Richest Americans of All Time

Mint.com

It's just an interesting infographic.

A Cool Tetris Art Creator

Click inside the white box.

Man Befriends 38 Lions

Cool PVC Instrument

For more videos from the creator:

http://www.youtube.com/user/snubbyj

Some Beautiful Chemical Reactions

Potassium, Diethyl Ether, and Water


Potassium Chlorate and ....a Gummy Bear?

On Drug Prohibition and the Lack of Logical Reasoning Behind It

This is just a logical analysis. It is likely incomplete, but filled with good points. If there are any refutations or other information on this, feel free to post in the comments.

Touchable Hologram

Brought to you by Tokyo University:



I dig that little elephant hologram!

Impressive Robotics (Part 1)

More than Meets the Eye


A Snake Robot


Dragonfly Robot


ASIMO


Rapid Motion-Sensing Robot Hand


Robotic Sunflower


Ball-Balancing Robot


Home-made Robot


University of Arizona Hexapod Robot


Ugobe Pleo Toy Robot

Vinni Puh

"While Eeyore frets ...
... and Piglet hesitates
... and Rabbit calculates
... and Owl pontificates
...Pooh just is."
~The Tao of Pooh
by Benjamin Hoff

~A 1969 Russian Version of Winnie the Pooh~
I find it to be much more well done, humorous, and philosophical than the US version. Winnie the Pooh was originally created by A. A. Milne in the 1920's and licensed to Disney in 1966. The original characters were actually meant to be psychological/philosophical archetypes.

Episode 1



Episode 2



Episode 3



Episode 4



"pam-param-pooroom-howlers
poom-pooroom-taram-yellers
poom-poopoompoom"
~Vinni Puh

Ana Somnia - lights out!

http://anasomnia.com/

I think this is a very creative and eerie feel to it. It works best if it's nighttime and you have a webcam.

Mandalas

"It became increasingly plain to me that the mandala is the center. It is the exponent of all paths. It is the path to the center, to individuation."
~Carl Jung

Mandalas are typically a Hindu or Buddhist sacred tradition. The idea is that they are a small representation of the universe. They are to be a spiritual/metaphysical representation of the universe, and as far as I understand not intended to actually represent the physical configuration of the universe (check some Hubble sites if you want to see something like this ;) ). I find it to be truly beautiful and intriguing artwork. Mandalas represent the intricacy and flexibility of human thought.

Buddhist Monks Create a Mandala in 6 Days


Tibetan Sand Mandala

Brilliant Physics Simulator

created by Lagoa Multiphysics

Non-Reflective Surfaces as Mirrors (Artist Daniel Rozin)

Daniel Rozin is an artist who makes reflective surfaces out of unreflective materials. This mirror is composed of 830 'wooden pixels.' A built in camera activates motors behind each of the wooden pieces, which through some (seemingly elaborate) method creates a visual representation. The other mirrors operate by similar principles

Other Mirrors

The Circles Mirror:


The Trash Mirror:


The Weave Mirror:

The Rust Mirror:

The rest of his art (or at least some portion of it) can be found on his site:

http://www.smoothware.com/danny/

Bug Trails

Above is a two second exposure to the flight of insects around a floodlight at Hurst, TX on September 19, 2009.

Some moths and bugs are attracted to bright lights because they use the Moon for orientation. Steady angle in relation to light source is possibly obtained via transverse orientation.

All of the information about all of this was pretty much obtained from:
http://epod.usra.edu/
http://www.abc.com
http://www.npr.org/

It's basically an e-Hamster

I would put it on the side with the fishes, but I'm not quite sure how.

A View of the Stars as Earth Rotates

Truly awe-inspiring in its magnitude and beauty.





"For small creatures such as we the vastness is bearable only through love."
~Carl Sagan

Check out the overview effect.

Extremophiles (radical, bro)

Searching for "Impossible" Life



Extremophiles are just lifeforms which thrive in extreme environments. They can be eukaryotic (animals, plants, fungi), but tend to be prokaryotic (archaebacteria, eubacteria). Most complex organisms that we know of would die under the conditions in which these beings survive. It has been proposed that perhaps the earliest life on Earth began in these extreme environments (as the early Earth would have been quite inhospitable for many species we know).

