Fractal Geometry

Summary

Fractal geometry is a branch of mathematics that studies irregular or fragmented shapes, such as clouds, trees, mountains, and coastlines. Applications are widespread, from the sciences and medicine to social sciences and the arts, and especially in human anatomy, ecology, physics, geology, economics, and computer graphics. Related areas include chaotic systems and turbulent systems.

Definition and Basic Principles

Fractal geometry is a workable geometric middle ground between the excessive geometric order of Euclid and the geometric chaos of general mathematics. According to French American mathematician Benoit Mandelbrot in Proceedings of the Royal Society of London (May 8, 1989), it is based on a form of symmetry previously underused, namely, “invariance under contraction or dilation”:

Fractal geometry is conveniently viewed as a [language that has proven] its value by its uses. Its uses in art and pure mathematics, being without “practical” application, can be said to be poetic. Its uses in various areas of the study of materials and [of] other areas of [engineering] are examples of practical prose. Its uses in [physical] theory, especially in conjunction with the basic equations of mathematical physics, combine poetry and high prose. Several of the problems that fractal [geometry] tackles involve old mysteries, some of them already known to primitive man, others mentioned in the Bible, and others familiar to every landscape artist.

Background and History

Fractal geometry was first defined in 1975 by Benoit Mandelbrot (1924–2010), a creative and prolific twentieth-century polymath who pioneered a synthesis of mathematical paradigms for interpreting rational surface roughness. Les Objets Fractals (1975), translated into English in 1977 under the title Fractals: Form, Chance, and Dimension, explored Mandelbrot's innovative approach to understanding the properties of mathematical and natural forms. The manuscript was revised and republished in 1983 as The Fractal Geometry of Nature, a classic work of twentieth-century mathematics. This book illustrates principles that reordered understanding of dimension, symmetry, and scalar transformations. As a result of rational inquiry into shapes and nonlinear physical phenomena, fractal analysis can be applied with exceptional effectiveness in pure and applied mathematics. Advances in computation during the late twentieth century made it possible to evaluate unique objects constructed after millions of iterations of a function. Repetitive calculations that are not feasible in human time were performed with novel speed and duration via the computer, producing extraordinary mappings of functions, the visual beauty of which astounded its practitioners. The results of fractal analysis transformed and aligned knowledge in many domains, including physics, religion, the mechanical and biological sciences, economics, statistics, music, and many fields of art.

Mandelbrot's investigations regarding the nature of roughness or nonlinearity in the structural composition and generation of mathematical and physical objects are products of the twentieth century. However, fractal analysis is synthetic and embraces systems of thought originating in the preceding centuries. In retrospect, the principles described by fractal theory provide a platform for integrating various human expressions regarding infinity. Throughout history, mathematicians intuitively understood the implicit beauty and symmetry of mathematical systems. Glimpses of the infinite were magnified and demonstrated in fractal forms with particular appeal to the human sense of vision. Moreover, the language of fractals can be grasped and used in fields requiring varying levels of mathematical and computational ability. Finally, fractal symmetry provides an accessible framework for understanding universal structures and dynamics.

The antecedents to fractal geometry can be seen in the following areas.

Projective Geometry. Projective geometry refers to a class of geometric properties describing invariance under projection. Scholarship suggests that this phenomenon was understood by early geometers—perhaps by Euclid, the ancient Greek mathematician considered the father of geometry, himself. Mathematicians of the seventeenth century are credited for realizing these properties in an attempt to make geometry more practical. French mathematicians Gérard Desargues (1591–1661), Blaise Pascal (1623–1662), and Philippe de la Hire (1640–1718) provided mathematical foundations for new applications in the nineteenth century. Subsequent formulations of the works of Desargues are fundamental to novel descriptions of space and time. French mathematicians Jean-Victor Poncelet (1788–1867) and Michel Chasles (1793–1880) continued Desargues' work. Similar innovations were made in analytic, non-Euclidean, and Riemannian geometry.

