Benoit Mandelbrot

The following sections are included:

• Contents
• Contributors
• Preface by Michael Frame
• Preface by Nathan Cohen

I remember Benoit (Fig. 1.1) always sitting at his desk (Fig. 1.2), years of sitting for many long hours, writing, crossing out, rewriting, cutting, pasting. While writing, he would often check one of his many dictionaries. It was important for him to use words with their exact meanings…

Benoit and I had one discussion about mortality (well, two, but the first, soon after we met, was humorous), though we never discussed religion. So why do I start my chapter with a Genesis quote? I want to try to sketch a picture of the brilliant, complex, and in an important way, very playful person who, for reasons I'll never understand, included me in his world, and who was for twenty years — a long time, but not nearly long enough — a dear friend. But about the Genesis quote I'll say this: the reference is to Rölvaag's novel Giants in the Earth, about pioneers in the Dakotas. I believe Benoit is better described as “pioneer” than as “explorer.” Yes, he explored many new ideas, clearly saw vistas of whole worlds unnoticed by others. But he did not simply mark the territory and move on. In each new area he stayed a while, looked carefully all over the place. He built things, surveyed the land for hordes of others, including the contributors to this volume. So pioneer it is…

B. Mandelbrot gave a new birth to the notions of scale invariance, self-similarity and non-integer dimensions, gathering them as the founding corner-stones used to build up fractal geometry. The first purpose of the present contribution is to review and relate together these key notions, explore their interplay and show that they are different facets of a single intuition. Second, we will explain how these notions lead to the derivation of the mathematical tools underlying multifractal analysis. Third, we will reformulate these theoretical tools into a wavelet framework, hence enabling their better theoretical understanding as well as their efficient practical implementation. B. Mandelbrot used his concept of fractal geometry to analyze real-world applications of very different natures. As a tribute to his work, applications of various origins, and where multifractal analysis proved fruitful, are revisited to illustrate the theoretical developments proposed here.

We provide a simple introduction to fractal transformations and fractal homeomorphisms. We introduce tri-affine iterated function systems on ℝ³, and illustrate associated fractal homeomorphisms applied to three dimensional graphical data sets. We also comment on the algorithms used.

To embrace Mandelbrot's whole work would necessitate knowledge and skills in many scientific fields. This is why this article deals only with the topics we are the most acquainted with: multiplicative processes and multifractal analysis. Nevertheless, to isolate them is somewhat artificial because in Mandelbrot's eyes all his research topics were closely related and the parts of the same design…

The loss of Benoit Mandelbrot has left us without one of the great scientific visionaries, not only of the twentieth century but of the entire history of science. His revolutionary interpretation of the world situates him at the height of genius like Archimedes, Kepler, Galileo, Newton or Einstein, geniuses who, armed only with a privileged imagination could discover what no one had seen before…

Benoit Mandelbrot always had a strong feeling that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this perspective? Here we discuss precisely what it means to say that a piece of music is fractal.

Self-similarity and fractals have opened new and important avenues for antenna and electronic solutions over the last 25 years. This primer provides an introduction to the benefits provided by fractal geometry in antennas, resonators, and related structures. Such benefits include, among many, wider bandwidths, smaller sizes, part-less electronic components, and better performance. Fractals also provide a new generation of optimized design tools, first used successfully in antennas but applicable in a general fashion.

A wideband invisibility cloak (IC) at microwave frequencies is described. Using fractal resonators in closely spaced (sub wavelength) arrays as a minimal number of cylindrical layers (rings), the IC demonstrates that it is physically possible to attain a ‘see through’ cloaking device with: (a) wideband coverage; (b) simple and attainable fabrication; (c) high fidelity emulation of the free path; (d) minimal side scattering; (d) a near absence of shadowing in the scattering. Although not a practical device, this fractal-enabled technology demonstrator opens up new opportunities for diverted-image (DI) technology and use of fractals in wideband optical, infrared, and microwave applications.

Fractional Brownian motion, introduced by Benoit Mandelbrot and John Van Ness in 1968, has had a major impact on stochastic processes and their applications. We survey a few of the many developments that have stemmed from their ideas. In particular we discuss the local structure of fractional and multifractional Brownian, stable and multistable processes, emphasising the ‘diagonal’ construction of such processes. In all this, the ubiquity and centrality of fractional Brownian motion is striking.

My first encounter with Benoit Mandelbrot was, in many ways, typical: He came to tell me I didn't know what I was talking about. And that was his message, for much of his life, to much of the financial world. It is a pity he wasn't heeded more quickly…

For a Borel measure and a sequence of partitions on the unit interval, we define a multifractal spectrum based on coarse Hölder regularity. Specifically, the coarse Hölder regularity values attained by a given measure and with respect to a sequence of partitions generate a sequence of lengths (or rather, scales) which in turn define certain Dirichlet series, called the partition zeta functions. The abscissae of convergence of these functions define a multifractal spectrum whose concave envelope is the (geometric) Hausdorff multifractal spectrum which follows from a certain type of Moran construction. We discuss at some length the important special case of self-similar measures associated with weighted iterated function systems and, in particular, certain multinomial measures. Moreover, our multifractal spectrum is shown to extend to a tapestry of complex dimensions for a specific case of atomic measures.