Some types of extremophiles:
Acidophile: Survives in extreme acidity (pH < 3)
Alkaliphile: Survives in extreme basicity (pH > 9)
Halophile: Survives in extreme salinity
Piezophile: Survives in extreme pressure
Thermophile: Survives in extreme heat (Temp > 60 Celsius)
-Hydrothermophile: Survives in extreme temperatures near hydrothermal vents
Psychrophile: Survives in extreme cold (Temp < 30 Celsius)

David Gallo on Deep Sea Ocean Life - WATCH THIS IF NOTHING ELSE!



Some articles:
Explanatory Article
Yellowstone Light-Harvesting Species
ARMAN Acidophile
Deep Subsurface Bacteria
Antarctic Deep Sea Methanophile
Life Below Antarctic Glacier (no light, no oxygen)
Deep Sea Microbes involved in Nitrogen Cycle?
Deep Ocean Biomass
& again


















^Multicellular Animal in Oxygen-Free Environment^
Radiation-Consuming Bacteria....just think of the practical uses!
So why not on other planets?
Extremophiles, Antarctica, and ET Life
Alaska as Model for Solar System Environments?
They can grow in moon soil
Ultramicrobacterium seems well suited for Exo-planetary life

Little Red Riding Hood for Engineers

Done as a reinterpretation of Little Red Riding Hood by graphics design student Tomas Nilsson

Check out his website:
http://www.tomas-nilsson.se/

Smithsonian Zoo Animal Webcams

I can't embed the cams or anything, but here's the main website for it:

http://nationalzoo.si.edu/animals/webcams/

My personal favorites are the Ferret Cam, the Microscope Cam, and the Gold Lion Tamarin Cam. The ferrets have recently had babies, so I believe I see them roamin' around in there from time to time.











Here's the zoo's main website:

http://nationalzoo.si.edu/

Mike Mitchell's Artwork

I very much enjoy this artist. He's got an interesting style, quite an imagination, and frequently give homage to many geeky topics.

Little Pilot Skullington



















Mr. Bunners the Rabbit Master



















Iron Giant



















The Dude Invitational



















Rocketbird



















Death of Neon



















Discover Magazine Einstein



















Reddit



















His main website:
http://sirmikeofmitchell.com/

Intricate Devices of Antiquity

ANTIKYTHERA MECHANISM


















Used to calculate motions of astronomical bodies. It's almost exactly precise, and it was created about 2,100 years ago by the Greeks (not to be confused with geeks)!!! The device's workings are based on the geocentric model of astronomy. Found in 1901 as part of a shipwreck in the town which shares the same name as the device, the original is on display at the National Archaeological Museum in Athens (http://www.namuseum.gr/wellcome-en.html). Devices of similar complexity to this appear to have been lost for some time and only recreated around the 1300's.

Check out wikipedia for a much better description:
http://en.wikipedia.org/wiki/Antikythera_mechanism

Michael Wright's Model:



Here's a simulation of the internal gear workings:


ASTROLABE



















It's another astronomical calculator (less intricate) developed in 150 BC. This primative version isn't quite as impressive as the 13th-century one shown in the next video.

Again, check wikipedia; it's much better:
http://en.wikipedia.org/wiki/Astrolabe

Tom Wujec demos the 13th-century Astrolabe:


Here's a spherical astrolabe, like those of the middle ages:














BAGHDAD BATTERY

These are most likely from the early centuries AD. There are a couple theories as to the use of these devices. Some suggest they weren't actually electrochemical cells at all, but that they were for storing scrolls, which decayed and then left acidic traces within. This theory is largely based on the lack of voltage which can be generated by a battery, but it's not completely unrealistic to assume that multiple batteries were connected by wires to achieve more usable voltages. I don't ascribe to this theory. I personally think they were used for electroplating (coating silver with gold in this case via a redox reaction) based on the different theories I've read. Other ideas are that they were used for electrical acupuncture or even a religious experience (shock). When Mythbusters tested 3 of these theories (electroplating, acupuncture, religious experience) by stringing a couple of the batteries together in series, all 3 were found to be plausible. Note that if these were indeed used as electrochemical cells, they precede the invention of batteries by over 1,000 years.

