The Unmeasurable Shore, Geometry and the Dynamic Coastline

“ Moreover, it turns out that the more rugged a coastline is, the higher fractal dimension it has: coasts with outlines that are very complex have higher fractal dimension than coasts that are relatively simple and smooth.
The most serious problem with using fractal dimension as a general measure of complexity is that it seems to chiefly be quantifying a fact about how complex an object’s spatial configuration is: the statistical self-similarity that both broccoli and coastlines show is a self-similarity of shape. ”

“ But as to those who keep at any considerable distance from the coast, there is no other method to be taken for finding the place than that of making it by the latitude, for there are so many ranges of mountains rising one upon the back of another within land, that no drawings of the appearance of the coast can be at all depended on when off {241} at sea, every little change of distance or variation of position bringing new mountains in view, and producing an infinity of different prospects, which render all attempts of delineating the aspect of the coast impossible. ”

“ The remarkable contrast between the scenery of the eastern and western coast-line of the British Islands arises partly from the preponderance of harder rocks on the west side, but probably in large measure upon the greater extent of the submergence of the western seaboard, whereby the sea has been allowed to penetrate far inland by fjords which were formerly glens and open straths.
The details of coast-scenery vary with the rock in which they are developed. Nowhere can the effects of each leading type of rock upon landscape be more instructively studied than along the sea-margin. ”

“ Where hard rocks form the coast-line the rate of waste is often exceedingly slow, and centuries may elapse without any apparent change being effected. When the rocks are of unequal hardness the coast-line becomes very irregular, the sea carving out bays and gullies in the softer portions, while the more durable masses stand out as capes and bold headlands. ”

“ The western shore of the Gulf of Carpentaria is somewhat higher, and from Limmen's Bight to the latitude of Groote Eylandt, is lined by a range of low hills. On the north of the latter place, the coast becomes irregular and broken; the base of the country apparently consisting of primitive rocks, and the upper part of the hills of a reddish sandstone; some of the specimens of which are identical with that which occurs at Goulburn and Sims Islands on the north coast, and is very widely distributed on the north-west. ”

“ Every division plane, whether due to bedding, to jointing, or to irregular fracture, is a source of weakness to the rock, and rock of great hardness may be so broken as to offer relatively little resistance. Inequalities of resistance, whatever their cause, give origin to inequalities of coastal configuration where wave-erosion is in progress. Given a coast of marked regularity and equal exposure, but composed of unequally resistant material, the waves will make it irregular by cutting most where the material is least resistant. ”

“ Still, there’s a clear parallel to standard statistical self-similarity: fractal dimension for normal physical objects roughly quantifies how much interesting spatial detail persists between magnification operations, and how much magnification one must do move from one level of detail to another. ”

“ The utter absence of any regularity or assimilation to the spheroidal figure, either in meridianal, equatorial or parallel lines, mountain ranges, sea beaches or courses of rivers, is fatal to mathematical accuracy in the more extended measurements. It is only by taking the mean of a great many measurements that an approximate accuracy can be obtained. ”

“ What aspects of fractal dimensionality does dynamical complexity incorporate? To begin, it might help to recall why fractal dimensionality by itself doesn’t work as a definition of complexity. Most importantly, recall that fractal dimensionality is a static notion—a fact about the shape of an object—not a dynamical one. We’re interested in systems, though, not static objects—science deals with how systems change over time. On the face of it, fractal dimensionality doesn’t have the resources to deal with this: it’s a geometrical concept properly applied to shapes. ”

“ When a considerable part of a coast-line formerly existing above tide-marks is found to gradually sink below the sea-level, the observer is probably apt to assume that the sea has simply altered its level. The idea of the sea being a constantly changing body is so widely entertained, and that of the land being a solid and immovable portion of the constitution of the earth, is also so deeply rooted in the popular mind, that it may take some little thinking to throw on the land the burden of the change and alteration. ”

“ Under examination, even our familiar term coastline proves to be only an abstraction with no corresponding reality in nature. Everywhere, whether on margin of lake or gulf, the actual phenomenon is a coast zone, alternately covered and abandoned by the waters, varying in width from a few inches to a few miles, according to the slope of the land, the range of the tide and the direction of the wind. ”

“ If the sea be imagined as rising 100 ft., a new coast-line, with bays and estuaries indented in the valleys, would appear at the new sea-level. If the sea sank once more to its former level, the 100-ft. contour-line with all its irregularities would be represented by the beach mark made by the sea when 100 ft. higher. If instead of receding the sea rose continuously at the rate of 100 ft. per day, a series of levels 100 ft. above one another would be marked daily upon the land until at last the highest mountain peaks appeared as islands less than 100 ft. ”

“ A subsidence of the land adjacent to the sea permits an extension of the waters landward; the sea will enter the valleys so as to form estuaries, bays, straits, etc., while the high land between the partially water-filled depressions will rise above the water-level and appear as peninsulas, capes, and islands. A bold, deeply sculptured coast when depressed will give origin to an intricate, and what may be termed a ragged shore-line; while a lower region crossed by large river-valleys would be changed to a system of broad estuaries. ”

“ The northwest coast, in common with all parts of the globe, has been subject to great and frequent oscillations of level, epochs of subsidence and upheaval being well marked in the Tertiary and post-Tertiary or latest geological age. These oscillations sunk the land below the surface of the ocean many thousand feet, raising it again to present elevations, as is shown by the abundance of fossil marine life on the summits of very high mountains. I have gathered shells of clams, identical with existing species, on the summit of Bald mountain, near Port Orford, 3,000 feet above the sea level. ”