Geometers have relied on the definitions and assumptions described in Euclid's Elements of Geometry (1893), a textbook that established time-honored principles for studying geometric properties and their mathematical relationships. These fundamental truths formed the basis of rational descriptions of shapes and measurements to the present day. During the nineteenth century, scholars found the ideal shapes of Euclidean geometry insufficient to analyze the natural phenomena described by science. In particular, Euclid's fifth postulate defied proof by the brightest minds. Investigations in the early 1800s by German mathematician Carl Friedrich Gauss, Hungarian mathematician János Bolyai, and Russian mathematician Nikolai Ivanovich Lobachevsky set the groundwork for the descriptions of the hyperbolic plane, a concept that would have stunning implications in descriptions of the mechanics of space and time, a field pioneered by Albert Einstein.

Cartography. In his 2002 essay “A Maverick's Apprenticeship,” Mandelbrot mentioned his familiarity with maps because of an uncle's tutoring. It is of note that well-established cartographic publishing houses and government agencies supported the works of many nineteenth- and twentieth-century mathematicians who made magnificent contributions to the field of cartography. Expanding centers of global trade, competition for world hegemony, and remarkable advances in science and technology created new demands for accurate systems of measurement. Novel uses of geometric projections stimulated inventive minds engaged in the proliferation of maps representing expanding enterprises in topographical land surveillance, marine navigation, meteorology, oceanography, climatology, geology, mineralogy, biogeography, and demography. Many two- and three-dimensional projections were developed to represent different sets of data visually. Accurate scalar measurements of coastlines and elevated land features were essential to mariners and land developers, who relied on visual data to guide their enterprises. Mathematics was an indispensable tool of the trade.

Symmetry. The adoption of the concept of symmetry was a very gradual process. It was an accepted condition of mathematical equations in the nineteenth century, but its laws and functions in relation to physical phenomena were poorly understood. In the field of physics, French mineralogist René Just Haüy's analysis of the geometric forms of crystals and subsequent studies of the symmetry of crystalline structure and properties were eventually applied to the study of other natural and dynamic systems. Mandelbrot's introduction of fractal theory helped to refine core concepts of symmetry, particularly those of proportion and scale. Symmetry is a universal property that unifies all natural and aesthetic phenomena.

Artists were among the first groups to embrace the fractal as an exquisite audiovisual medium for creative expression. The processes of fractal symmetry have stimulated profound intellectual and technical affinities among mathematicians, scientists, theologians, software designers, composers, architects, graphic artists, and writers who understand the universal principles of harmony that fractals represent. Fractals have been increasingly found in both modern and ancient works of art, including much ancient African art. Additionally, some neuroscientists suggest that viewing certain fractal patterns can induce stress reduction.

Monsters. German mathematician Karl Weierstrass's proof of a non-differential function replicated earlier work by Bernhardt Riemann and opened the door for further analysis of nondifferentiable curves. Functional analysis and formal logic were essential practices of the mathematician's craft. Graphic representations of complex functions were not in practice, and, in the case of some functions, their representations simply could not be mapped in human time. Subsequently, many leading mathematicians rebuffed the anomalous solutions of nondifferential functions, banishing them to the periphery of mathematics. As Mandelbrot explained in his introduction to The Fractal Geometry of Nature, many symbolic solutions of nondifferential functions were anomalous to the tradition of differential curves and thus dismissed as “pathological” and belonging to a “gallery of monsters” and relegated to the periphery of mathematics.

Mandelbrot also pointed out that these unique mathematical concepts are now the fundamental tools defining natural phenomena in the world around us. He referenced particular mathematicians who made singular contributions to the theory of fractals. Their theories and the objects named after them are standards of the canon of fractal geometry.

In 1883, German mathematician Georg Cantor, one of the founders of point-set topology, introduced the Cantor set (also known as the triadic Cantor dust), a self-similar disconnected function. In 1890, Italian mathematician Giuseppe Peano defined the function of a space-filling curve, and in 1891, German mathematician David Hilbert provided variations of the Peano curve and included graphic representations of the models described. In 1915, Polish mathematician Waclaw Sierpinski introduced the Sierpiński triangle (also called gasket or sieve).

