Misplaced Pages

#345654

40-588: The Damariscotta River begins at the outlet of Damariscotta Lake , at Damariscotta Mills, a village straddling the boundary between the towns of Newcastle and Nobleboro . Damariscotta Lake extends 12 miles (19 km) north into the town of Jefferson and is fed from tributaries originating as far north as Washington and Somerville, Maine . From the lake's outlet, the Damariscotta River drops 50 feet (15 m) over just 0.1 miles (0.16 km) through Damariscotta Mills before reaching tidewater, at an arm of

80-444: A coastline, and using mathematical integers within these practical limitations to calculate the length to a meaningful level of precision. These practical solutions to the problem can resolve the problem for all practical applications while it persists as a theoretical/mathematical concept within our models. The coastline paradox is often criticized because coastlines are inherently finite, real features in space, and, therefore, there

120-495: A decade after Richardson completed his work, Benoit Mandelbrot developed a new branch of mathematics , fractal geometry , to describe just such non-rectifiable complexes in nature as the infinite coastline. His own definition of the new figure serving as the basis for his study is: I coined fractal from the Latin adjective fractus . The corresponding Latin verb frangere means "to break:" to create irregular fragments. It

160-435: A given coastline has this property of self-similarity, then no matter how great any one small section of coastline is magnified, a similar pattern of smaller bays and promontories superimposed on larger bays and promontories appears, right down to the grains of sand. At that scale the coastline appears as a momentarily shifting, potentially infinitely long thread with a stochastic arrangement of bays and promontories formed from

200-422: A matter of fierce debate; furthermore, the problem extends to demarcating territorial boundaries , property rights , erosion monitoring , and the theoretical implications of our geometric modelling . To resolve this problem, several solutions have been proposed. These solutions resolve the practical problems around the problem by setting the definition of "coastline," establishing the practical physical limits of

240-408: A smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve . Benoit Mandelbrot showed that D is independent of ε . The basic concept of length originates from Euclidean distance . In Euclidean geometry, a straight line represents the shortest distance between two points . This line has only one length. On the surface of

280-411: A sphere, this is replaced by the geodesic length (also called the great circle length), which is measured along the surface curve that exists in the plane containing both endpoints and the center of the sphere. The length of basic curves is more complicated but can also be calculated. Measuring with rulers, one can approximate the length of a curve by adding the sum of the straight lines which connect

320-508: A surface varies depending on the measurement resolution. Shortly before 1951, Lewis Fry Richardson , in researching the possible effect of border lengths on the probability of war, noticed that the Portuguese reported their measured border with Spain to be 987 km (613 mi), but the Spanish reported it as 1,214 km (754 mi). This was the beginning of the coastline problem, which

360-592: A useful model in everyday measurement—is a matter of philosophical speculation, and may or may not reflect the changing realities of "space" and "distance" on the atomic level (approximately the scale of a nanometer ). Coastlines are less definite in their construction than idealized fractals such as the Mandelbrot set because they are formed by various natural events that create patterns in statistically random ways, whereas idealized fractals are formed through repeated iterations of simple, formulaic sequences. More than

400-562: Is not a well-defined measure . Damariscotta Lake is a lake in Lincoln County, Maine . Damariscota Lake, which covers 4,300 acres (1,700 ha), encompasses part of the towns of Jefferson , Nobleboro , and Newcastle . Damariscotta Lake State Park in Jefferson occupies 19 acres at the far northern end of the lake. The lake is known for its annual run of alewives and the dam and fish ladder built in 1807 at Damariscotta Mills at

440-407: Is a mathematical uncertainty inherent in the measurement of boundaries that are irregular. The prevailing method of estimating the length of a border (or coastline) was to lay out n equal straight-line segments of length l with dividers on a map or aerial photograph. Each end of the segment must be on the boundary. Investigating the discrepancies in border estimation, Richardson discovered what

SECTION 10

#1732855362346

480-522: Is a quantifiable answer to their length. The comparison to fractals, while useful as a metaphor to explain the problem, is criticized as not fully accurate, as coastlines are not self-repeating and are fundamentally finite. The source of the paradox is based on the way we measure reality and is most relevant when attempting to use those measurements to create cartographic models of coasts. Modern technology, such as LiDAR , Global Positioning Systems and Geographic Information Systems , has made addressing

520-484: Is because one would be laying the ruler along a more curvilinear route than that followed by the yardstick. The empirical evidence suggests a rule which, if extrapolated, shows that the measured length increases without limit as the measurement scale decreases towards zero. This discussion implies that it is meaningless to talk about the length of a coastline; some other means of quantifying coastlines are needed. Mandelbrot then describes various mathematical curves, related to