Wikipedia: http://en.wikipedia.org/wiki/Baghdad_Battery

The Deeply Enlightened Carl Sagan

Look again at that dot. That’s here. That’s home, That’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every ‘superstar,’ every ‘supreme leader,’ every saint and sinner in the history of our species lived there — on a mote of dust suspended in a sunbeam… There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we’ve ever known.
~Pale Blue Dot



So truly touching and beautiful (please read):
http://www.csicop.org/si/show/ann_druyan_talks_about_science_religion/



The Cosmos is all that is or ever was or ever will be. Our feeblest contemplations of the Cosmos stir us — there is a tingling in the spine, a catch in the voice, a faint sensation, as if a distant memory, of falling from a height. We know we are approaching the greatest of mysteries.
~Cosmos

The receipt of a message from an advanced civilization will show that there are advanced civilizations, that there are methods of avoiding the self-destruction that seems so real a danger of our present technological adolescence… Finding a solution to a problem is helped enormously by the certain knowledge that a solution exists. This is one of many curious connections between the existence of intelligent life elsewhere and the existence of intelligent life on Earth.
~The Dragons of Eden

This is the time when humans have begun to sail the sea of space. The modern ships that ply the Keplerian trajectories to the planets are unmanned. They are beautifully constructed, semi-intelligent robots exploring unknown worlds.
~Cosmos

The human brain seems to be in a state of uneasy truce, with occasional skirmishes and rare battles. The existence of brain components with predispositions to certain behavior is not an invitation to fatalism or despair: we have substantial control over the relative importance of each component. Anatomy is not destiny, but it is not irrelevant either.
~The Dragons of Eden

When our genes could not store all the information necessary for survival, we slowly invented brains. But then the time came, perhaps ten thousand years ago, when we needed to know more than could conveniently be contained in brains. So we learned to stockpile enormous quantities of information outside our bodies. We are the only species on the planet, so far as we know, to have invented a communal memory stored neither in our genes nor in our brains. The warehouse of that memory is called the library.

A book is made from a tree. It is an assemblage of flat, flexible parts (still called “leaves”) imprinted with dark pigmented squiggles. One glance at it and you hear the voice of another person — perhaps someone dead for thousands of years. Across the millennia, the author is speaking, clearly and silently, inside your head, directly to you. Writing is perhaps the greatest of human inventions, binding together people, citizens of distant epochs, who never knew one another. Books break the shackles of time, proof that humans can work magic.
~Cosmos

I worry that, especially as the Millennium edges nearer, pseudoscience and superstition will seem year by year more tempting, the siren song of unreason more sonorous and attractive. Where have we heard it before? Whenever our ethnic or national prejudices are aroused, in times of scarcity, during challenges to national self-esteem or nerve, when we agonize about our diminished cosmic place and purpose, or when fanaticism is bubbling up around us — then, habits of thought familiar from ages past reach for the controls. The candle flame gutters. Its little pool of light trembles. Darkness gathers. The demons begin to stir.
~The Demon-Haunted World

All inquiries carry with them some element of risk. There is no guarantee that the universe will conform to our predispositions. But I do not see how we can deal with the universe — both the outside and the inside universe — without studying it. The best way to avoid abuses is for the populace in general to be scientifically literate, to understand the implications of such investigations. In exchange for freedom of inquiry, scientists are obliged to explain their work. If science is considered a closed priesthood, too difficult and arcane for the average person to understand, the dangers of abuse are greater. But if science is a topic of general interest and concern — if both its delights and its social consequences are discussed regularly and competently in the schools, the press, and at the dinner table — we have greatly improved our prospects for learning how the world really is and for improving both it and us.
~Broca's Brain

We must stop pretending we’re something we are not. Somewhere between romantic, uncritical anthropomorphizing of the animals and an anxious, obdurate refusal to recognize our kinship with them — the latter made tellingly clear in the still-widespread notion of ‘special’ creation — there is a broad middle ground on which we humans can take our stand.
~Shadows of Forgotten Ancestors

Because the word ‘God’ means many things to many people, I frequently reply (to people who ask ‘Do you believe in God?’) by asking what the questioner means by ‘God.’ To my surprise, this response is often considered puzzling or unexpected: ‘Oh, you know, God. Everyone knows who God is.’ Or ‘Well, kind of a force that is stronger than we are and that exists everywhere in the universe.’ There are a number of such forces. One of them is called gravity, but it is not often identified with God. And not everyone does know what is meant by ‘God.’ … Whether we believe in God depends very much on what we mean by God.