In 1919, German mathematician Felix Hausdorff developed the concept of fractional dimension, a measure theory originated by Greek mathematician Constantin Carathéodory in 1914. Russian mathematician Abram Samoilovitch Besicovitch also developed the idea of fractional dimension between 1929 and 1934. Taken together, the concepts defined by Hausdorff and Besicovitch were used by Mandelbrot to define what he termed the “fractal dimension” of a surface.

In 1938, French mathematician Paul Lévy introduced another geometric object, Lévys Dragon, which demonstrated a triangular set that could be configured (or tiled) to fill a curved space. Other contributors to fractal theory mentioned by Mandelbrot are Bohemian mathematician Bernard Bolzano, French mathematician Henri-Leon Lebesgue, American mathematician William Fogg Osgood, and Russian mathematician Pavel Samuilovich Urysohn. Finally, the works of French mathematicians Gaston Maurice Julia, Henrí Poincaré, and Pierre Joseph Louis Fatou are memorialized in Mandelbrot's graphic representations of the Julia set and the Mandelbrot set described below.

Unusual Data Sets and Nonlinear Phenomena. During the late 1950s and 1960s, Mandelbrot was an employee at IBM, and computer technologies were in their infancy. One of the first problems Mandelbrot was asked to resolve was that of “noise” during data transmission across telephone wires. Occasional errors were problematic. Mandelbrot graphed the sequence of erratic transmissions and noticed a regular, self-similar series that reminded him of the phenomena described by Georg Cantor many years before. Mandelbrot realized that the binary nature of electronic data transmission permitted intermittent switching of signals that would prove disastrous to the system IBM was interested in marketing. The recognition of these signal patterns confirmed his deep-seated hunches about the nature of the “monsters” that lurked in the works of several turn-of-the-century mathematicians. These transmission data sets illustrated the powerful concepts of reiteration and self-similarity.

However, not all natural phenomena are self-identical, nor are they fixed or static. At the time, Mandelbrot was aware that nonlinear mechanics were an important component of dynamic systems, particularly in the study of turbulence and galaxy formation. Edward Lorenz, a meteorologist at the Massachusetts Institute of Technology, was pioneering new investigations in chaos theory relative to the unreliable prediction of weather patterns. Better working concepts were needed to explain the mechanics of difficult nonlinear phenomena such as turbulence and clustering. Mandelbrot recognized the potential of the Hausdorff-Besicovitch dimension to describe irregular phenomena and chose the coastline of Great Britain as a teaching rubric for working through the concept of fractional dimension. In subsequent studies of scalar proportions, he recognized that the shape of the coastline was similar to the form of the Koch curve (also called Koch snowflake, Koch star, or Koch island) developed by Swedish mathematician Helge von Koch in 1904. Manipulating the parameters of the curve, Mandelbrot mathematically duplicated, with exceptional accuracy, the shapes of various landforms.

Self-Similar Systems: The Julia Set and the Mandelbrot Set. In 1918, Gaston Maurice Julia and Pierre Joseph Louis Fatou published independent works describing the process of iterating rational functions. Mandelbrot's uncle gave him original reprints of the articles, suggesting they were worth pursuing, but he did not follow up on them until much later, when employed at IBM.

Access to computers gave him an exceptional opportunity to test the reiterative models described by Julia and Fatou. Instead of simply solving equations, he ran them as a reiterative feedback loop, using the results of one calculation to serve as the inputs of the next run. After millions of iterations, he mapped the results on a grid and was astounded by the results, now known as a Julia set. Working with the graphics of different functions, Mandelbrot devised his own complex quadratic polynomial, such that f(x) = x² + c, starting at x = 0, does not go into infinity, and then ran it. The result was a point set called the Mandelbrot set. The Mandelbrot set is unique in its graphic mapping of recursive self-similarity applicable at all magnitudes examined. Its manifestation had a profound and immediate effect on scholars and artists worldwide, who found in the magnificent representations of nonlinear functions a new and incontrovertible paradigm for understanding the universal shapes and forms of the natural world.

