Fractals are a mathematical subject, so a full understanding of fractals requires mathematics. A person can look at a rainbow, a waterfall or a cloud and enjoy the beauty of these objects without understanding the physics behind any of them. And fractals are the same way. However, in my view fractals become even more enjoyable as one understands the mathematics behind them.
This is hardly an exhaustive list of reference materials on the subject. A complete literature search for the author Mandelbrot will produce a couple hundred works on a variety of subjects (Mandelbrot was a prolific author). A complete subject search and/or citation search would produce even more. This is an attempt to list a few representative examples of works to show what fractals are and how they apply to specific areas in astronomy, however I have included references on physics as well.
The level of math in the following works varies tremendously. Some of these works discuss areas of science where fractals can be applied even though the author of the work in question did not know about fractals at the time the book or article was written. Other works discuss areas of science where the author was fully aware of fractals. Still others discuss fractals without reference to science at all. You should assume all of these references (except letters to the editor) are mathematical unless I’ve indicated otherwise.
------. 1986. “A Fractal Universe?” Science News, 129 (April 5), p. 217.
This non-mathematical and short article discusses the nature of space itself. Physicists have proposed that space consists of 11 or 12 dimensions. In order to explain why we observe a three dimensional world, they propose the remaining dimensions are “folded” up. Now suppose these extra dimensions are in fact “fractal dimensions.” A serious examination of this possibility was conducted. If this were the case, slight discrepancies would be observed in the laws of nature. Observations were made of the Lamb shift (a change in spectral lines when hydrogen atoms are exposed to a magnetic field) and planetary orbits. These observations were used in an attempt to answer this question. The conclusion: the difference between a three dimensional universe and the real universe is very small if it exists at all. So if you are looking for fractals in the universe, this may not be the place to look.
Aharony, A. and Feder, J., editors. 1989. Fractals in Physics: Essays in Honour of Benoit B. Mandelbrot; Physica D, 38.
A series of articles, mainly on applications of DLAs and percolation theory. There is an article on multifractals and another on the distribution of galaxies.
Arthur, D. W. G. 1954. “The Distribution of Lunar Craters.” Journal of the British Astronomical Association, 64 (February), pp. 127-132.
Craters on the moon (or other bodies) have a distribution of sizes. Some craters are small, some are medium size and still others are large. Data on craters were a motivation for later work by Mandelbrot and others who proposed a fractal distribution for craters on the moon (though I don’t know if Mandelbrot read this article before or after he did his work).
Avnir, David et al. 1998. “Is the Geometry of Nature Fractal?” Science, 279 (January 2), pp. 39-40.
A letter to the editor. The authors argue that fractals are not a good model for natural phenomenon. See [Mandelbrot, Benoit B. et. al., 1998] for a rebuttal and further discussion.
Avron, J. E. & Simon, B. 1981. “Almost Periodic Hill’s Equation and the Rings of Saturn.” Physical Review Letters, 46 (April 27), pp. 1166-1168.
Data from the Voyager space craft showed that the rings of Saturn were arranged in complex patterns of gaps and ringlets. The arrangement is a good example of a chaotic process. This article was a first attempt to quantify these patterns.
Bahcall, Neta A. 1997. “Large Scale Structure in the Universe” in Unsolved Problems in Astrophysics. John Bahcall and Jeremiah P. Ostriker, editors. Princeton, New Jersey: Princeton University Press.
Barnsley, Michael. 1988. Fractals Everywhere. Boston: Academic Press.
A very detailed and rigorous description of how an IFS is generated. Includes many color as well as black and white images.
Barnsley, Michael, Devaney, Robert L., Mandelbrot, Benoit B., Peitgen, Heinz-Otto, Saupe, Dietmar and Voss, Richard F. 1988. in Heinz-Otto Peitgen and Dietmar Saupe, editors. The Science of Fractal Images. New York: Springer-Verlag.
Discusses aggregate replacement fractals, DLAs, 1/f and power law noises, Brownian motion, probability distribution functions, computer algorithms for generating fractals, chaotic dynamical systems, Julia sets, the Mandelbrot set, IFS and the Chaos game.
Buchler, J. Robert and Eichhorn, Heinrich, editors. 1987. The Annals of the New York Academy of Sciences, Volume 497: Chaotic Phenomena in Astrophysics. New York: The New York Academy of Science.
Several different examples of the use of chaos theory in astrophysics. This includes galactic models, the Zeeman effect, variable stars, large scale structure of the universe and models of the interstellar medium.