My deeply held belief is that if a god of anything like the traditional sort exists, our curiosity and intelligence are provided by such a god. We would be unappreciative of those gifts (as well as unable to take such a course of action) if we suppressed our passion to explore the universe and ourselves. On the other hand, if such a traditional god does not exist, our curiosity and our intelligence are the essential tools for managing our survival. In either case, the enterprise of knowledge is consistent with both science and religion, and is essential for the welfare of the human species.
~Broca's Brain

Those who raise questions about the God hypothesis and the soul hypothesis are by no means all atheists. An atheist is someone who is certain that God does not exist, someone who has compelling evidence against the existence of God. I know of no such compelling evidence. Because God can be relegated to remote times and places and to ultimate causes, we would have to know a great deal more about the universe than we do now to be sure that no such God exists. To be certain of the existence of God and to be certain of the nonexistence of God seem to me to be the confident extremes in a subject so riddled with doubt and uncertainty as to inspire very little confidence indeed. A wide range of intermediate positions seems admissible, and considering the enormous emotional energies with which the subject is invested, a questioning, courageous and open mind seems to be the essential tool for narrowing the range of our collective ignorance on the subject of the existence of God.
~Broca's Brain

We have held the peculiar notion that a person or society that is a little different from us, whoever we are, is somehow strange or bizarre, to be distrusted or loathed. Think of the negative connotations of words like alien or outlandish. And yet the monuments and cultures of each of our civilizations merely represent different ways of being human. An extraterrestrial visitor, looking at the differences among human beings and their societies, would find those differences trivial compared to the similarities. The Cosmos may be densely populated with intelligent beings. But the Darwinian lesson is clear: There will be no humans elsewhere. Only here. Only on this small planet. We are a rare as well as an endangered species. Every one of us is, in the cosmic perspective, precious. If a human disagrees with you, let him live. In a hundred billion galaxies, you will not find another.
~Cosmos

Each of us is a tiny being, permitted to ride on the outermost skin of one of the smaller planets for a few dozen trips around the local star… The longest-lived organisms on Earth endure for about a millionth of the age of our planet. A bacterium lives for one hundred-trillionth of that time. So of course the individual organisms see nothing of the overall pattern — continents, climate, evolution. They barely set foot on the world stage and are promptly snuffed out — yesterday a drop of semen, as the Roman Emperor Marcus Aurelius wrote, tomorrow a handful of ashes. If the Earth were as old as a person, a typical organism would be born, live, and die in a sliver of a second. We are fleeting, transitional creatures, snowflakes fallen on the hearth fire. That we understand even a little of our origins is one of the great triumphs of human insight and courage.
~Shadows of Forgotten Ancestors

The cannabis experience has greatly improved my appreciation for art, a subject which I had never much appreciated before. The understanding of the intent of the artist which I can achieve when high sometimes carries over to when I'm down. This is one of many human frontiers which cannabis has helped me traverse. There also have been some art-related insights - I don't know whether they are true or false, but they were fun to formulate. For example, I have spent some time high looking at the work of the Belgian surrealist Yves Tanguey. Some years later, I emerged from a long swim in the Caribbean and sank exhausted onto a beach formed from the erosion of a nearby coral reef. In idly examining the arcuate pastel-colored coral fragments which made up the beach, I saw before me a vast Tanguey painting. Perhaps Tanguey visited such a beach in his childhood.

A very similar improvement in my appreciation of music has occurred with cannabis. For the first time I have been able to hear the separate parts of a three-part harmony and the richness of the counterpoint. I have since discovered that professional musicians can quite easily keep many separate parts going simultaneously in their heads, but this was the first time for me. Again, the learning experience when high has at least to some extent carried over when I'm down. The enjoyment of food is amplified; tastes and aromas emerge that for some reason we ordinarily seem to be too busy to notice. I am able to give my full attention to the sensation. A potato will have a texture, a body, and taste like that of other potatoes, but much more so. Cannabis also enhances the enjoyment of sex - on the one hand it gives an exquisite sensitivity, but on the other hand it postpones orgasm: in part by distracting me with the profusion of image passing before my eyes. The actual duration of orgasm seems to lengthen greatly, but this may be the usual experience of time expansion which comes with cannabis smoking.
~Marijuana Reconsidered Article