560-415: 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

600-431: Is different from Wikidata Pages using infobox body of water with auto short description Coordinates on Wikidata Articles using infobox body of water without alt Articles using infobox body of water without image bathymetry Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from

640-426: Is fundamentally different from the measurement of other, simpler edges. It is possible, for example, to accurately measure the length of a straight, idealized metal bar by using a measurement device to determine that the length is less than a certain amount and greater than another amount—that is, to measure it within a certain degree of uncertainty . The more precise the measurement device, the closer results will be to

680-480: 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. A key property of some fractals is self-similarity ; that is, at any scale the same general configuration appears. A coastline is perceived as bays alternating with promontories. In the hypothetical situation that

720-440: Is important in local commerce for tourism, Oyster and Mussel Farming as well as other forms of aquaculture, clamming, marine worming and fishing. Damariscotta Mills is a place where locals gather every year (May–June) to celebrate the “Annual Alewife Festival” which sees millions of alewives returning to the Damariscotta River. This festival is attracting more people through the years. The centerpiece of this annual alewife festival

760-405: Is longer for the same ε . D is approximately 1.02 for the coastline of South Africa , and approximately 1.25 for the west coast of Great Britain. For lake shorelines, the typical value of D is 1.28. The coastline paradox describes a problem with real-world applications, including trivial matters such as which river , beach , border , coastline is the longest, with the former two records

800-403: Is now termed the "Richardson effect": the sum of the segments monotonically increases when the common length of the segments decreases. In effect, the shorter the ruler, the longer the measured border; the Spanish and Portuguese geographers were simply using different-length rulers. The result most astounding to Richardson is that, under certain circumstances, as l approaches zero, the length of

840-472: Is the newly rebuilt/renovated Damariscotta Stone Fish Ladder. This fish ladder was constructed in 1807 to bypass the Damariscotta Mills Dam but the fish ladder was falling into dis-repair by 2007. With local community support and financing this unique stone fish ladder was fully re-constructed between 2007 and 2013. This reconstitution has once again allowed alewives to increase into the millions along

SECTION 20

#1732855362346

880-457: Is therefore sensible ... that, in addition to "fragmented" ... fractus should also mean "irregular". In " How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension ", published on 5 May 1967, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals , although Mandelbrot does not use this term in

920-595: 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). The paper does not claim that any coastline or geographic border actually has fractional dimension. Instead, it notes that Richardson's empirical law

960-411: The fractal curve –like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension . Although the "paradox of length" was previously noted by Hugo Steinhaus , the first systematic study of this phenomenon was by Lewis Fry Richardson , and it was expanded upon by Benoit Mandelbrot . The measured length of the coastline depends on the method used to measure it and

1000-523: The Damariscotta. 43°49′N 69°34′W  /  43.82°N 69.57°W  / 43.82; -69.57  ( Damariscotta River ) This article related to a river in Maine is a stub . You can help Misplaced Pages by expanding it . Damariscotta Lake Coordinates : 44°08′49″N 69°29′10″W  /  44.147°N 69.486°W  / 44.147; -69.486 From Misplaced Pages,

1040-508: The Richardson effect is: where L , coastline length, a function of the measurement unit ε , is approximated by the expression. F is a constant, and D is a parameter that Richardson found depended on the coastline approximated by L . He gave no theoretical explanation, but Mandelbrot identified D with a non-integer form of the Hausdorff dimension , later the fractal dimension. Rearranging

1080-411: The calculation of infinitesimally small distances. The following animation illustrates how a smooth curve can be meaningfully assigned a precise length: Not all curves can be measured in this way. A fractal is, by definition, a curve whose perceived complexity does not decrease with measurement scale. Whereas approximations of a smooth curve tend to a single value as measurement precision increases,

1120-411: The coastline approaches infinity . Richardson had believed, based on Euclidean geometry, that a coastline would approach a fixed length, as do similar estimations of regular geometric figures. For example, the perimeter of a regular polygon inscribed in a circle approaches the circumference with increasing numbers of sides (and decrease in the length of one side). In geometric measure theory such

1160-443: The degree of cartographic generalization . Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be taken into consideration when measuring, and hence no single well-defined perimeter to the landmass. Various approximations exist when specific assumptions are made about minimum feature size. The problem

1200-419: The expression yields where Fε must be the number of units ε required to obtain L . The broken line measuring a coast does not extend in one direction nor does it represent an area, but is intermediate between the two and can be thought of as a band of width 2 ε . D is its fractal dimension, ranging between 1 and 2 (and typically less than 1.5). More broken coastlines have greater D , and therefore L

