Fractals Form Chance And Dimension Pdf

[Tarja Rabito]

Animportant factor in the analysis of deformation and failure of rock masses along discontinuities is the roughness of the jointing surfaces. The fractal dimension is proposed as a method of objectively quantifying the roughness profile of such discontinuities. With this method much less engineering judgement and experience are required in determining the surface roughness,sb than is required for other methods such as the widely used joint roughness coefficient (JRC) suggested by the International Society for Rock Mechanics [1] (Int. J. Rock. Mech. Min. Sci. & Geomech. Abstr. 15, 319-368, 1978). The fractal dimension, suggested by Mandelbrot [2] (Fractals-Form, Chance and Dimension, Freeman, San Francisco), describes the degree of variation of a curve, a surface or a volume has from its topological ideal. In this application, the length of the surface profile is measured stepwise along the curve with rulers of different lengths. From these data, the fractal dimension of the surface profile can be determined. An empirical equation was found relating the fractal dimension to the JRC value. Application of the fractal dimension to three independent studies (field and laboratory) all showed excellent agreement between the surface roughness and the fractal dimension, the rougher the surface, the higher the fractal dimension. The peak shear stress on large-block laboratory direct shear tests increased as the fractal dimension of the shear surface increased, which again verifies the usefulness of the concept. 1990.

An illustration of how deeply the concept of fractals has entered into popular culture is that it makes an appearance in the song "Let it go" from the film Frozen, in the line "My soul is spiraling in frozen fractals all around". So Helen was trying to sing the word at two! (It may be some time before she's up to understanding Hausdorff dimension, but self-similarity in natural phenomena is something she might be able to appreciate now, or at least in a few years.)

Thinking of this, it occurs to me that I'm actually part of the very first "fractal generation", among the first to grow up with the concept. The term fractal was coined by Mandelbrot in a 1975 book in French, translated in 1977 as Fractals: Form, Chance and Dimension. My father had a copy of this, presumably bought sometime before it was updated in 1982 as The Fractal Geometry of Nature (my review), and I read it in my early teens. It is a relatively non-mathematical presentation, making the argument that some previously obscure mathematics had a unity of its own and was a fundamental tool for understanding and representing natural phenomena and not just a source of pathological examples. I had spent a lot of time trying to create maps with convincing coastlines and rivers, for Dungeons and Dragons worlds, so the discussion of coastlines resonated. And the computer we had (an early IBM-cline running a DOS-clone) was capable of generating Mandelbrot sets and we spent some time with that. So I internalised Mandelbrot's message and concepts of fractal structure have been one of my tools for thinking about the world ever since.

I never really followed up on the mathematics, however. At some point I borrowed Falconer's The Geometry of Fractal Sets (1985) from the mathematics library at Sydney Uni (or possibly had my father borrow it for me when I was still at school), but that was too dense for me at the time, launching straight into measure theory. And I looked at nothing in the twenty-five years since, until I was inspired by reading Berger's Geometry Revealed. I now have copies of The Fractal Geometry of Nature (since my copy of Form, Chance and Dimension is in storage in Sydney), Peitgen et al's Chaos and Fractals (my review), Falconer's Fractal Geometry: Foundations and Applications (my review), and Kirillov's A Tale of Two Fractals, and am working my way through them steadily.

My intellectual peregrinations, my reading and seminar-attending and so forth, have a fractal-like structure, both in depth, with local points of relatively focus and broad swathes of shallow survey, and temporally, with some topics bubbling into focus briefly and others retaining my interest on longer time-frames.

In this final interview shot by filmmaker Erol Morris, Mandelbrot shares his love for mathematics and how it led him to his wondrous discovery of fractals. His work lives on today in many innovations in science, design, telecommunications, medicine, renewable energy, film (special effects), gaming (computer graphics) and more.

Fractal geometry is being used in the biological sciences to accurately model the human lung, heartbeats and blood vessels, neurological systems and countless other physiological processes. Doctors and researchers are now using the mathematics behind fractal geometry to build models that they hope will identify microscopic patterns of diseases and abnormalities earlier than ever before.

