intentionally built with self-similar patterns for religious and other cultural reasons.  The patterns seen in the layouts of these villages differ from the branch-like patterns and convoluted boundaries that we have focused on in class, but there is some self-similarity present from the large scale to smaller scales so they can be considered fractals.  These are not perfect mathematical fractals as they only scale down a finite number of times, but they still provide an interesting example of self-similar patterns.  They also loosely relate to Philip Ball’s comments about the layout of cities in chapter 2 of Branches.

The part of the video that specifically relates to these villages is from 3:00 to about 6:30 into the video.  Also, in the first three minutes there is a general discussion of fractals that includes references to the Cantor set and the Koch curve. 

More details about the shapes of the villages, the occurrence of self-similar patterns in African art, and about fractals in general can be found at…     

http://csdt.rpi.edu/african/African_Fractals/homepage.html