1240-956: The 💕 Lake in Lincoln County, Maine Damariscotta Lake [REDACTED] View of Damariscotta Lake in Nobleboro, Maine [REDACTED] [REDACTED] Damariscotta Lake Location Lincoln County, Maine Coordinates 44°08′49″N 69°29′10″W  /  44.147°N 69.486°W  / 44.147; -69.486 Type Lake Primary outflows Damariscotta River Max. length 10 miles (16 km) Max. width 1.9 miles (3.1 km) Surface area 4,381 acres (1,773 ha) Average depth 30 feet (9.1 m) Max. depth 114 feet (35 m) Shore length 21.4 miles (34.4 km) Surface elevation 52 feet (16 m) Islands Spectacle Islands Shore length

Damariscotta River - Misplaced Pages Continue

1280-925: The lake's southern tip. References [ edit ] ^ "Damariscotta Lake" . Geographic Names Information System . United States Geological Survey , United States Department of the Interior . ^ "Damariscotta Lake Survey" (PDF) . Lincoln County Lake Survey Maps . Maine Department of Inland Fisheries and Wildlife . Retrieved June 19, 2020 . ^ "Damariscotta Lake" . Lakes of Maine . Retrieved June 19, 2020 . ^ "Damariscotta Lake Region" . Midcoast Conservancy . Retrieved June 19, 2020 . ^ "Damariscotta Lake State Park" . Department of Agriculture, Conservation and Forestry . Retrieved June 19, 2020 . ^ Ken Textor (February 13, 2002). "Focus on: Damariscotta Lake, Maine" . New England Boating . Lighthouse Media Solutions. Archived from

1320-466: The measured value for a fractal does not converge. As the length of a fractal curve always diverges to infinity, if one were to measure a coastline with infinite or near-infinite resolution, the length of the infinitely short kinks in the coastline would add up to infinity. However, this figure relies on the assumption that space can be subdivided into infinitesimal sections. The truth value of this assumption—which underlies Euclidean geometry and serves as

1360-535: The original on March 22, 2015 . Retrieved September 5, 2014 . ^ Julia McLeod; Mary Sheldon; Marilyn Speckmann (2011). Damariscotta Lake . Arcadia Publishing. p. 7. ISBN   978-0-7385-7509-4 . Retrieved from " https://en.wikipedia.org/w/index.php?title=Damariscotta_Lake&oldid=1077670740 " Categories : Lakes of Maine Lakes of Lincoln County, Maine Hidden categories: Pages using gadget WikiMiniAtlas Articles with short description Short description

1400-417: The paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals. Empirical evidence suggests that the smaller the increment of measurement, the longer the measured length becomes. If one were to measure a stretch of coastline with a yardstick , one would get a shorter result than if the same stretch were measured with a 1-foot (30 cm) ruler . This

1440-445: The paradox much easier; however, the limitations of survey measurements and the vector software persist. Critics argue that these problems are more theoretical and not practical considerations for planners. Alternately, the concept of a coast "line" is in itself a human construct that depends on assignment of Tidal datum which is not flat relative to any Vertical datum , and thus any line constructed between land and sea somewhere in

1480-426: The points: Using a few straight lines to approximate the length of a curve will produce an estimate lower than the true length; when increasingly short (and thus more numerous) lines are used, the sum approaches the curve's true length, and that length is the least upper bound or supremum of all such approximations. A precise value for this length can be found using calculus , the branch of mathematics enabling

1520-660: The river known as Salt Bay. The tidal Damariscotta flows southward between Newcastle, Edgecomb and Boothbay on the west and Damariscotta , Bristol and South Bristol on the east, reaching the Atlantic Ocean between Linekin Neck on the west and Inner Heron Island on the east. It is a navigable river for nearly its entire 19-mile (31 km) length, to the bridge between Newcastle and Nobleboro ( 44°03′36″N 69°31′30″W  /  44.060°N 69.525°W  / 44.060; -69.525  ( Damariscotta River head ) ). It

1560-496: The small objects at hand. In such an environment (as opposed to smooth curves) Mandelbrot asserts "coastline length turns out to be an elusive notion that slips between the fingers of those who want to grasp it". There are different kinds of fractals. A coastline with the stated property is in "a first category of fractals, namely curves whose fractal dimension is greater than 1". That last statement represents an extension by Mandelbrot of Richardson's thought. Mandelbrot's statement of

1600-404: The true length of the edge. When measuring a coastline, however, the closer measurement does not result in an increase in accuracy—the measurement only increases in length; unlike with the metal bar, there is no way to obtain an exact value for the length of the coastline. In three-dimensional space, the coastline paradox is readily extended to the concept of fractal surfaces , whereby the area of

#345654