Describing Chaos. The dynamics of complex systems are notoriously difficult to characterize, much less to control. Nevertheless, Mandelbrot's application of the concepts of the random walk, Brownian motion, diffusion-limited aggregation, galaxy clusters, fractal attractors, percolation, fractal nets and lattices, L-systems, box counting, and multi-fractal surface dimensions created new conceptual models for rationalizing chaotic systems. Novel cluster patterns are used in the research design of global feedback systems to regulate chaotic systems. In May 2001, researchers at the Fritz Haber Institute of the Max Planck Society in Germany studying catalytic reactions announced the controlled design of repeating chemical clusters with fractal-like patterns. These patterns and formations anticipated new methods of controlling chemical systems turbulence.

Fractals and the Natural Sciences. The first conference on Fractals in Natural Sciences was held in Budapest, Hungary, from August 30 to September 2, 1993. The major subjects covered included topics related to DNA sequencing and biorhythms; complex bacterial colony formation; correlations of galaxy distribution; fractal tectonics, erosion, and river networks; the geometry of large diffusion-limited aggregates; the use of wavelets to characterize fractals beyond fractal dimension; fixed-scale transformations; diffusion properties of dynamical systems; transfers across fractal electrodes; crack branching; the fractal structure of electrodeposits; the vibrations of fractals; experiments with surface and kinetic roughness; self-organized criticality; granular segregation and pattern formation; and nonlinear ocean waves. The proceeds of the conference were published by the World Scientific Publishing Company in 1994 and provide an excellent starting point for understanding the immediate reception of fractal theory in the biological sciences, chemistry, earth science, and physics.

How It Works

The Feedback Loop. The iterated feedback loop is a fundamental concept of fractal geometry. A feedback loop is a transformational system where the input of a cycle (or mathematical function) gives a different output, which becomes the input of a new transformation. Feedback processes are fundamental tools for studying natural systems. Examples of iterated mathematical systems can be found in history over thousands of years. However, the capability of the feedback loop to generate complex systems was not fully understood until electronic and computational technologies were developed. The iterated feedback system is a standard process for generating geometric fractals. The function of an iterated object can be linear or nonlinear and include affine, projective, and Möbius transformations. It is an essential component of the description of chaotic systems.

Fractal Dimension. Describing the dimensions of a given space was a topic that reordered the landscape of mathematics and physics in the nineteenth and twentieth centuries. Euclidean space is three-dimensional, a definition that refers to the range of motion of objects in a physical space. D = 1 refers to points on a straight line, D = 2 refers to points located on a plane, and D = 3 refers to points located within a cube.

Smoothness is an essential feature of Euclidean geometry. Idealized figures such as the cube, the sphere, and the triangle are abstractions that facilitate human understanding of fundamental spatial relationships. These concepts have had enormous use in the applied arts and sciences and are basic technology tools. However, natural forms such as flowers, mountains, or riverbeds defied classic mathematical descriptions. Mandelbrot's breakthrough was the realization that Euclidean parameters could not be used to define natural forms because they are irregular. Roughness is an essential characteristic of the natural world, requiring a different set of mathematical tools for analysis.

Mandelbrot used the fraction to describe non-integer dimensions that pertain to natural surfaces. The dimensions (D) of natural objects are fractional composites of the straight line, the plane, and the cube. That fractal dimension is summarized in the equation D = log(N )/ log(r), where N refers to the number of pieces and the magnification factor and r is a scaling factor indicating the roughness of an object. When these values are coordinated on a log-log plot, the steepness of the line's slope (gradient) indicates the fractal dimension or roughness of the object.

