‘How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,’ pp. 636-638 in: Science, New Series, Vol. 56, No. 3775, May 5, 1967. [With:] RICHARDSON, Lewis Fry. ‘The problem of contiguity: An appendix to Statistics of Deadly Quarrels,’ pp. 139-187 in: General Systems: Yearbook of the Society for the Advancement of General Systems Theory. Ann Arbor, MI: The Society, 1961.

Washington, DC: American Association for the Advancement of Science, 1967.

First edition, journal issue in original printed wrappers, of Mandelbrot’s first paper on fractals (a term he coined in 1975). “Today Mandelbrot’s paper on the coast of Britain is famous in the history of mathematics” (historyofscience.com). “Mandelbrot had come across the coastline question in an obscure posthumous article by an English scientist, Lewis F. Richardson, who groped with a surprising number of the issues that later became part of chaos [theory] … Wondering about coastlines, Richardson checked encyclopaedias in Spain and Portugal, Belgium and the Netherlands, and discovered discrepancies of 20% in the estimated lengths of their common frontiers … [Mandelbrot] argued [that] … the answer depends on the length of your ruler. Consider one plausible method of measuring. A surveyor takes a set of dividers, opens them to a length of one yard, and walks them along the coastline. The resulting number of yards is just an approximation of the true length, because the dividers skip over twists and turns smaller than one yard, but the surveyor writes the number down anyway. Then he sets the dividers to a smaller length – say, one foot – and repeats the process. He arrives at a somewhat greater length, because the dividers will capture more of the detail and it will take more than three one-foot steps to cover the distance previously covered by a one-yard step. He writes this new number down, sets the dividers at four inches, and starts again … Common sense suggests that, although these estimates will continue to get larger, they will approach some particular final value, the true length of the coastline … if a coastline were some Euclidean shape, such as a circle, this method of summing finer and finer straight-line distances would indeed converge. But Mandelbrot found that as the scale of measurement becomes smaller, the measured length of a coastline rises without limit” (Gleick, pp. 94-96). A copy of Richardson’s ‘obscure’ article, posthumously published in 1961 although written in the 1920s, accompanies Mandelbrot’s article here. Richardson proposed, in section 7 (‘Lengths of land frontiers or seacoasts’) of his article, that the measured length of the coastline should be proportional to G1 – D, where G is the length of the ruler and D is a number, possibly fractional, greater than or equal to 1. On p. 636 of his article, Mandelbrot notes that: “Such a formula, of an entirely empirical character, was proposed by Lewis F. Richardson [in the offered paper] but unfortunately it attracted no attention.” Mandelbrot suggests that D should be regarded as the dimension of the coastline – it is now known as the ‘fractal dimension’. “Although the key concepts associated with fractals had been studied for years by mathematicians, and many examples, such as the Koch ‘snowflake’ curve were long known, Mandelbrot was the first to point out that fractals could be an ideal tool in applied mathematics for modeling a variety of phenomena from physical objects to the behavior of the stock market. Since its introduction in 1975, the concept of the fractal has given rise to a new system of geometry that has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics. Many fractals possess the property of self-similarity, at least approximately, if not exactly. A self-similar object is one whose component parts resemble the whole. This reiteration of details or patterns occurs at progressively smaller scales and can, in the case of purely abstract entities, continue indefinitely, so that each part of each part, when magnified, will look basically like a fixed part of the whole object … This fractal phenomenon can often be detected in such objects as snowflakes and tree barks. All natural fractals of this kind, as well as some mathematical self-similar ones, are stochastic, or random; they thus scale in a statistical sense” (Britannica).

“The paper examines the coastline paradox: the property that the measured length of a stretch of coastline depends on the scale of measurement … Th[e] discussion implies that it is meaningless to talk about the length of a coastline; some other means of quantifying coastlines are needed. Mandelbrot discusses an empirical law discovered by Lewis Fry Richardson (1881-1953), who observed that the measured length L(G) of various geographic borders was a of the measurement scale G. Collecting data from several different examples, Richardson conjectured that L(G) could be closely approximated by a function of the form

L(G) = MG1 – D

where M is a positive constant and D is a constant, called the dimension, greater than or equal to 1 [now known as the ‘fractal dimension’]. Intuitively, if a coastline looks smooth it should have dimension close to 1; and the more irregular the coastline looks the closer its dimension should be to 2. The examples in Richardson’s research have dimensions ranging from 1.02 for the coastline of South Africa to 1.25 for the West coast of Britain.

“Mandelbrot then describes various mathematical curves, related to the ‘Koch snowflake,’ which are defined in such a way that they are strictly self-similar. Mandelbrot shows how to calculate the Hausdorff dimension of each of these curves, each of which has a dimension D between 1 and 2 (he also mentions but does not give a construction for the space-filling ‘Peano curve,’ which has a dimension exactly 2). He notes that the approximation of these curves with segments of length G have lengths of the form G1 – D. The The resemblance with Richardson's law is striking. The paper does not claim that any coastline or geographic border actually has fractional dimension. Instead, it notes that Richardson's empirical law is compatible with the idea that geographic curves, such as coastlines, can be modelled by random self-similar figures of fractional dimension.