Burns, Jack O. 1998. “Stormy Weather in Galaxy Clusters.” Science, 280 (April 17), pp. 400-404.
Chandrasekhar, S. 1943. “Stochastic Problems in Physics and Astronomy.” Reviews of Modern Physics, 15, pp. 1-89.
Covers two topics: namely Brownian motion and the gravitational field from a random distribution.
Coleman, P. H. and Pietronero, L. 1992. “The Fractal Structure of the Universe.” Physics Reports, 213 (May), pp. 311-389.
This article discusses the distribution of galaxies within the universe. Galaxies tend to associate into clusters, and clusters of galaxies form into superclusters. There are also immense voids were no galaxies are present (the galaxy clusters are concentrated within films along the edges of these voids). If this clustering persists from the largest scales down to the scale of individual galaxy clusters, it is possible that the entire universe can be modeled as a fractal. Coleman and Pietronero make a good case for this (but are not in a position to prove it yet).
Devaney, Robert L. 1992. A First Course in Chaotic Dynamical Systems: Theory and Experiment. Reading, Massachusetts: Addison-Westley.
Covers much of the same ground as [Peitgen, Jürgens and Saupe, 1992] but is shorter, more approachable and less rigorous.
Diacu, Florin and Holmes, Philip. 1996. Celestial Encounters: The Origins of Chaos and Stability. Princeton, New Jersey: Princeton University Press.
Discusses the history of celestial mechanics and how the problem of stability of the solar system has been addressed. This covers chaos theory, KAM theory and the various definitions of the word “stability.”
Dickson, F. P. 1968. The Bowl of Night; the Physical Universe and Scientific Thought. Netherlands: Eindhoven.
Feder, Jens. 1988. Fractals. New York: Plenum Press.
Discusses fractal dimensions, aggregate replacement fractals, multifractal measures, DLAs, percolation and Brownian motion.
Fleischmann, M., Tildesley D. J., and Ball, R. C., editors. 1989. Fractals in the Natural Sciences. Princeton, New Jersey: Princeton University Press.
This book contains a set of 64 papers on biology, chemistry, geology and physics showing how fractals apply in a wide variety of scientific applications. This book assumes you already understand fractal geometry.
Gamow, G. 1954. “Modern Cosmology.” Scientific American, 190 (March), pp. 54-63.
Gardner, Martin. 1967a. “Mathematical Games: An Array of Problems That Can Be Solved With Elementary Mathematical Techniques.” Scientific American, 216 (March), pp. 124-127, 129.
This article consists of a set of problems. Problem number 3 relates to fractals. See [Gardner, 1967b] for the answer.
------. 1967b. “Mathematical Games: The Amazing Feats of Professional Mental Calculators and Some Tricks of the Trade.” Scientific American, 216 (April), pp. 116-120, 122.
The end of this article has the solutions to the problems posed in the previous month’s article [Gardner, 1967a].
------. 1973. “Mathematical Games: Fantastic Patterns Traced by Programmed ‘Worms.’ ” Scientific American, 229 (November), pp. 116-123.
This article is not about fractals, but is a very cute example of how a simple set of rules can produce very complex patterns.
------. 1976. “Mathematical Games: In Which ‘Monster’ Curves Force Redefinition of the Word ‘Curve.’ ” Scientific American, 235 (December), pp. 124-133.
A good, easy introduction to aggregate replacement fractals.
------. 1989. Mathematical Magic Show. Washington, D. C.: The Mathematical Association of America, pp. 207-209, 215-220, 292-293.
A problem related to fractals.
Gleick, James. 1987. Chaos: Making of a New Science. New York: Viking.
This book attempts to explain chaos theory and its relationship to science and fractals from a totally non-mathematical basis. Chaos theory is a rich tapestry: While Gleick gives us a reasonable introduction to the subject, he presents only part of the tapestry. To find the rest, you need to go elsewhere: After you are finished reading Gleick, I would suggest [Schroeder, 1991].
Haldane, J. B. S. 1985. “On Being the Right Size.” in John Maynard Smith, editor. On Being the Right Size. Oxford: Oxford University Press.
Haldane originally wrote this classic article in 1928. It concerns the question of why animals are the size they are and not larger or smaller.
[Peitgen, Jürgens and Saupe, 1992, pp. 138-146] refers to this article. Mathematically we scale fractals by making them larger or smaller by a fixed amount. When we do this with animals, we do not find strict self-similarity. However that statement doesn’t tell us anything about the occurrence of fractals in natural settings.
