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Fractals - Patterns and Chaos

Posted: March 24th, 2015, 2:20 pm
by ExSan

Fractals - Patterns and Chaos

Posted: March 30th, 2015, 11:34 am
by ExSan

Fractals - Patterns and Chaos

Posted: April 1st, 2015, 11:24 pm
by ExSan
Strobe-Animated Artichoke?an Example of a Naturally Occurring "bloom"

Fractals - Patterns and Chaos

Posted: April 2nd, 2015, 10:11 pm
by ExSan
From OT: Random photo'sQuoteOriginally posted by: outrunQuoteOriginally posted by: tagomaQuoteOriginally posted by: outrunQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: outrunSee how your brain processes patterns by blinking at different speeds while looking at this GIF. MUST.... SEND..... OUTRUN..... MONEY!!!!!!!WESTERN UNIONMUST ... FORWARD ... MONEY ... TO ... GS ... ACCOUNT ...oh no! I nearly made it, and then "poof ..it's gone"ppauper will probably post again the tv cartoon showing a bank customer make some fast money lost money investment.That's one of my favorite South Park episodes!

Fractals - Patterns and Chaos

Posted: April 3rd, 2015, 9:12 am
by ExSan
QuoteOriginally posted by: outrunthe first thing my son said after blinking was "I only see 3 distinct patterns".. I think he's right!I can see circles, lines and there is also rotation, and shifting ?

Fractals - Patterns and Chaos

Posted: April 3rd, 2015, 11:53 am
by Traden4Alpha
QuoteOriginally posted by: ExSanQuoteOriginally posted by: outrunthe first thing my son said after blinking was "I only see 3 distinct patterns".. I think he's right!I can see circles, lines and there is also rotation, and shifting ?Hmmm... I think I see six. There is a 3-cycle with: one circle in the center; then two circles at an angle on the left side; then two circles at an angle on the right side. But the 3-cycle has two alternating variants : one in which the centers of circle(s) are dark and the outer ring has a light path and the second in which the center of the circle in light and the outer ring path is dark.

Fractals - Patterns and Chaos

Posted: April 10th, 2015, 12:19 pm
by ExSan
The Montecarlo MethodQuoteAll too often statements are made that are supported by ?the Monte Carlo Method?. Media pundits, financial consultants, and many others sight this mysterious prognosticator, this mathematical crystal ball as the source of the given information. They present the result as having been ?scientifically analysed? or ?mathematically predicted?. While those statements are not blatantly false, they are somewhat deceptive. The Monte Carlo Method is a statistical tool and as with all statistical tools, they can be misused or poorly interpreted.The concept originated with the applied mathematician Stanislaw Ulam. (Ulam and Edward Teller are credited with the design of the first thermonuclear weapon.) While working on the Manhattan Project, Ulam became ill. During his recovery he played Canfield solitaire. During his recovery, he began thinking about the probability that any given deal would result in a win. He devoted a substantial amount of time to manually calculate the probabilities and then wondered if there was not a better, faster method for obtaining the results. He determined that repeated trials should generate near accurate information. While the card example may appear trivial, Ulam determined that this method could be used to simulate tests at the Manhattan Project. This was a critical issue as there was a very limited supply of Uranium available and each test could prove expensive. So exactly what does the Monte Carlo Method actually do? As much as this may disappoint the media, it does not provide a specific answer to a problem; it provides probabilities of various outcomes. For example, it cannot tell you that the next roll of two dice will total ?7?. However, it does inform us that there is a 16.7% probability that ?7? will be rolled and, ?7? has the highest success rate among all of the possible outcomes.Basically there are a variety of inputs to a function and the results are recorded. As the number of runs increase, the number of times this action is repeated, the expected accuracy increases. This sum of the statistical trials can be examined and the probabilities of any particular event can be determined.Consider the roll of two dice again. We roll the dice 100 times and record the results. After 100 rolls, our test indicates ?7? appears 21% of the time. After 1000 trials, ?7? has dropped to 18% of the results. After 5000 trial runs, it rests at just under 17%. This is a simplified example but it should reveal the procedure. The Monte Carlo Method is (and has been) used in a variety of sciences, business plans, in analyzing possible sports strategies (should we tie with a field goal or go for the touchdown, 2 point shot for the tie or 3 point shot for the win, etc.). The most important concept to remember is that the method provides probabilities not assurances. In the long term ?7? will appear more often than any other roll but don?t bet it all on that very next roll - there is only a 16.7% chance that it will be a ?7?.

Fractals - Patterns and Chaos

Posted: April 10th, 2015, 9:01 pm
by ExSan
From Chemistry to Mathematics To build a chemical database, we must grow mathematical trees. Growing mathematical trees is just like growing biological trees: it starts from the root. Cayley?s formula Π_i=1~∞ {(1-x^i)^(-b_i)}, which is an infinite product of simple terms, is the magic used in generating tree structures. We collect terms of the same power and obtain the counts for trees of the same size. The collection process is simple: Trees are grouped by Young diagrams, which are ways of partitioning a natural number. (Tune in for another post.) Then, trees are de-rooted to obtain root-free trees. (Root-free trees are constructed by combining rooted trees through the use of dissimilarity characteristic theorem, an extended form of the Euler characteristic theorem.)

Fractals - Patterns and Chaos

Posted: April 12th, 2015, 9:24 am
by ExSan
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Fractals - Patterns and Chaos

Posted: April 13th, 2015, 12:09 pm
by ExSan
mandelbulb.com

Fractals - Patterns and Chaos

Posted: April 14th, 2015, 10:18 am
by ExSan

Fractals - Patterns and Chaos

Posted: April 20th, 2015, 11:01 am
by ExSan
QuoteOriginally posted by: outrunDeepest Mandelbrot zoom ever=this is a zoom in, great!what could we expect if we zoom out ?

Fractals - Patterns and Chaos

Posted: April 20th, 2015, 6:24 pm
by tagoma
Amazing!