The truth may be puzzling. It may take some work to grapple with. It may be counterintuitive. It may contradict deeply held prejudices. It may not be consonant with what we desperately want to be true. But our preferences do not determine what's true. We have a method, and that method helps us to reach not absolute truth, only asymptotic approaches to the truth — never there, just closer and closer, always finding vast new oceans of undiscovered possibilities. Cleverly designed experiments are the key.
~Skeptical Inquirer Article

I would love to believe that when I die I will live again, that some thinking, feeling, remembering part of me will continue. But much as I want to believe that, and despite the ancient and worldwide cultural traditions that assert an afterlife, I know of nothing to suggest that it is more than wishful thinking.
The world is so exquisite with so much love and moral depth, that there is no reason to deceive ourselves with pretty stories for which there's little good evidence. Far better it seems to me, in our vulnerability, is to look death in the eye and to be grateful every day for the brief but magnificent opportunity that life provides.
~In the Valley of the Shadow

In science it often happens that scientists say, "You know that's a really good argument; my position is mistaken," and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.
~CSICOP Conference

It is sometimes said that scientists are unromantic, that their passion to figure out robs the world of beauty and mystery. But is it not stirring to understand how the world actually works — that white light is made of colors, that color is the way we perceive the wavelengths of light, that transparent air reflects light, that in so doing it discriminates among the waves, and that the sky is blue for the same reason that the sunset is red? It does no harm to the romance of the sunset to know a little bit about it.
~Pale Blue Dot

A Section From Broca's Brain:
http://www.stephenjaygould.org/ctrl/sagan_science.html


http://www.skeptic.com/
http://en.wikiquote.org/wiki/Main_Page

Chaos Theory (Part 2)

An attractor is a set in phase space that a particular dynamical system (weather patterns in the case of the Lorenz attractor) evolves towards over time. Any point which approaches the attractor will remain close to that attractor. The attractor can be a point, a line, a surface, or for many strange attractors, a fractal. That any strange attractor has a fractal set is to say that there exists some manifold with which the attractor can intersect to produce the Cantor set. A fractal is a geometric shape that exhibits both irregularity and self-similarity. Exhibiting self-similarity means that there are portions of the set, which if magnified a certain amount, is exactly the same set as the one which you magnified. Size-invariance is a continuous version of self-similarity. It must also exhibit irregularity, however, as a straight line is simple enough to be described by Euclidean geometry (e.g. high school geometry), and is not a fractal. Chaos theory typically deals with strange attractors. Pictures are really a much better way of understanding fractals and self similarity, so here are some cool ones!

First, the Cantor Set:












Other Famous Examples of Self-Similarity/Fractals

Mandelbrot Set














Mandelbrot Set's Correspondence with Logistic Map



















Sierpinski Triangle



















This site has a cool visualization:
http://serendip.brynmawr.edu/playground/sierpinski.html

Sierpinski Carpet



















Menger Sponge















& Mengerubik Cubesponge
http://forgetomori.com/2009/science/mengerubik-cubesponge/

Dragon Curve











Peano Curve
















Julia Set (of sin(z)...there are Julia sets of many functions)




















Koch Snowflake



















Koch Curve Self-Similarity








Brownian Tree (crystal growth)













Brownian Motion



















Lichtenberg Figures and Lightning




































(notice how it seems to choose one pathway after branching out - very interesting)


Ferns (real and fractal generated)





































Trees Too?
http://www.gskinner.com/blog/assets/InteractiveElm.html
http://www.webcalc.net/calc/0467.php

Artistic Representation of a Brain as Fractal



















Benoit Mandelbrot: Fractals and the art of roughness


Ron Eglash on African Fractals (from TED.com)


Another vid (visit the link):
PBS NOVA - Hunting the Hidden Dimension

- Watch more Videos at Vodpod.

Types of Strange Attractors



Lorenz Attractor














http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/Lorenz/
http://local.wasp.uwa.edu.au/~pbourke/fractals/lorenz/lorenz.m4v
http://crossgroup.caltech.edu/chaos_new/Lorenz.html

Julia Attractor (really cool!)
http://www.openprocessing.org/visuals/?visualID=8058

Rossler Attractor














Tamari Attractor



















Henon Attractor













Also look up: flow, Poincare mapping, bifurcations, catastrophe theory, self-organization

Fluid dynamics has also found much of chaos theory to be of use, especially in studies of turbulence, which I still consider to be unsolved.

