# Sculpting of a fractal river basin

In mathematicsa fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set. Fractal geometry lies within the mathematical branch of measure theory.

One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two the ratio of the new to the old side length raised to the power of two the dimension of the space the polygon resides in. Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two the ratio of the new to the old radius to the power of three the dimension that the sphere resides in.

However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer.

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Analytically, fractals are usually nowhere differentiable. Starting in the 17th century with notions of recursionfractals have moved through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard BolzanoBernhard Riemannand Karl Weierstrass[8] and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.

There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals. The consensus is that theoretical fractals are infinitely self-similar, iteratedand detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. The word "fractal" often has different connotations for the lay public as opposed to mathematicians, where the public is more likely to be familiar with fractal art than the mathematical concept.

The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background. The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure.

If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive e.

The difference for fractals is that the pattern reproduced must be detailed.

### Sculpting of a Fractal River Basin

This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived.

Now, consider the Koch curve. This number is what mathematicians call the fractal dimension of the Koch curve; it is certainly not what is conventionally perceived as the dimension of a curve this number is not even an integer!

The fact that the Koch curve has a fractal dimension differing from its conventionally understood dimension that is, its topological dimension is what makes it a fractal. This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable ".

In a concrete sense, this means fractals cannot be measured in traditional ways. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve.

The result is that one must need infinite tape to perfectly cover the entire curve, i. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way.

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Bytwo French mathematicians, Pierre Fatou and Gaston Juliathough working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors i.

Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered the Julia set, for instance, could only be visualized through a few iterations as very simple drawings.

In [11] Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations.

These images, such as of his canonical Mandelbrot setcaptured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented geometric shape that can be split into parts, each of which is at least approximately a reduced-size copy of the whole"; [1] this is generally helpful but limited.

Authors disagree on the exact definition of fractalbut most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in. One point agreed on is that fractal patterns are characterized by fractal dimensionsbut whereas these numbers quantify complexity i.British Wildlife is the leading natural history magazine in the UK, providing essential reading for both enthusiast and professional naturalists and wildlife conservationists.

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Fine Art Wood Sculpture, Wood Electrification (Lichtenberg fractal figure)

Fractal River Basins considers river basins and drainage networks in the light of their scaling and multi-scaling properties, and the dynamics responsible for their development.

The hydrology of river basins, and prediction of their growth, demands knowledge of a range of temporal and spatial scales. The core of Fractal River Basins is the search for the hidden order of these temporal and spatial variabilities in river basins, despite variations in size, climate and geology.

The commonality of branching networks to other natural phenomena will make this book applicable to a wide range of disciplines. Hydrologists and geomorphologists will find that this book opens up the important topic of the fractal structure of networks at an accessible level. Mathematicians and physicists will appreciate the application of the theory to this aspect of the earth sciences.

Comprehensive, well illustrated and with many real-world examples Fractal River Basinswill be useful to researchers and students alike. A view of river basins 2. Fractal characteristics of river basins 3. Multifractal characteristics of river basins 4. Optimal channel networks: minimum energy and fractal structures 5. Self-organized fractal river networks 6. On landscape self-organization 7.

Geomorphological hydrologic response 8. Montgomery, Nature "[ Cambridge University Press has done its usual excellent job of production.

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It may also be useful as a source of introductory material and examples for a rather specialized course on complex systems. English Deutsch.

A view of river basins 2. Fractal characteristics of river basins 3. Multifractal characteristics of river basins 4. Optimal channel networks: minimum energy and fractal structures 5. Self-organized fractal river networks 6. On landscape self-organization 7. Geomorphological hydrologic response 8.

View PDF. Save to Library. Create Alert. Launch Research Feed. Share This Paper. Cudennec, Y. Species survival and scaling laws in hostile and fluctuating environments. Rocha, Wagner Figueiredo, … A. Maritan Citation Type. Has PDF. Publication Type. More Filters. Highly Influenced. Research Feed.

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Network robustness assessed within a dual connectivity perspective. Potential-driven flows through self-similar trees.

DOI: Banavar and F. Colaiori and A. Flammini and A. Giacometti and A. Maritan and A. BanavarF. The principle of reparametrization invariance is used to derive a dynamical equation for the erosion of the landscape of the drainage basin of river networks.

The stationary solutions of the equation are found to have scaling behavior that is consistent with observational data. Our analytic prediction of the main stream profile is confirmed by numerical results and is amenable to direct observational verification. View PDF.

Save to Library. Create Alert. Launch Research Feed. Share This Paper. Dynamics of erosion-sedimentation in River networks. Chang Figures from this paper. Citation Type. Has PDF. Publication Type. More Filters. Research Feed. View 1 excerpt, cites background. Theoretical and Experimental Studies of Dendritic Metacommunities. View 1 excerpt, cites methods. Models of Fractal River Basins.

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Jayanth Banavar. Amos Maritan. Sculpting of a Fractal River Basin. The stationary solutions of the equation are found to have scaling behavior that is consistent with observational data.

Our analytic prediction of the main stream profile is confirmed by numerical results and is amenable to direct observational verification. Fb, Ak Drainage basins of rivers evolve into striking fractal quantitative explanation of the observed facts. We also forms as a result of erosional processes [1]. Soil height predict the scaling of the main stream profile that may be maps [2] of such self-organized landscapes have been deduced from observational data and would provide a test used to study scale-free algebraic distributions of several of our theory.

River basins around the The constant coefficient b can be set equal to 1 on world are found to have values of t, g, and h in the defining the time units appropriately. In the Monge range 1. The stationary solution of with 7 is obtained on setting the right-hand side equal to zero p and leads to Eq.

The second equation in 9 balances the uplift. The first of these nuity of the solution and exists for any 0m cyL. In two dimensions, the mainstream is topologically one dimensional with the key difference that a is no longer proportional to x but on the average to x 1yh where h is the Hack exponent and x is the upstream mainstream length. We over samples starting from different randomly chosen have confirmed that the scaling form holds extremely well initial conditions are plotted together with the analytical result in stationary solutions of the two dimensional erosion [Eq.

The value of h s0. A direct integration of the two dimensional equation df s1. Collapse of profiles along the mainstream correspond- FIG.The effects of erosion, avalanching and random precipitation are captured in a simple stochastic partial differential equation for modelling the evolution of river networks.

Our model leads to a self-organized structured landscape and to abstraction and piracy of the smaller tributaries as the evolution proceeds. An algebraic distribution of the average basin areas and a power law relationship between the drainage basin area and the river length are found. The physics of river network evolution arises from an interplay of the structured landscape governing the water flow with the erosional effects of the water feeding back into further sculpting of the landscape.

### Fractal River Basins: Chance and Self-Organization

Extensive studies of the fractal characteristics of real river networks have been carried out. Documents: Advanced Search Include Citations. Abstract The effects of erosion, avalanching and random precipitation are captured in a simple stochastic partial differential equation for modelling the evolution of river networks. Powered by:.West Ham 1-0 Chelsea: SIX THINGS YOU MISSED - Michy.

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