Hartmann, W. K. 1977. “Cratering in the Solar System.” Scientific American, 236 (January), pp. 84-86, 89-99.
I would make the same comment as I made for [Arthur, 1954]. However this covers craters on a number of objects, not just the moon.
Hastings, Harold M. and Sugihara, George. 1994. Fractals: A User’s Guide to the Natural Sciences. Oxford: Oxford University Press.
Hoyle, F. 1953. “On the Fragmentation of Gas Clouds into Galaxies and Stars.” Astrophysical Journal, 118, pp. 513-528.
Hurd, Alan J. 1988. “Resource Letter FR-1: Fractals.” American Journal of Physics, 56, pp. 969-975.
This is a bibliography of fractal articles.
Körner, T W. 1989. Fourier analysis. New York: Cambridge University Press.
Pages 38 through 45 describe Karl Weierstrass’ function, which is generally considered to be the first fractal to be discovered.
Krantz, Steven G. 1989. “Fractal Geometry.” The Mathematical Intelligencer, 11 (Fall 1989), pp. 12-16.
Mandelbrot has argued that fractal geometry offers a new way of looking at some of the sciences. While fractals are not a panacea, they offer the possibility of much better understanding in some areas. Krantz is less optimistic and wrote this criticism of fractal geometry. If you read this letter, you will not be surprised that Mandelbrot did not agree: Mandelbrot wrote a rebuttal [Mandelbrot, 1989].
Kuhn, Thomas S. 1973. The Structure of Scientific Revolutions. 2nd edition. Chicago: University of Chicago Press.
Kuhn discusses his view of scientific change: Science alternates between periods of “normal science” and “revolutionary science.” In Kuhn’s view normal science proceeds until a crisis (an accumulation of data that is not explained by the theories of the day) eventually triggers a paradigm shift (a new way of looking at the world). The paradigm shift causes a period of revolutionary science along with new theories that better explain the data. At this point we return to normal science again.
I believe viewing scientific phenomenon as fractals was a paradigm shift that occurred in the 1970s, but the crisis itself dates back to the late 1800’s.
Labini, Francesco Sylos. 1998. Scale Invariance of Galaxy Clustering: Application to the Most Recent Data. Talk given at the 5ème. Colloque Cosmologie Observatoire de Paris (Fundamental Problems in Classical, Quantum and String Cosmology). June 3-5, 1998.
See comment under Mandelbrot, 1998a.
Mandelbrot, Benoit B. 1967. “How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.” Science, 156 (May 5), pp. 636-638.
One of the earliest examples of an application of fractals to the natural sciences. Here Mandelbrot brings mathematics to the subject of geology (specifically, the exact shape of islands and other coastlines). Even though the word fractal hadn’t been invented yet, you can see fractal geometric thinking at work here.
------. 1977. Fractals: Form, Chance and Dimension. San Francisco: W. H. Freeman and Company.
Mandelbrot first used the word “fractal” in his 1975 book Les Objets Fractals: Forme, Hasard et Dimension. Since it was written in french, it did not reach a wide audience. Fractals: Form, Chance and Dimension was the first place an English speaking audience was exposed to the word “fractal.” My first encounters with the fractal concept included this book (which I read in 1977 or 1978). Mandelbrot covers simple fractals and Brownian motion (he doesn’t cover more complex fractals such as Julia sets or DLAs). It is superseded by his 1985 book.
------. 1985. The Fractal Geometry of Nature. New York: W. H. Freeman and Company.
This book is an update of Mandelbrot’s 1977 book. It has more images and more examples. Unlike the 1977 book there are more complex fractals such as Julia sets and it is the first place where a wide audience saw the Mandelbrot set. Note, printings of this book before Fall 1985 do not show the Mandelbrot set in as much detail as later printings. (New printings of this book have a “publisher’s note” on the bottom right corner of page 188). It is partly a book on pure mathematics and partly a book on how the mathematics of fractals can be applied to various sciences. This book contains some ideas about diffusion, but the term DLA is not used. Mandelbrot had the idea for multifractals much earlier, but the word multifractal does not appear in this book.
------. 1986. “Letters: Multifractals and Fractals.” Physics Today, 39 (September), pp. 11, 13.
A very brief explanation of fractals and multifractals (written in response to another letter which asked “where are the fractals in science, which supposedly are everywhere?”).
