Monthly Archives: September 2010

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

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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

Mandelbrot Set

While technically a fractal, the Mandelbrot Set is a good example of emergent order from chaos. The order is more evident in a fractal than in a traditional chaotic system, but nonetheless it is chaotic.

Stock market chaos

The stock market is an example of chaos theory because initial conditions can have an impact on the flow of the market.  There are specific events that determine what a stock may do such as a company scandal or a hot new product on the market.  The stock price for the next day can sometimes be determined by the stock price from the previous day but a prediction for days or weeks in advance can be near impossible because of the uncertainty of the market.  If a stock market chart is viewed for an entire year the movement will look chaotic because more than likely there will be no consistent pattern for the movement of the stock.

Media Hunt #1: Kaleidoscope

The images produced by a kaleidoscope are a good example of a chaotic system.  The kaleidoscope very clearly shows sensitivity to initial conditions because if just one little colored piece moves, the resulting patterns will be drastically different.  In addition, a pattern once formed is not easy to replicate.  Even if you try and turn the end of the kaleidoscope back the other direction, it will not create the same patterns seen before.  There are bounds within a kaleidoscope, such as how many little colored pieces there are and the size of the area they can move in, which means that the kaleidoscope does not make purely random patterns, but is in a chaotic system.  

Media Hunt #1 Chaotic Pendulum AGAIN!!!!!

So we have seen tied chaotic pendulums (Riddick Hall) and also the magnetic pendulum built out of Magnetix pieces (WordPress).

Well this is a computer simulated one:

The Basics: They are releasing a pendulum at a certain displacement and then subjecting it to an oscillating force that induces the pendulum to swing (damping field)

http://www.myphysicslab.com/pendulum2.html

Not to sure on the blogging thing on how to embed the demo but you can alter the mass the damping field(oscillating force) and its behavior
But it also plots the angular velocity of the pendulum on a chart.

To make a live demo of these you would have to attach the RIGID pendulum to an electric motor that was given an oscillating amount of current to act as the damping force.

Just the slightest change in the system’s initial conditions yields a different but an obvious replication(repeating pattern) that proves it to not be random but also aperiodic. The only concerning thing is the Physics derivation at the end that yields an equation, may not be chaotic then however to those that don’t understand it the angular velocity appears to be chaotic

[Sorry it is not a video but the one I wanted to do was taken already, so i found the next best thing: AN INTERACTIVE VIDEO]

Media Hunt #1: Chua’s Circuit

Chau’s Circuit is a simple electronic circuit that demonstrates chaotic behavior.  The circuit was developed by Leon O. Chua in 1983, and its ease of construction has made it a very popular, real-world example of chaotic systems.  In order to construct a chaotic, autonomous circuit, the circuit must include three components:  1) one or more non-linear elements, 2) one or more locally active resistors, and 3) three or more energy-storage units.  Chua’s Circuit is the simplest type of chaotic circuit.  Its energy storage elements contain two capacitors (C1 and C2) and an inductor (L1).  It also includes an active resistor (R), as well as a non-linear resistor made of two linear resistors and two diodes.  Below is a sample picture of the circuit, with the far right containing a negative impedence converter.

The equations to describe Chua’s Circuit are as follows:

\frac{dx}{dt}=\alpha [y-x-f(x)]
\frac{dy}{dt}=x-y+z
\frac{dz}{dt}=-\beta y
Where x(t), y(t), and z(t) represent the voltage differences across C1 and C2 as well as the intensity of the current in L1, respectively.  The function f(x) describes the response of the non-linear resistor, and alpha and beta are determined by the particular values of the circuit’s components.  When these three equations are used to graph (x,y,z) in 3-d space, a chaotic attractor can be observed for certain values of f(x).  This example is extremely similar to the Lorenz’ Attractor, in both concept and results, due to the non-linear equations the circuit is based off of.  Below is an image of experimental results with Chua’s Circuit, as well as a link to an applet that models the chaotic attractor.
Chua\’s Circuit Applet

Hurricane Earl


Hurricane Earl

Hurricane Earl's planned path

Seen above is Hurricane Earl, early in its path on Wednesday, September 1, 2010.  The image shows the path which meteorologists at the National Oceanic and Atmospheric Administration predicted.  It depicts chaos, following that it is not accurately predictable, sensitive to initial conditions, and deterministic.

