Math and Nature

General discussions of forests and trees that do not focus on a specific species or specific location.

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ryandallas
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Math and Nature

Post by ryandallas » Sun May 22, 2016 4:39 pm

The more I learn about math, the more I wish I had taken it seriously when I was in school. Because math is so good at describing the natural world. (Mathematical Platonists even believe that mathematical structures are real things, not just human constructs. I find myself inching towards that conclusion.)

Here's a neat language call the l-system, which a botanist developed in the 60s. It's basically more logic than math, but you can insert mathematics into the language.

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

And here's a mountain fractal:

http://www.mactech.com/articles/mactech ... index.html

--Ryan

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Don
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Re: Math and Nature

Post by Don » Sun May 22, 2016 10:43 pm

Ryan-
And how about those Fibonacci Sequences!?
http://jwilson.coe.uga.edu/emat6680/par ... nature.htm
Don Bertolette - President/Moderator, WNTS BBS
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Lucas
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Re: Math and Nature

Post by Lucas » Tue May 24, 2016 12:32 pm


Click on image to see its original size

A cheap one dimensional laugh but I still laughed.
We travel the Milky way together, trees and men. - John Muir

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Lucas
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Re: Math and Nature

Post by Lucas » Tue May 24, 2016 12:33 pm

http://www.theatlantic.com/science/arch ... ts/462150/

All this suggests that the Babylonians had a mathematics worth copying, which was why the Greeks did copy it and thereby rooted these number systems in Western tradition. The latest indication of Babylonian mathematical sophistication is the discovery that their astronomers knew that, in effect, the distance traveled by a moving object is equal to the area under the graph of velocity plotted against time. Previously it had been thought that this relationship wasn’t recognized until the fourteenth century in Europe. But since historian Mathieu Ossendrijver of the Humboldt University in Berlin found the calculation described in a series of clay tablets inscribed with cuneiform writing in Babylonia during the fourth to the first centuries B.C.E., where it was used to figure out the distance traveled across the sky by the planet Jupiter.


http://www.theatlantic.com/magazine/arc ... on/426855/

“For American students who have an appetite to learn math at a high level, something very big is happening.”
We travel the Milky way together, trees and men. - John Muir

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Lucas
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Re: Math and Nature

Post by Lucas » Tue May 24, 2016 12:36 pm

We travel the Milky way together, trees and men. - John Muir

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Lucas
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Re: Math and Nature

Post by Lucas » Tue May 24, 2016 12:40 pm

I sucked at math and physics in school.

You might know, that now, I regret it, big time. Some of the most amazing ideas are revealed by it, like is the whole universe, a big math equation or, is this 3D universe, the event horizon around a black hole in a 4D universe?

Comes in handy for tree measuring, too.
We travel the Milky way together, trees and men. - John Muir

Joe

Re: Math and Nature

Post by Joe » Tue May 24, 2016 1:48 pm

I suggest reality is what It is- our theories and models of it are a pale representation of it. It almost certainly is infinitely big, infinitely old, infinitely complex- not to mention that 94% of it is dark matter and dark energy- which will be stuff to figure out for future generations.

But, metaphysically, to me- it's all a dream in Buda's mind- one of an infinite number of dreams.

Or, as the British American Zen philosopher, Alan Watts, once named one of his books, which blew me away back in the '60s- the title of which is, "This is It".
Joe

ryandallas
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Re: Math and Nature

Post by ryandallas » Thu May 26, 2016 3:28 pm

"Ryan-
And how about those Fibonacci Sequences!?
http://jwilson.coe.uga.edu/emat6680/par ... nature.htm"

Very nice! I like that more than computational methods.

http://www.theatlantic.com/science/arch ... ts/462150/

"All this suggests that the Babylonians had a mathematics worth copying, which was why the Greeks did copy it and thereby rooted these number systems in Western tradition. The latest indication of Babylonian mathematical sophistication is the discovery that their astronomers knew that, in effect, the distance traveled by a moving object is equal to the area under the graph of velocity plotted against time. Previously it had been thought that this relationship wasn’t recognized until the fourteenth century in Europe. But since historian Mathieu Ossendrijver of the Humboldt University in Berlin found the calculation described in a series of clay tablets inscribed with cuneiform writing in Babylonia during the fourth to the first centuries B.C.E., where it was used to figure out the distance traveled across the sky by the planet Jupiter."

Very cool! Also, Archimedes proposed an inchoate but ingenious derivative called "exhaustion".

"You might know, that now, I regret it, big time. Some of the most amazing ideas are revealed by it, like is the whole universe, a big math equation or, is this 3D universe, the event horizon around a black hole in a 4D universe?

Comes in handy for tree measuring, too."

I was decent, but I never excelled. But I'm thinking of taking math classes at a community college.

"I suggest reality is what It is- our theories and models of it are a pale representation of it."

I think that was Bertrand Russell's view--that mathematical laws are purely linguistic, nothing but tautologies. I used to embrace that view, but now I reject it. Imagine I threw a ball. Should I say, "We can *describe* the ball's trajectory as having been parabolic"? Or should I say, "The ball's trajectory *was* parabolic"? I endorse the latter statement. Of course, philosophers have been debating this for centuries.