------. 1989. “Some ‘Facts’ that Evaporate Upon Examination.” The Mathematical Intelligencer, 11 (Fall 1989), pp. 17-19.
This is a rebuttal to [Krantz, 1989]. The ‘facts’ are several statements made by Krantz in his letter.
------. 1990. “Negative Fractal Dimensions and Multifractals.” Physica A, 163, pp. 306-315.
------. 1998a. Fractality, Lacunarity and the Near-Isotropic Distribution of Galaxies. Talk given at the 5ème. Colloque Cosmologie Observatoire de Paris (Fundamental Problems in Classical, Quantum and String Cosmology). June 3-5, 1998.
I did not attend this or any of the other talks at this colloquium. Information about previous colloquia can be found in the Proceedings of the Second (1994) and Third Colloquia (1995) published by World Scientific (H J de Vega and N. Sánchez, Editors). The Proceedings of the Fourth Colloquium (1997) will appear soon by the same publisher. My assumption is the fifth colloquium will also be published by World Scientific. Additional information is available at http://www.obspm.fr/chalonge .
------. 1998b. Seventh Ta-You Wu Lecture Given at the University of Michigan: “Fractals and Scale-Invariant Roughness in the Sciences.” October 7, 1998.
------. 1998c. Seminar Given at the University of Michigan: “Wild Variability in Physics: Turbulence, 1/f Noise, and Galaxies.” October 8, 1998.
Mandelbrot, Benoit B., et. al. 1998. “Letters: Is Nature Fractal?” Science, 279 (February 6), pp. 783, 785-786.
A set of three letters to the editors of Science regarding the topic “Is Nature Fractal?” The first is a rebuttal by Mandelbrot to a previous article [Avnir et al, 1998].
Mandelbrot, Benoit B. and Evertsz, Carl J. G. 1990. “Letters: The Potential Distribution around Growing Fractal Clusters.” Nature, 39 (November 8), pp. 143-145.
A letter briefly explaining DLAs.
Merritt, David. 1996. “Chaos and the Shapes of Elliptical Galaxies.” Science, 271 (January 19), pp. 337-340.
Novak, Miroslav M., editor. 1994. Fractals in the Natural and Applied Sciences: Proceedings of the Second IFIP Working Conference on Fractals in the Natural and Applied Sciences. Amsterdam: Elsevier Science BV.
This book contains a set of papers explaining the use of fractals in astronomy, biology, chemistry, geology, materials science, medicine and physics. This book assumes you already understand fractal geometry.
Pebbles, P. J. E. 1980. The Large-Scale Structure of the Universe. Princeton, N. J: Princeton University Press.
Peitgen, Heinz-Otto, Jürgens, Hartmut and Saupe, Dietmar. 1991a. Fractals for the Classroom (Part 1). New York: Springer-Verlag.
The two parts of this book are almost identical to [Peitgen, Jürgens and Saupe, 1992]. There are some slight differences in content however. For example, unlike the 1992 book there is no appendix on multifractals. The 1992 book seems to be more complete than the two parts of this book.
------. 1991b. Fractals for the Classroom (Part 2). New York: Springer-Verlag.
See comment under [Peitgen, Jürgens and Saupe, 1991a].
------. 1992. Chaos and Fractals. New York: Springer-Verlag.
This book is 984 pages. Portions are difficult to follow; however if you really want to know exactly how fractals work from a mathematical basis, this book is absolutely essential. The authors explain in great detail how the various types of fractals are constructed. There is also a lot of historical information on the development of fractal geometry. Unfortunately the book is a little weak on the more useful types of fractals namely multifractals and DLAs and doesn’t spend much time on scientific applications. There is a good section on Brownian motion.
Includes 40 color prints and many black and white diagrams.
Peterson, Ivars. 1998. The Mathematical Tourist. New York: W. H. Freeman and Company.
Chapter 5 covers fractals, chapter 6 covers chaos and chapter 7 covers cellular automata.
Peitgen, Heinz-Otto, Jürgens, Hartmut, Saupe, Dietmar, Maletsky, E., Perciante, T. and Yunker, L. 1991. Fractals for the Classroom, Strategic Activities, Volume 1. New York: Springer-Verlag.
------. 1992. Fractals for the Classroom, Strategic Activities, Volume 2. New York: Springer-Verlag.
Pietronero, Luciano. 1998. Scale Invariance of Galaxy Clustering: General Concepts. Talk given at the 5ème. Colloque Cosmologie Observatoire de Paris (Fundamental Problems in Classical, Quantum and String Cosmology). June 3-5, 1998.