“ Everywhere throughout the world we read the same tale of subsidence and accumulation, of upheaval and denudation. The ancient sedimentary deposits which form the major portion of our land-surfaces, are the — 61 — waste materials derived from the demolition of plains, plateaux, and mountains of elevation. In some mountain-regions we read the evidence of successive epochs of uplift, separated by long intervening periods of erosion, followed by depression and accumulation of newer sediments over the denuded surface. Thus the Alps began to be elevated towards the close of Palæozoic times. ”

“ Not only are inequalities necessary to the existence of land, but these inequalities must be renewed from time to time, or the land area would soon, geologically speaking, be covered by the sea. The renewal has been made again and again in geological history by movements that have increased the inequalities in the surface of the lithosphere. With each such movement, apparently, the oceans have withdrawn more completely within the basins, and the continents have stood forth more broadly and relatively higher, until again worn down. ”

“ And, more powerful than any of these agents of denudation, the waves and the tides are still at work along every coast-line, eating away into the cliffs, undermining gradually and submerging acre after acre, and making with the refuse a shingly, or a sandy, or a muddy beach—the nucleus of a new geological formation.
Of all denuding agents, there can be no doubt that, to the land exposed to them, the waves of the sea are by far the most powerful. ”

“ The erosion of the land is constantly going on, every shower carries its tribute of earthy matter toward the sea, and every wave that strikes against a beach or cliff does some work toward the grinding of shells, pebbles or stone. Thus, everywhere around our continents there is a continuous deposition of beds of earthy matter, and it is this which, when elevated into new land, has given us our chronological series of geological formations. ”

“ But we cannot watch the slow upheaval of a long line of coast, where the fisherman hardly knows at the end of a lifetime whether the sea has drawn back or his own landmarks have been moved; we are all unable to note how new continents are now being formed in the ocean's stillest [Pg 7] depths, from whose hardened and uplifted strata future ages may dig out the relics of so much that has been dear and precious to us ”

“ A Progressive Change is evidently taking place in the boundary line of the coast—the sea making considerable invasions on the south side, which is exposed to the resistless currents of the ocean; while on the north it is found to be more gradually receding, from the accumulation of sand and shingle drifted and deposited by the less impetuous tides of the Solent Channel.—About Brixton, for instance, between Blackgang Chine and the Freshwater Cliffs, the loss of land has been estimated (from the successive removals of paths and hedges,) to exceed 200 feet in breadth in less than a century ”

“ The particular features of the coast are for the most part unnoticed. Long distances, embracing from two hundred to six hundred miles each, are passed over with little or no remark. Islands, rivers, capes, bays, and other land or seamarks, by which navigators usually describe their progress along an unknown coast, are almost entirely unmentioned. For a distance of over two thousand miles, adopting the narrowest limits possible assigned to the discovery, only one island, one river, and one bay are attempted to be described, and not a single cape or headland is referred to. ”

“ The surveys of such localities will only hold good in their general features; in the shore-lines and in the landmarks by which a vessel may approach and feel her way in; the more frequented harbors of this nature require reëxamination from time to time.
Several nations have provided for a trigonometrical survey of their coasts only, in advance of geodetical operations embracing their entire domain. ”

“ Granting that the geoid or mean surface of the ocean is a uniform spheroid, the distribution of land and water approximately indicates a division of the surface of the globe into two areas, one of elevation and one of depression. The increasing number of measurements of the height of land in all continents and islands, and the very detailed levellings in those countries which have been thoroughly surveyed, enable the average elevation of the land above sea-level to be fairly estimated, although many vast gaps in accurate knowledge remain, and the estimate is not an exact one. ”

“ It is the coast as a human habitat that primarily concerns anthropo-geography. A careful analysis of the multifarious influences modifying one another in this mingled environment of land and water reveals an intricate interplay of geographic forces, varying from inland basin to marginal sea, from marginal sea to open ocean, and changing from one historical period to another—an interplay so mercurial that it could find only a most inadequate expression in the rigid mathematical formula of Carl Ritter. ”

“ We could develop a geometry in which distances which seem subjectively equal to our eye are called equal, and upon this assumption we would be able to develop a self-consistent system or science. Such a geometry, however, would have an extremely complex and impractical structure for objective purposes (as, for example, land measurement) , and so we strive to develop a science as free as possible from subjective factors. Historically, we have before us a process of this sort in the astronomy of Ptolemy and that of Copernicus. ”

“ We determine the length of the great arc on the surface of the earth by triangulation, which depends upon a base measured with exactitude. But whatever precision may be brought to the measure of the angles, the inevitable errors can, by accumulating, cause the value of the arc concluded from a great number of triangles to deviate appreciably from the truth. We recognize this value, then, only imperfectly unless the probability that its error is comprised within given limits can be assigned. The error of a geodetic result is a function of the errors of the angles of each triangle. ”

“ Mathematical space is amorphous; it possesses no intrinsic metrics, and our choice of standards of measurement is largely arbitrary. As a result, absolute shape, size and straightness are meaningless concepts. But in physical space, it is a matter of common knowledge that men have no difficulty in agreeing at least approximately on the sameness of two shapes or of two sizes. They agree that a stone remains undeformed when displaced, hence declare the stone to be rigid, whereas they recognise that an object behaving like a worm is of the squirming variety. ”

“ When we revert to experience for an understanding of congruence, we find it exemplified in the rigid bodies of nature, whose geometrical dispositions yield, more or less precisely, the results of pure Euclidean geometry. If we idealise congruence, as thus defined, and express it mathematically, we may say that perfectly rigid bodies are those whose measurements would yield Euclidean results with absolute precision. ”