"It almost certainly is infinitely big, infinitely old, infinitely complex- not to mention that 94% of it is dark matter and dark energy- which will be stuff to figure out for future generations."

And there may be a near-infinite number of multiverses, too.

"But, metaphysically, to me- it's all a dream in Buda's mind- one of an infinite number of dreams. Or, as the British American Zen philosopher, Alan Watts, once named one of his books, which blew me away back in the '60s- the title of which is, "This is It".

Interesting! So you're an Idealist?

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Lucas
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Re: Math and Nature

Post by Lucas » Mon May 30, 2016 9:19 am

A small mystery with big implications? L

https://www.sciencenews.org/article/mat ... ext=191681

Mathematicians find a peculiar pattern in primes

Final digit in consecutive numbers is not as random as expected

BY RACHEL EHRENBERG 3:58PM, MARCH 18, 2016
prime numbers

PRIME TIME Prime numbers are loathe to repeat the final digit of the prime that precedes them, a bias that has mathematicians scratching their heads.

CHRIS/FLICKR (CC BY-NC-SA 2.0)

Magazine issue: Vol. 189, No. 8, April 16, 2016, p. 7

Prime numbers, divisible only by 1 and themselves, hate to repeat themselves. They prefer not to mimic the final digit of the preceding prime, mathematicians have discovered.

“It is really, really bizarre,” says Stanford University postdoctoral researcher Robert Lemke Oliver, who, with Stanford number theorist Kannan Soundararajan, discovered this unusual prime predilection. “We are still trying to understand what is at the heart of this,” Lemke Oliver says.

Generally speaking, primes are thought to behave much like random numbers. So whenever some kind of order is revealed, it gives mathematicians pause.

“Any regularity you can show about primes is beguiling, because there may lurk there some new structure,” says number theorist Barry Mazur of Harvard University. “Revealing some bit of architecture where we thought there was none may lead to inroads into the structure of the mathematics.”

Once primes get into the double digits, they must end in either a 1, 3, 7 or 9. Mathematicians have long known that there are roughly the same number of primes ending with each digit; each appears as the final number about 25 percent of the time. The prime number theorem in arithmetic progressions proved this distribution about 100 years ago, and the still unsolved Riemann hypothesis predicts that the rates rapidly approach 25 percent. This property has been tested for many millions of primes, says Soundararajan.

And without any reason to think otherwise, mathematicians just assumed that the distribution of those final digits was basically random. So given a prime that ends in 1, the odds that the next prime ends in 1, 3, 7 or 9 should be roughly equal.

“If there’s no interaction between primes, that’s what you would expect,” says Soundararajan. “But in fact, something funny happens.”

Do not repeat
Mathematicians have long assumed that the final digits of consecutive prime numbers are distributed randomly. So for 100 million primes, a prime ending in 1 should be followed by a prime ending in 1, 3, 7, or 9 roughly 6.25 million times per digit (dotted lines in graphs). But in reality, the final digit of the prime is biased against repeating the final digit of the prime that came before it.


SOURCE: R.J. LEMKE OLIVER AND K. SOUNDARARAJAN/ARXIV.ORG 2016; CREDIT: E. OTWELL
Despite each final digit appearing roughly the same amount of the time, there’s a bias in the order in which these final digits appear. A prime that ends in 7, for example, is far less likely to be followed by a prime that also ends in 7 than a prime that ends in 9, 3 or 1.

The discovery of the final digit bias has “no conceivable practical use,” says Andrew Granville, a number theorist at the University of Montreal and University College London. “The point is the wonder of the discovery.”

The peculiar pattern had been noted previously by two separate teams of researchers, but the Stanford duo is the first to articulate a mathematical explanation for the pattern, which they posted online March 11 at arXiv.org. When the researchers crunched the numbers, their predictions based on the hypotheses fit the results remarkably, says Granville, who calls the work “rigorous, refined and delicate.”

You might think this “anti-sameness” bias follows naturally from the order of numbers. After all, 67 is followed by 71, which is followed by 73. But this explanation doesn’t fit the data, says Lemke Oliver, who ran computer calculations out to 400 billion primes.“The bias is way too large,” he says. What’s more, the bias isn’t equal for the nonrepeating final digits. So among the first hundred million primes, for example, a prime that ends in 3 is followed by a prime that ends in 9 about 7.5 million times, whereas it is followed by a prime that ends in 1 about 6 million times. A final 3 is followed by a final 3 a mere 4.4 million times.

Yet as the number of primes approaches infinity, the bias slowly disappears. This restoration of seeming randomness makes sense mathematically, but the slow rate at which the bias disappears is notable.

“It would almost be perverse if it didn’t even out,” says Lemke Oliver. “It would bother me a little.”
We travel the Milky way together, trees and men. - John Muir

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Lucas
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Re: Math and Nature

Post by Lucas » Mon May 30, 2016 9:21 am

BTW trees are fractal. A reiterating pattern, trunk to limb to twig.
We travel the Milky way together, trees and men. - John Muir

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