News 14 Carolina has similar information in a video at the link below:

http://charlotte.news14.com/content/top_stories/629808/hurricane-earl-maintains-category-4-strength

From this information, it is clear that the hurricane would stay within certain bounds, not being extreme or random.  Sensitivity to initial conditions is demonstrated from the shape of its planned path, being able to vary very little shortly after the storm’s information was recorded.  A wide range of possibilities was open for the hurricane in the time periods farther away, which depended on other weather conditions in the Atlantic.  With the number of variables involved (but not depicted) in determining the path of hurricanes, it is safe to say that hurricanes are chaotic in nature.

Media Hunt #1: Chaotic Spirals

These pictures are from Wolfram Demonstrations Project (http://demonstrations.wolfram.com/SpiralExplorer/). They are demonstrations of what J.P. Davis explains in his book Spirals: From Theodorus to Chaos. Essentially, the spirals are created by plugging in complex numbers for a and b. When a=1 and b=i, you get the discrete spiral. However, when a and b are changed slightly, different spirals result (as evidenced above).  This is an example of order arising from chaos because the dots are plotted chaotically but always end up creating a spiral (although the specifics of the spiral differ). The system is dependent on the initial conditions of what a and b are.

Zombie Outbreak Simulator

Simulator

I remember stumbling upon this game a while ago, and only now realize how well it fits the chaotic model we’ve learned. In this game, you are given a Google Maps picture of Washington D.C. and can control the starting number of zombies and numbers of civilians, along with the speed of zombies and other factors. The outcome of each iteration is the same (all zombies) but the speed at which it reaches this ending is sensitive to the starting conditions. The general progression can be determined (outwards from the initial) but the speed and exact shape are unpredictable.

The original game, “Zombie Outbreak Simulator,” wont load, but the newer version, ” Class 3 Outbreak,” is about the same. There have been some updates, such as the cops that can be controlled, but the fundamentals of the game still remain chaotic. The zombies now appear in random locations, so you cant control the initial locations, but the progression is still the same. The police squads can be used to add another variable to the how the game goes, but the outcome will still be 100% zombie.

Media Hunt 1: Arnold’s Cat Map

It’s Nilknarf! (who is that weird guy?)  All I know is that he’s really weird.  Anyway, Arnold’s cat map is a really cool chaotic map (somewhat akin to a function, like the logistic map) that occurs by mapping onto a torus on itself, with the help of a transformation matrix in modulo 1 (sorry for the jargon; just think of it as moving pieces of an image).  Arnold (the same kid who was on “Magic Schoolbus”… okay, not really) demonstrated his map by taking a picture of a cat, dividing it into sectors to be rearranged, and letting his transformation loose.  At first, the images appeared to change in random ways, producing ugly, decidedly non-catlike pictures, but after a number of transformations (which vary, depending on the divisions of the picture), the feline was returned to its original state.  Here’s a picture of the map!

Hope today’s installment was wacky and fun!  Join us next week for tips on how to make sure your roommate isn’t a space alien.

(image from wikipedia)

Media Hunt # – Rossler Attractor

Similar to the Lorenz attractor, the Rossler attractor models three non-linear differential equations that demonstrate chaotic behavior. The attractor, however, still has a form that involves symmetry and follows a certain pattern. The equations were only theoretical and did not have a specific application, but it was later found that they helped in studying chemical reactions (more specifically in modeling the equilibrium). The graph begins with a spiral close to one plane (and around a certain point), and then begins to rise and twist in the direction orthogonal to that original plane. The chaos occurs when the constants are set to a = 0.2, b = 0.2, and c = 5.7, but other values can produce similar chaotic results.

Media Hunt #1: Emergent Order on a Vibrating Plate

The below video, which I found on YouTube, is an example of what is known as a Chladni plate.  A metal plate is covered with salt, set up on top of a loudspeaker, and vibrated at various frequencies.  These vibrations cause the grains of salt to organize into patterns.  The patterns arise because salt gathers at the nodes of the standing waves that form in the metal plate at resonant frequencies.  Notice that at higher frequencies the lines of salt that form are closer together because the standing waves have a shorter wavelength so the nodes are closer together.  This is a variation of an experiment done famously by German physicist Ernst Chladni in the 1780s.  He used a violin bow instead of a speaker and sand instead of salt but the concept was the same.  Chladni was actually repeating the experiment of Robert Hooke, who had done the same thing a century earlier using flour on a glass plate, but for some reason it was Chladni’s name that stuck.

