Previously we wrote about randomly generated attractors. That time we have used Wolfram CDF technology to power the interactive applet. This technology has a serious drawback that you have to have installed specific additional software to be able to use it. As of now we have replaced the old app with HTML5-based interactive applet. This applet can be run on almost any modern web browser without a need to have any additional software preinstalled.
Main topic: Fractals
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line…” were the words using which B. Mandelbrot attempted to lure his readers into the fractal geometry of nature. This of course is an obvious truth, but the message behind it is truly fascinating. So what is fractal and what significance does it have?
Word fractal originated from latin word fractus (en. “composed of pieces”). Thus fractal ought to be composed of smaller parts. And thus the most fascinating property of fractals lies within it, as each smaller part fully or partially resembles every other smaller and larger parts. Mathematicians call it self-similarity and it’s inherent property of any fractal.
Self-similarity of fractals seems to be appealing to both – scientists and people far from science. Former group are interested in fractals due to purely scientific reasons – fractals give philosophical insight into the underlying rules of nature. Some of philosophical ideas resemble ideology behind the cellular automata, while some relate to the ideas of non-linearity within dynamical chaos. Mesmerizing beauty of fractals also attracts ordinary people, some of who see fractals as a form of art.
Thus in this section we will attempt mesmerize you with the beauty, both visual and philosophical, and the variety of fractals.
In classical physics differential equations is the main tool to mathematically describe dynamical systems. Having obtained the mathematical descriptions of the system we should be able to predict the evolution of the system. It is noted that the evolution of the classical systems is pretty trivial – no matter what the initial condition is the system will “find” the stable state. Usually dissipative forces (such as frictions) are to be blamed for this. Though not all systems are so simple… Continue reading “Randomly generated strange attractors” »
Recent hurricane, which struck east coast of the USA, has very interesting symmetry properties. This natural phenomenon obeys the golden ratio! Well at least such information has been circulating on the science.memebase.com! Similar properties are also observed in some fractals such as Penrose tiling (we have not yet discussed this fractal on Physics of Risk, thus we’d like to recommend reading an article on the Wikipedia). Continue reading “Hurricane Sandy” »
Previously (ex. while speaking about the multifractality of time series) we have already discussed that fractals are observed in many daily phenomena. Interestingly enough we can eat fractals for our breakfast! Recently we have found an article 1 which studies fractal structures, 1/f-like noise in ham! Continue reading “Fractals in pork!” »
Recently the largest Lithuanian anime community has launched its 2011 anime awards. In context of Physics of Risk I have found one very interesting nominee – Fractale. It is nominated as the best adventure and best science fiction anime of the year, though so far it is far behind the leaders.
First thing each viewer see is memorable opening sequence, which is rich of strange patterns and fractals. The view is nice, interesting and very sophisticated. Continue reading “Aleksejus Kononovicius: Fractale – anime on fractals” »
One of the conclusions of fractal geometry is a fact that fractals unlike traditional Euclidean shapes lack characteristic scale. Those “fractured” objects are self-similar – defining geometry is clearly visible on multitude of scales. It is known that self-similarity is observed not only in formally defined geometric objects, such as Sierpinsky triangle or Koch snowflake, but also in the surrounding nature. One of my most favorite examples is a comparison of tree, its branches and a leaf (for more inspiring examples see introduction of Fractals section) – they all have branching structure and something green filling the extra space in between.
The interesting thing, in context of the topic in focus, is that one can extend fractal formalism beyond formal or natural geometric shapes. It is also noticed that some of the natural processes exhibit fractal features in their time series! It is known that geoelectrical processes 1, heartbeat 2 and even human gait 3 time series posses this feature. While financial market, frequently analyzed on Physics of Risk website, time series are also no exception 4, 5. Though the aforementioned time series are much more complex – they exhibit not monofractality (single manner self-similar behavior as the aforementioned formal geometric fractals do), but multifractality! Continue reading “Multifractality of time series” »
Newton-Raphson, sometimes just Newton or Newton-Fourier, method is an approximate method in mathematical analysis for finding local roots of very complex functions (such as polynomials with large powers). Recall that root of the function is defined as a solution of . The essence of this method is to linearize function at the guessing point. The point where linearized function passes the abscissa axis is assumed to be a more precise estimate of the actual root. Continue reading “Newton-Raphson method” »