“Near the end of the paper Mandelbrot briefly discusses how one might approach the study of fractal-like objects in nature that look random rather than regular. For this he defines statistically self-similar figures and says that these are encountered in nature.

“The paper is important because it is a ‘turning point’ in Mandelbrot’s early thinking on fractals. It is an example of the linking of mathematical objects with natural forms that was a theme of much of his later work” (Wikipedia).

“When the present paper was published it was mainly misunderstood. Years later Mandelbrot had this to say about the origin of the paper:

‘By the mid-1960s my record of publications was substantial but presented a serious flaw. Those publications' topics ranged all too widely and were perceived as an aimless juxtaposition of studies of noise, turbulence, galaxy clustering, prices and river discharges. Few persons realized that, to the contrary, I did not deserve to be criticized for immature aimlessness but for increasingly acute single-mindedness. As early as 1956 … then increasingly and more seriously in my works on finance and on noise, I had somehow latched on the process of renormalization and found it useful in very diverse contexts. Unfortunately, the nature and worth of that concept was not appreciated until much later when it was rediscovered quite independently in the statistical physics of critical phenomena that arose in 1972.

‘More specifically, nearly all my works were linked by the ubiquity of ‘power-law’ relations, each endowed with an important exponent. Superficially those exponents seemed both formal and mutually unrelated. But in fact I knew how to interpret them geometrically as ‘the’ fractal dimension of suitable sets. Furthermore, this interpretation gave to my work a profound unity that promised further growth. But I soon found out that mention of a fractal dimension in a paper or a talk led all referees and editors to their pencils, and some audiences to audible signs of disapproval. Practitioners accused me of hiding behind formulas that were purposefully incomprehensible. Few mathematicians knew any of the flavors of fractal dimension; if asked, they were worse than useless in explain this notion to those I was trying to convert …

‘Fortunately, I stumbled one day upon Richardson’s empirical data on coastline lengths, and recognized instantly that a study of coastlines might lend itself to a ‘Trojan horse’ manoeuver. Indeed, everyone has knowledge of geography, but no one I knew professionally had a vested professional interest in facts and theories concerning coastlines and relief. The manoeuver succeeded. Everyone was wonderfully objective and receptive to the seemingly wild idea contained this paper, and as a result, became more receptive to the use of fractal dimension in fields that really matter to me.’

“In his 1975 French book, Les objets fractals: Forme, hasard et dimension, Mandelbrot first coined the term ‘fractal.’ He revised, expanded and translated these ideas in his 1977 English language book, Fractals: Form, Chance and Dimension … In 1999 American Scientist magazine stated that Mandelbrot’s 1977 book was ‘one of the hundred most influential science books’ of the 20th century … Regarding Mandelbrot’s 1977 book Freeman Dyson wrote in 1978:

‘Fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] … an historical role … in the development of pure mathematics. A great revolution of ideas separates the classical mathematics of the 19thcentury from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton. Modern mathematics began with Cantor’s set theory and Peano’s space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded … as ‘pathological,’ … as a ‘gallery of monsters,’ akin to the cubist paintings and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitations imposed by its natural origins.

‘Now, as Mandelbrot points out … Nature has played a joke on the mathematicians. The 19th century mathematicians may have been lacking in imagination, but Nature was not. The same pathological structures that the mathematicians invented to break loose from the 19th century naturalism turn out to be inherent in familiar objects all around us (Dyson, ‘Characterizing Irregularity,’ Science 200, 4332 (1978), 677-78)” (historyofscience.com).

Lewis Fry Richardson (1881-1953) “while undeservedly little known, had a fundamental (often posthumous) role in twentieth-century science. Even though his name is linked to several important results in fluid dynamics, in meteorology, and in numerical analysis (let us just recall his stability criterion for fluids, his idea of a scale-dependent diffusion coefficient, and the algorithm that bears his name, which is still used to integrate differential equations), physicists and mathematicians themselves often don’t know who he was. His originality was too often mistaken for eccentricity, but some of the ideas and methods that he conceived would be rediscovered only decades later” (Vulpiani, p. 121).

https://www.historyofscience.com/pdf/The%20Scope%20of%20Benoit%20Mandelbrot%27s%20Work%20and%20its%20Influence.pdf. Gleick, Chaos, 1998. Vulpiani, ‘Lewis Fry Richardson: scientist, visionary and pacifist,’ Lettera Matematica 2 (2014), pp. 121-128.

[Mandelbrot:] Large 8vo, pp. 573-678. Original printed wrappers (address label on rear wrapper, a little soiled)). [Richardson:] 4to, pp. [v], [3], 206, [2, blank]. Original stiff printed wrappers (a little soiled).

Item #5204

Price: $2,000.00