As a word of caution, I would suggest lowering the volume on your computer before playing this video.  The sounds used to vibrate the plate, especially the high-pitched ones, are a little annoying.

Media Hunt 1: Swinging Atwood’s Machine

This is a video of Swinging Atwood’s Machine which consists of two different masses hanging on a string over a pulley.  The string and the pulley are considered massless.  In Atwood’s Machine both the masses only move in one dimension, but in Swinging Atwood’s Machine one mass moves in two dimensions creating the swinging pattern.  The path of movement has a general pattern, but also has a very chaotic pattern as it goes at very different angles each time.  The more you watch the mass the more order seems to emerge, however it is still has sensitivity to initial conditions as the location it begins affects the path greatly.  It also can only be predicted on a general level, such as which direction it will travel next.

Langton’s Ant

Langton’s Ant lives on a grid of colored squares. At each step, the ant turns left or right depending on the color of its current square. Additionally, the ant changes the color of the square it just left. The system displays interesting behavior with even the simplest rules (2 possible colors) With this minimal rule set, the ant moves with no discernible pattern for around 10,000 steps, then falls into an infinitely repeating pattern of 104 steps (see the video). More complicated rules result in other interesting steady state behaviors. The way that Langton’s ant modifies its own world, thereby affecting its own future behavior, is reminiscent of nonlinear systems, where “playing the game has a way of changing the rules.”

For more information, see http://mathworld.wolfram.com/LangtonsAnt.html

The Tamari Attractor

The Tamari Attractor is a similar model to the Lorenz Attractor. It is a chaotic representation using 3 differential equations, however, its applications are in economics. The parameters are as follows: inertia, productivity, printing (of money), adaptation, exchange rate, indexation (linking), expectations, unemployment and interest. Using these parameters, the Attractor outputs GDP, M1 and CPI – 3measures of a country’s economic well-being. Using these 3 outputs, the system can predict the future economic success of a country, and to some extent, the political conditions of a country. The picture about shows the Tamari equations’ prediction for various countries based on data from 1960 to 2008.

Below is a picture of the Tamari Attractor “nest”:

Conway’s Game of Life

The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970. The “game” is a zero-player game, meaning that its evolution is determined by its initial state, requiring no further input. One interacts with the Game of Life by creating an initial configuration and observing how it evolves. In the above video, random initial conditions are applied to the hardware setup and the final result is a clear discernible pattern pattern. Also, the flashy lights look cool!

For more information please refer to: http://en.wikipedia.org/wiki/Conway’s_Game_of_Life

Media Hunt #1: Magnetic Pendulum

The magnetic pendulum displayed in the video above is an example of a “simple” chaotic system. At least, simple in the sense that it is easy to make and test. In class, we have come to learn and agree that sensitivity to initial conditions is one defining property of any chaotic system. This basic property is easily displayed in the magnetic pendulum. After testing each position on the grid, the experiment is executed once more and this time different results are obtained. Since the exact release position and release force was not provided on the second attempt there was no way for it to obtain the same results as the first run. If the experiment was attempted several more times it is highly unlikely that the first run results will ever be repeated. In the video, we are also shown a computer simulation of the magnetic pendulum. The first run of the computer program also do not match the results of the experiment during the first trial. If the simulation was run twice without any of the variables (such as the release position and the release force) changed, then it is likely that both outcomes will be the same.

A similar program that was used in the video can be found here and a bit of a mathematical explanation of the pendulum behavior can be found here.

The Tilt-a-Whirl

The Tilt-a-Whirl exhibits chaotic behavior that is very similar to that of the waterwheel and of the pendulum in the atrium of Riddick.  All of the “cars” are located around the edges of a large round platform which rotates at a constant speed in a constant direction.  This platform goes up and down over (usually three) hills.  Each car is also on its own track which allows it to rotate freely.  This is where the chaotic movement comes in to play.  As the cars go up and down over the hills, you never actually know which way the car will turn.  Sometimes you never even know if your car will move.  As you can tell from the video, some of the cars also rotate a lot quicker than the others.  This reminds me of the waterwheel because the ride starts from the same position every time.  However, you never get the same path.  The first hill on one turn may send you spinning in one direction and the other time will send you spinning in the other direction.  This unpredictability is what makes this ride fun and a staple at amusement parks.

More info: http://www.sciencenewsforkids.org/pages/puzzlezone/muse/muse0501.asp