Back in the early 1990s I was researching topics for possible science TV documentaries and reading Ian Stewart's book Does God Play Dice? [1] I came across the chapter he called The Gingerbread Man. It was about the Mandelbrot set. Reading that chapter I could see that here was the perfect subject for a film. The Mandelbrot set is a fractal (Fig. 13.1) and fractal geometry is the geometry of nature (Figs. 13.2 to 13.4). So here I had the stunning visual beauty of the set itself and the natural world to point my camera at. It was a science story I could tell in pictures. It wasn't the Big Bang or gravity or quantum mechanics, it was something you could actually see and which could be filmed directly without recourse to endless and expensive CGI…

Demetri Maritn prepared this palindromic poem as his project for Michael Frame's fractal geometry class at Yale. Notice the first, fourth, and seventh words in the second and next-to-second lines are palindromes, the first two and last two lines are palindromes, the middle line, “Be still if I fill its ebb” minus its last letter is a palindrome, and the entire poem is a palindrome…

The following sections are included:

• Introduction
• Reading time series
• Four bin multifractals
• Binomial multifractals
• Finding the f(α) curvature from time series
• Building time series with prescribed memory
• Transition matrix variations under moving bin boundaries
• Interpreting the curvature of the f(α) curves
• Curvature surfaces of the data
• Interpreting the curvature plots
• Conclusion
• Appendix
• Bibliography

The following sections are included:

• A natural teacher
• The fractal geometry workshops
• First meetings
• The NSF grant
• The website
• The lab manual
• The software
• The lesson plans
• The DVD
• Satellite programs
• Conclusion
• Bibliography

Throughout my time studying and practising architecture, I have been deeply interested in understanding aesthetics from a scientific perspective. In university, I had read The Fractal Geometry of Nature and was drawn in by the visual basis of Mandelbrot's mathematics. At the time, I felt that there was some deep connection between fractal geometry and architectural aesthetics…

The paper gives probabilists who are unfamiliar with the Ising model a coherent, integrated explanation of why the Ising model displays long-range dependence at critical temperature. The Ising model in two dimensions involves spins σ_j,k = ±1 located at every node (j,k) of the lattice, with nearest neighbor interactions. We shall focus on the covariances and between the spin at the origin and the spin located either on the axis at (0,N) or located on the diagonal at (N,N), when the temperature equals a critical value. Using a recent formulation of the so-called ‘Szegö's theorem’, we explain why these covariances decrease to zero like N^−1/4 as N → ∞, thus at a slow enough rate so as to exhibit long-range dependence.

Chance plays a role in the affairs of men. The importance of chance was Mandelbrot's theme. These comments, delivered at Yale's memorial meeting for Benoit, show some of the uncertainties surrounding his key publications. It is Benoit we are celebrating, and I am sure that had he been a classicist or philologist a similar celebration would be being held, with other participants, but with similar stories and tributes. At this memorial, Ralph Gomory spoke of Benoit's courage in pursuing his insights. His enthusiasm and good will were important as well. How did his seminal books come to be published? Here is the story…

Benoit Mandelbrot had a profound influence in the physics community by connecting scaling and geometry. One example is the Lévy flight. In this memorial issue, in honor of Benoit, I review some personal history and ideas on the mathematics of Lévy flights and then introduce Lévy drives for applications to physical problems.

The following sections are included:

• Introduction
• The Noah and Joseph effects and the non-Gaussian and non-Brownian issues of the financial theory
• The first model of Mandelbrot (1962): α-stable motion with paretian tails
• The second model of Mandelbrot (1965): fractional brownian motion with aperiodic cycles
• The third model of Mandelbrot (1967): time changed Brownian motion with stochastic clock
• Appendix: a tale of fat tails
• Bibliography

The following sections are included:

• Introduction
• Enters Benoit Mandelbrot
• How Mandelbrot's concepts affected the way we look at lung structure
• The “fractal leap” in morphological sciences
• Bibliography

The theoretical allometry relation (AR) between the size of a network Y and a property of the network X is of the form X = aY_b and has been known for nearly two centuries. The allometry coefficient a and allometry exponent b have been fit by various data sets over that time. The ubiquity of ARs in biology, sociology, ecology and indeed in virtually all the other science disciplines entreats science to find the origin of ARs. Data analysis indicates that the empirical AR is obtained with the replacements X → 〈X〉 and Y → 〈Y〉 and the brackets denote an average over an ensemble of realizations of the network. It has been shown that the empirical AR cannot usually be derived from the theoretical one by simple averaging due to the fractal statistics of the fluctuations. Consequently we hypothesize that a possible origin of AR is the Principle of Minimum Entropy Generation (PMEG). We establish in support of this hypothesis that if the fluctuations in the allometry variables have fractal statistics then the PMEG entails the AR between a complex host network and a subnetwork strongly coupled to it.

Any of the arts may produce exemplars that have fractal characteristics. There may be fractal painting, fractal poetry, and the like. But these will always be specific instances, not necessarily displaying intrinsic properties of the art-medium itself. Only music, I believe, of all the arts possesses an intrinsically fractal character, so that its very nature is fractally determined. Thus, it is reasonable to assert that any instance of music is fractal…

The Yale University mathematics department hosted a memorial for Benoit on April 29 and 30, 2011. The first day of the meeting consisted of three technical talks on some aspects of fractals, Benoit's principal intellectual legacy. Bernard Sapoval spoke on fractals in physics, Peter Jones on fractals in mathematics, and Nassim Taleb on fractals in finance…

One afternoon Benoit invited me to a master's tea in his college at Yale. He and a group of colleagues discussed opera, the distribution of galaxies, Hokusai's art, Stoppard's Arcadia, many other topics. Despite the conversation's swerves, Benoit contributed detailed opinions to each new direction. When he excused himself to speak with someone on the other side of the room, a diminuitive Cuban poet who joined the group late turned to me, clearly Benoit's guest, and asked, “Who is that enormous man?” Before I could reply, another said, “Oh, that's Benoit Mandelbrot. He reinvented geometry.” “Ah, a scientist. Then why does he know opera and art?”…

The following section is included:

• Index