A Karmen Vortex Street













Learn More
http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/
http://www.hypertextbook.com/chaos/
http://www.stsci.edu/~lbradley/seminar/index.html
http://www.scholarpedia.org/article/Chaos
http://www.calresco.org/ - a link of links
http://math.rice.edu/~lanius/frac/
http://www.imho.com/grae/chaos/chaos.html
http://www.societyforchaostheory.org/
http://www.exploratorium.edu/complexity/CompLexicon.html
http://www.csuohio.edu/sciences/dept/physics_from_www/kaufman/yurkon/chaos.html
http://classes.yale.edu/fractals/
http://www.egregium.us/
http://www.theory.org/

About Fractals and Fractal Art Specifically:
http://www.chaospro.de/
http://local.wasp.uwa.edu.au/~pbourke/fractals/
http://content.techrepublic.com.com/2346-10877_11-56937-1.html?tag=content;leftCol
http://www.enchgallery.com/
http://mitpress.mit.edu/books/FLAOH/cbnhtml/slides.html

More Intricate Articles and Research:
http://physics.mercer.edu/petepag/combow.html
http://www.zfm.ethz.ch/~leine/toys.htm
http://vigo.ime.unicamp.br/2p/PendulaProject.html
http://www.math.bme.hu/~bnc/wada/
http://www.matternews.com/research/Researchers_move_closer_to_understanding_chaotic_motion_of_a_solid_body_in_a_fluid.asp
http://www.osti.gov/accomplishments/prigogine.html

College Web Pages
http://www.santafe.edu/
http://www.cscs.umich.edu/
http://wwwrsphysse.anu.edu.au/nonlinear/
http://www-chaos.umd.edu/

Complexity in Social Science
http://www.irit.fr/COSI/

Catastrophe Theory
http://en.wikipedia.org/wiki/Catastrophe_theory
http://l.d.v.dujardin.pagesperso-orange.fr/ct/eng_index.html

Most Links I have found on this subject are incorrect, misleading, incorrect interpretations, disconnected, or spiritual. I utilized Lorenz's 'The Essence of Chaos', Gleick's 'Chaos: Making a New Science', and Schroeder's 'Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise' as references for my own writings. I didn't intend to copy anything directly, but I apologize if I did. Some of it may be due to the fact that sometimes things can't be said precisely without copying. I don't claim to have come up with any of the actual ideas expressed in either part of this post. There was just so much cool stuff out there on this subject I could hardly get myself to finish this article! Please check out the links and explore this more. I think chaos theory will yield very interesting physics research in years to come. I took many videos and pictures from all over the web. If I am violating any copyright laws or anything of the sort, please let me know and I will remove whatever is necessary. Also, if anybody finds incorrect statements with particular things which have been said, please let me know as I have tried very hard to express these concepts accurately.

Chaos Theory (Part 1)

The main idea of chaotic systems is that they appear very complex and seemingly random, but are determined by precise laws. The problem with chaotic systems is that they are extremely sensitive to very slight changes. For all practical purposes, measurement becomes an impossible means of ascertaining a representation of the future of a system. The generality of chaotic systems will be elaborated upon more accurately below, but it may be useful to keep some real-life examples in mind such as the falling of a leaf from a tree, motion of a ping pong ball floating down a stream, cream mixing with coffee while stirring, or stock market fluctuations.

A deterministic system is a system in which a finite set of fixed laws governs the transition of one state of the system to another, a state being one instance of the entire system with all dynamic variables specified (e.g. positions and momenta). The laws determine precisely how the system evolves in time or throughout space. In other words, given the laws and the initial state (or any specified state) of the system, all other states of the system could be correctly determined. The problem with such a system would still be in measuring precisely the variables.

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”
~Pierre-Simon Laplace

The alternative to a deterministic system is an indeterministic system. An indeterministic system is a system in which there is no finite number of laws which precisely specifies the time evolution of a system. Another way of saying this is that the states of a system are disconnected, in that there is no true dependence of any state on another given state.