Technologists and statisticians working with graphic variables describing biological and physical phenomena immediately recognized similarities in the results of time-series representations and the graphic plots of fractal analytics. Heartbeats, physical motions, geological processes, and the biological progression of species in a particular ecosystem could be defined using fractal nomenclature. Institutions worldwide use fractal analysis as a standard tool for interpreting physical systems. The BENOIT fractal analysis software provides methods for measuring data sets that are chaotic and not amenable to traditional analytics. These include methods for plotting self-similar and self-affine fractal characteristics. Self-similar methods apply calculations of box dimension, perimeter-area dimension, information dimension, mass dimension, and the ruler method. Self-affine analytics require tools that calculate the scalar properties of the data aligned on the horizontal and vertical axes at constant sampling intervals. These include measurements of fractal Brownian motion, fractional Gaussian noise, power-spectral analysis, variogram analysis, and wavelet analysis.

Iterated Function Systems. Iterated function systems allow natural objects to be recreated using mathematical descriptions collected into data sets with accompanying computation rules. This process is also known as fractal image compression, and it is the technology that makes computer-generated landscapes and visual effects in film possible. Very simply, a table of numbers (or matrix formulation) is created to describe the desired affine transformations. These follow the conventional order of scalings, reflections, rotations, and translations.

Applications and Products

Fractals have applications in anything with a rhythm or pattern—cardiovascular systems, germ growth, wireless signals, computer functioning, geomorphology, linguistics, and economics. In his introduction to The Fractal Geometry of Nature, Mandelbrot wrote, “More generally, I claim that many patterns of nature are so irregular and fragmented, that, compared with Euclid—a term used in this work to denote all of standard geometry—nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of natural patterns is for all practical purposes infinite.”

Biological Diversity. Understanding biodiversity and the power laws that govern its rich complexity is one of the great challenges of science in the twenty-first century. Scaling relationships in nature provides a powerful paradigm for understanding how biological systems evolve and their relationship to the physical landscape. Environmental factors have a measurable influence on diffusion distributions of species.

Earth Sciences. Computational molecular modeling is slowly transforming the metrics of geochemistry, making it possible to analyze the geometric structures and properties of materials at microscopic levels. Materials such as clays, composed of fine-grain silicates, are considered complex systems, the dynamics of which are still poorly understood. Water, too, is a substance the surface of which is difficult to characterize. Surface chemistries at the edge of a molecule are often different from that of a flat surface, a reality that challenges accurate manipulations of surface dynamics.

Chemical and Biochemical Processes. In physical chemistry and biochemistry, the fractal dimension of surface porosities is a remarkable statistic for the evaluation of processes affecting adsorption rates, chemical clustering, dispersion and uptake algorithms, spectroscopy, photochemistry, estimates of agglomeration dynamics and electrolyte deposition, the behavior of electrolytic dendrites, and the study of disordered systems and catalytic rates. Controlling for nanoscale surface roughness is an important part of developing effective thermal conductors for use in microelectronics. Similar systems analyses of the properties of granular surfaces are of particular importance to the understanding of the chemistry of interstellar dust particles and how ice crystals adhere to them, essential processes in the evolution of stars and galaxies.

Self-Organized Nanostructures. One of the remarkable properties of nanoparticles (NPs) is their capacity for self-organization. In the early twenty-first century, researchers noted that nanoparticles interact to form rings, linear chains, and hyperbranched (dendritic) structures. Of particular interest is the step-growth organization of NPs into polymer-like aggregates with varying types of isomers. These properties suggest the potential use of nanostructures in a variety of products. In 2008, researchers at the University of Wisconsin–Madison reported the creation of nanotrees, spiraling branched objects. Scientists are developing nanoforests with an array of structures. These have great promise for revolutionizing the production of high-performance integrated circuits, biosensors, solar cells, light-emitting diodes (LEDs), and lasers.