See comment under Mandelbrot, 1998a.
Prusinkiewicz, P. and Lindenmayer, A. 1985. The Algorithmic Beauty of Plants. New York: Springer-Verlag.
This book has many examples of how L-systems (a type of fractal) can be used to produce images that strongly resemble different types of plant life. The book includes a number of color prints.
Schiling, Gouert. 1998. “Did Galaxies Bloom in Clumps?” Science, 279, (January 23), p. 479.
Schroeder, Manfred. 1991. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman and Company.
While there are frequent equations, the mathematics in this book is comparatively lightweight. If you don’t mind a little algebra, this is a readable book. If you want something more rigorous, you might try [Devaney, 1992] or [Peitgen, Jürgens and Saupe, 1992] instead.
Stauffer, Dietrich. 1985. Introduction to Percolation Theory. London: Taylor and Francis Ltd.
de Vaucouleurs, G. 1956. “The Distribution of Bright Galaxies and the Local Supergalaxy.” in Vistas in Astronomy II, London: Pergamon, pp. 1584-1606.
Work that led up to [Coleman and Pietronero, 1992].
------. 1970. “The Case for a Hierarchical Cosmology.” Science, 167 (February 27), pp. 1203-1213.
Work that led up to [Coleman and Pietronero, 1992].
------. 1971. “The Large-Scale Distribution of Galaxies and Clusters of Galaxies.” Publications of the Astronomical Society of the Pacific, 83, pp. 113-143.
de Vega, Héctor J. 1998. Fractals and Scaling Laws in Galaxy Distributions and in the Interstellar Medium: New Field Theory Approach. Talk given at the 5ème. Colloque Cosmologie Observatoire de Paris (Fundamental Problems in Classical, Quantum and String Cosmology). June 3-5, 1998.
See comment under Mandelbrot, 1998a.
Vicsek, T., Shlesinger, M. and Matsushita, M., editors. 1994. Fractals in Natural Sciences. Singapore: World Scientific Publishing Company.
Voss, R. F. and Clarke, J. 1975. “ ‘1/f Noise’ in Music and Speech.” Nature, 258 (November 27), pp. 317-318.
The work of Voss and Clarke showed that music and human speech contain certain mathematical patterns that other sounds do not possess. These patterns are expressed by distributions that comply with a 1/f curve. Such distributions are now known as “1/f distributions.” Since this article was written, a variety of phenomenon in other sciences (including economics) have been successfully modeled using 1/f distributions. 1/f distributions have a strong relationship to multifractals.
Watson, Andrew. 1997. “Cosmologists Celebrate the Death of Defects.” Science, 278, (October 24), p. 574.
Wilson, Jack M. and Redish, Edward F. 1989. “Using Computers in Teaching Physics.” Physics Today. 42 (January), pp. 34-41.
Examples using computers in physics classrooms. Some of these examples involve fractals.
Wolfram, Stephen. 2002. A New Kind of Science. Champaign, Illinois: Wolfram Media, Inc.
This is a long book, it has 1197 pages not including the preface or index. However it is not as difficult as the length might suggest. The book is divided into two parts, the main text and notes. The main text starts with the idea of mathematical models and their relevance to science, but instead of using models based on differential equations (as is common practice in modern science), he uses discrete models based on cellular automata. He explains what cellular automata are and explores how they behave under a variety of circumstances. He discovers that some cellular automata produce regular pattern, some are capable of producing fractals and others produce patterns that look essentially random (even though in all cases the underlying rules are very simple).
Next he explores how models based on cellular automata (and related mathematical structures) can be applied to a variety of sciences, including physics and biology. He suggests (but is unable to prove) that all of phenomena of nature can be explained with simple models. However the fact that simple models are used does not mean that nature is predictable, to the contrary some models produce patterns that seem to be impossible to predict without running the model. According to Wolfram, nature is performing a computation, similar to computations taking place in the human brain or in a computer. There are no computations in the universe, now or in the future, that cannot be transform into a computation performed by cellular automata. While you might disagree with some of Wolfram’s ideas, this is likely to trigger a lot of discussion.
The notes are not as easy to follow. He makes continual references to Mathematica functions, which may be difficult to follow if you aren’t familiar with Mathematica. There are also many references to somewhat obscure ideas in mathematics, physics and computer science. However if you think you disagree with Wolfram, you need to read the notes.