Posts Tagged: Mathematics


26
Jan 11

The Secret Life of Chaos

A nice video on Chaos Theory. The video comes in 6 parts. All 6 parts are below.

part 1 of 6

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part 2 of 6

http://www.youtube.com/watch?v=wHu8iaLs9i4

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part 3 of 6

http://www.youtube.com/watch?v=Oj–pxcFUjg

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part 4 of 6

http://www.youtube.com/watch?v=AMbua0BGfFE

part 5 of 6

http://www.youtube.com/watch?v=wc0IYJm5-mE

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part 6 of 6

http://www.youtube.com/watch?v=-1x-7ZLKhjw


14
Feb 10

Cliodynamics

Interesting work by Peter Turchin called: Cliodynamics.

Seems to be somewhat along the same lines as Oswald Arnold Gottfried Spengler’s “Der Untergang des Abendslandes” (i.e., “The Decline of the West“) and Abū Zayd ‘Abdu r-Raḥman bin Muḥammad bin Khaldūn Al-Hadrami’s “Muqaddimah“.

What is cliodynamics?

Empires rise and fall, populations and economies boom and bust, world religions spread or wither… What are the mechanisms underlying such dynamical processes in history? Are there ‘laws of history’? We do not lack hypotheses to investigate – to take just one instance, more than two hundred explanations have been proposed for why the Roman Empire fell. But we still don’t know which of these hypotheses are plausible, and which should be rejected. More importantly, there is no consensus on what general mechanisms explain the collapse of historical empires. What is needed is a systematic application of the scientific method to history: verbal theories should be translated into mathematical models, precise predictions derived, and then rigorously tested on empirical material. In short, history needs to become an analytical, predictive science (see Arise cliodynamics).

Cliodynamics (from Clio, the muse of history, and dynamics, the study of temporally varying processes) is the new transdisciplinary area of research at the intersection of historical macrosociology, economic history/cliometrics, mathematical modeling of long-term social processes, and the construction and analysis of historical databases. Mathematical approaches – modeling historical processes with differential equations or agent-based simulations; sophisticated statistical approaches to data analysis – are a key ingredient in the cliodynamic research program (Why do we need mathematical history?). But ultimately the aim is to discover general principles that explain the functioning and dynamics of actual historical societies.

(Link)


12
Jan 10

Golden Ratio and Eye Scanning Speed

A theory about the or a reason people find the golden ratio beautiful….

The Egyptians supposedly used it to guide the construction the Pyramids. The architecture of ancient Athens is thought to have been based on it. Fictional Harvard symbologist Robert Langdon tried to unravel its mysteries in the novel The Da Vinci Code.

“It” is the golden ratio, a geometric proportion that has been theorized to be the most aesthetically pleasing to the eye and has been the root of countless mysteries over the centuries. Now, a Duke University engineer has found it to be a compelling springboard to unify vision, thought and movement under a single law of nature’s design.

Also know the divine proportion, the golden ratio describes a rectangle with a length roughly one and a half times its width. Many artists and architects have fashioned their works around this proportion. For example, the Parthenon in Athens and Leonardo da Vinci’s painting Mona Lisa are commonly cited examples of the ratio.

Adrian Bejan, professor of mechanical engineering at Duke’s Pratt School of Engineering, thinks he knows why the golden ratio pops up everywhere: the eyes scan an image the fastest when it is shaped as a golden-ratio rectangle.
[...]

(Emphasis mine.)

(Link)


9
Jan 10

The Golden Ratio and Quantum Mechanics

Researchers [...] have for the first time observed a nanoscale symmetry hidden in solid state matter. They have measured the signatures of a symmetry showing the same attributes as the golden ratio famous from art and architecture.
[...]
On the atomic scale particles do not behave as we know it in the macro-atomic world. New properties emerge which are the result of an effect known as the Heisenberg’s Uncertainty Principle. [...]
[...]
When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal pattern. [...] We have tuned the system exactly in order to turn it quantum critical.”

By tuning the system and artificially introducing more quantum uncertainty the researchers observed that the chain of atoms acts like a nanoscale guitar string. [...] “Here the tension comes from the interaction between spins causing them to magnetically resonate. For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture.” [...] “It reflects a beautiful property of the quantum system — a hidden symmetry. Actually quite a special one called E8 by mathematicians, and this is its first observation in a material,” [...]

[...] “Such discoveries are leading physicists to speculate that the quantum, atomic scale world may have its own underlying order. Similar surprises may await researchers in other materials in the quantum critical state.”

(Link)