Scientists have recently shown that the distribution of large branches to smaller branches in a single tree exactly replicates the distribution of large trees to smaller trees in an entire forest. Research is currently underway to use this information to measure how much carbon dioxide a single forest is capable of processing. From there, scientists will be able to apply their findings to every forest on earth, quantifying how much carbon dioxide the entire world can safely absorb.

Today, we have merely scratched the surface of what fractal geometry can teach us. Weather patterns, stock market price variations and galaxy clusters have all proven to be fractal in nature, but what will we do with this insight? Where will the rabbit hole take us? The possibilities, like the Mandelbrot set, are infinite.

Benoit Mandelbrot was an intellectual jack-of-all-trades. While he will always be known for his discovery of fractal geometry, Mandelbrot should also be recognised for bridging the gap between art and mathematics, and showing that these two worlds are not mutually exclusive. His creative approach to complex problem solving has inspired peers, colleagues and students alike, and instilled in IBM a strong belief in the power of perspective.

NARRATOR: You can find it in the rain forest, on thefrontiers of medical research, in the movies, and it's all over the world ofwireless communications. One of nature's biggest design secrets has finally beenrevealed.

Well,it might surprise a lot of people that ExxonMobil would be interested inlithium ion battery technology applied to hybrid electric vehicles. Our newbattery separator film is a true breakthrough that's going to enable thedeployment of more hybrid vehicles, faster. This means a tremendous reductionin greenhouse gases, the equivalent of removing millions of cars from the road.I think this is the most important project that I've worked on in my career.

LORENCARPENTER: I would get the data from themand make pictures from various angles, but I wanted to be able to put amountain behind it, because every Boeing publicity photo in existence has amountain behind it. But there was no way to do mountains. Mountains hadmillions and millions of little triangles or polygons or whatever you want tocall it, and we had enough trouble with a hundred. Especially in those dayswhen our machines were slower than the ones you have in your watch.

LORENCARPENTER: In 1978, I ran into this bookin a bookstore: Fractals: Form, Chance and Dimension, by BenoitMandelbrot, and it has to do with the fractal geometry of nature. So I boughtthe book and took it home and read it, cover to cover, every last little word,including the footnotes and references, twice.

NARRATOR: In his book, Mandelbrot said that many forms innature can be described mathematically as fractals: a word he invented todefine shapes that look jagged and broken. He said that you can create afractal by taking a smooth-looking shape and breaking it into pieces, over andover again.

Themethod is dead simple. You start with a landscape made out of very roughtriangles, big ones. And then for each triangle, break it into, into fourtriangles. And then do that again, and then again and again and again.

LORENCARPENTER: The pictures were stunning. Theywere just totally stunning. No one has had ever seen anything like this. And I just opened awhole new door to a new world of making pictures. And it got the computergraphics community excited about fractals, because, suddenly, they were easy todo. And so people started doing them all over the place.

KEITHDEVLIN: The key tofractal geometry, and the thing that evaded anyone until, really, Mandelbrotsort of said, "This is the way to look at things, is that if you look on thesurface, you see complexity, and it looks very non-mathematical." WhatMandelbrot said was that..."think not of what you see, but what it took toproduce what you see."

BRIANENQUIST (University of Arizona): If we look at each of the nodes, the branchingnodes of this tree, what you'll actually see is that the pattern of branching isvery similar throughout the tree. As we go from the base of the tree to higherup, you'll see we have mother branches then branching then into daughterbranches.

Ifwe take this one branch and node and then go up to a higher branch or node,what we'll actually find is, again, that the pattern of branching is similar. Again,this pattern of branching is repeated throughout the tree, all the way,ultimately, out to the tips where the leaves are.

NARRATOR: You see self-similarity in everything from a stalkof broccoli, to the surface of the moon, to the arteries that transport bloodthrough our bodies. But Mandelbrot's fascination with these irregular-lookingshapes put him squarely at odds with centuries of mathematical tradition.

KEITHDEVLIN: Mandelbrotcame along and said, "Hey, guys, all you need to do is look at these patternsof nature in the right way, and you can apply mathematics. There is an orderbeneath the seeming chaos. You can write down formulas that describe clouds andflowers and plants. It's just that they're different kinds of formulas, andthey give you a different kind of geometry."

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