A deterministic system can also be called a dynamical system. However, sometimes a dynamical system is considered a more ‘real’ version in which determinism is just a very close approximation. Dynamical systems can have randomness as long as the randomness does not appreciably change the system’s behavior (i.e. the system would appear almost exactly the same if that randomness could be removed somehow).

A dynamical system can be either linear or nonlinear. A linear system is one in which the slight change of a system’s state will only cause changes which are proportional to that change. In a nonlinear system, changing the initial state does not create proportional alterations of the later states. Within a very short duration of time, the system may be unrecognizable from what it would have been if the initial condition wasn’t changed. Take note that when I speak of the initial conditions being changed; a more practical interpretation is that the experimenter imperfectly measured the initial conditions (the change being the difference between the measurement and reality).

Given perfect measurements of a system’s state, the later state of the system can be calculated perfectly for both linear and nonlinear systems. In reality, however, we are rarely afforded with this benefit. Let us say that we are 0.0000000001 meters off in measuring the initial state of a particle in our system. In a linear system this would not make a huge difference. Measuring a later state of the particle would yield a result that is not much more than 0.0000000001 meters off of where the particle would really be.

The problem arises with nonlinear systems. Take the same example as before, where we imperfectly measured our particle’s position by 0.0000000001 meters. Unlike a linear system, it is very likely that in a nonlinear system we could still figure out where the particle will be at some point in the future. The actual future state would be far different from the future state which we calculated.

Another way of making this distinction is that accurate and precise measurements are exponentially more important when dealing with nonlinear systems.

A finite number of variables specifies the state of each particle in the system. A useful way to represent a system is with a phase space. A phase space maps all possible states of a system's motion. A phase space basically represents all possible states a system can take within an n-dimensional grid. A phase space has the same number of dimensions as variables necessary for specifying the system’s state (# particles X number of variables to specify state of each particle). Basically, every degree of freedom is an axis of the entire grid. The portion which is 'colored in' is the phase space. The unshaded parts represent combinations of the variables which the system cannot be in. One point in the n-dimensional phase space is specified by n coordinates (one for each variable).

Here's a picture of a particular system (the Van der Pol oscillator) represented as a 2-D phase space:














(3, -2) is a state of the system which never naturally occurs. (3, 1), however, looks as though it is a state which the system can take. Order of the coordinates does matter. I've taken this ordering to be (x-axis, y-axis). If you still don't understand phase space completely, don't worry. Pictures of attractors in Part 2 will help (plus the video at the bottom of this site and the wikipedia description).

An orbit is a phase-space representation of a sequence of states in time. There are two types of orbits. A periodic orbit is one that repeats the past behavior after a certain fixed period of time (its period). An aperiodic orbit is an orbit where the close repetition of a previous state does not remain close for any appreciable duration of time. It is a nonlinear in that sense. The future cannot be precisely determined if the initial measurement is not 100% correct.

Aperiodic orbits exhibit what is known in chaos theory as sensitive dependence. This can be pictured as saying that small changes in the state of the system will cause very large changes in the overall time evolution of the system. Another way of looking at it is that there are many other orbits which approach the aperiodic orbit, but almost none of them will remain nearby as time progresses. These other orbits can be thought of as the theoretical prediction of motion of the imperfect measured particle (the imperfect measurement being at the point where it approaches the aperiodic orbit). The butterfly effect is the original name given to sensitive dependence by Edward Lorenz.

“Predictability: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?”
~ Edward Lorenz

Chaos theory, then, is the study of nonlinear dynamical (deterministic) systems or the study of systems with aperiodic orbits. The main point of all this is that non-perfect measurements of the state of a chaotic system can’t be used as accurate determiners of the future state of that system. A more practical way to figure out the future of a chaotic system is to just wait until that future moment is the present, and measure it. This ensures the highest amount of accuracy. A full chaotic system is one in which most or all orbits the system can take are aperiodic. A limited chaotic system is one in which most orbits are periodic or nearly periodic, with a few special orbits being aperiodic.

A good documentary about chaos theory:
http://video.google.com/videoplay?docid=-7171768157657288194#

Pendula of all sorts seems to exhibit chaotic behavior:






A cool simulation:
http://www.myphysicslab.com/pendulum2.html