Epigenetics. Evidence of self-organization at the nanoparticle level has provided new insights aligning the structural self-organization of life forms from the symmetries of atomic structure through the allometric scalings of organisms to the forces arranging the patterns of stars. Nevertheless, continuing research into the dynamics of the epigenome reinforces the enormous complexity of the pathways of genetic inheritance, suggesting that genetic expression is influenced by a recursive and scaling array of multidimensional patterns affecting phenotype expression. Fractional dimension analytics are a valuable tool of bioinformatics and are used to measure the complexity of self-similar biological organisms and their relationship to other organisms in their environment. Time is an essential dimension of epigenetic models.

Archaeology. Fractals can be used in archaeology to discover ancient sites that reveal history. For example, in 2012, German archaeologists used fractals to learn more about the area surrounding the ancient Egyptian royal necropolis of Dahshur. They created a digital topography of the land, which shows areas of ancient river runoff in fractal patterns. Areas lacking these patterns indicate that humans altered the landscape, enabling archaeologists to better determine the extent of the Dahshur necropolis complex.

Technology. In the twenty-first century, fractals are used to develop cell phones and other communication devices. Antennas designed using fractal geometry allow manufacturers to reduce antenna sizes while improving the device's range and reliability in receiving and transmitting signals. Computer graphics heavily rely on fractals to compress large files. These innovations in image data compression technology are evident in the computer-generated landscapes of movies such as Star Wars: Episode VI—Return of the Jedi (1983) and Star Trek II: The Wrath of Khan (1982). Software programs ranging from free to thousands of dollars allow users to create fractal images, conduct complex mathematical operations, or enhance artificial intelligence. TruSoft International's BENOIT fractal analysis system enables users to measure the fractal dimension and the Hurst exponent of data sets using eleven methods. Similar software is increasingly used in statistical mechanics to plot galaxy, star, and space gas distributions.

Careers and Course Work

Fractal geometry applies to a broad range of academic subjects and careers. Dynamic geometry software programs, such as Geometer's Sketchpad, provide interactive tools for learning fractal structures at the high school level. At the university level, introductory fractals and chaos theory courses have been designed for and taught to undergraduate liberal arts majors. Chaos games, iterated function systems, fractional dimension, cellular automata, and artificial life are basic topics addressed. Familiarity with fractal software programs is essential for careers in which the product is the visual image, including photography, fashion design, graphic design, urban landscape design, the technologies of film and sound, medical imaging, scientific illustration, laser optics, and land topography mapping technologies. Courses include fractal geometry and chaotic dynamics, chaos and fractals, and chaos, fractals, and dynamics.

Upper-level undergraduate and graduate courses in fractal mathematics require training in writing mathematical proofs, advanced calculus, metric topology, and measure theory. Fractal mathematics is an essential feature of internet engineering design, economics, mechanical and electrical engineering, the physical and life sciences, computer-game design, electronics, telecommunications, hydrology, geography, demography, and other statistical disciplines. Aspirants can work in archeology, trading, geospatial data analysis, or quantum physics.

Social Context and Future Prospects

In 1993, Benoit Mandelbrot was awarded the Wolf Foundation Prize for Physics. On April 25, 2003, he shared the Japan Prize for Science and Technology of Complexity with James A. Yorke, professor of mathematics and physics at the University of Maryland. Both men were cited for creating the science and technology of universal concepts in complex systems, namely fractals and chaos. These structures underlie complex phenomena in a wide range of fields, demonstrating the importance of understanding the behavior of systems as a whole instead of the reduction of phenomena into discrete elements for observation. Both the Wolf Foundation Prize and the Japan Prize were given in recognition of unique contributions to the progress of science and technology and the promotion of peace and prosperity to humankind.

Fractal theory continues to integrate human knowledge, providing a workable interface that links very different cognitive domains to examine complex physical phenomena in unusual ways. Computer technologies and interactive software programs search for ways to engage the evolutionary capacity of the human eye to assist in the processing of multiple and continuous streams of data. Fractal mappings and time series analyses are important new components of human logic. These are built on new realms of thinking about and modeling geometric shapes, setting the stage for the creation and manipulation of workable multidimensional geometries and design technologies.

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