Coastline paradox
Encyclopedia
The coastline paradox is the counterintuitive
observation that the coastline of a landmass
does not have a well-defined length. This results from the fractal
-like properties of coastlines. It was first observed by Lewis Fry Richardson
.
More concretely, the length of the coastline depends on the method used to measure it. 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 limit to the size of the smallest feature that should not be measured around, and hence no single well-defined perimeter to the landmass. Various approximations exist when specific assumptions are made about minimum feature size.
For practical considerations, an appropriate choice of minimum feature size is on the order of the units being used to measure. If a coastline is measured in kilometers, then small variations much smaller than one kilometer are easily ignored. To measure the coastline in centimeters, tiny variations of the size of centimeters must be considered. However, at scales on the order of centimeters various arbitrary and non-fractal assumptions must be made, such as where an estuary
joins the sea, or where in a broad tidal flat the coastline measurements ought to be taken.
Extreme cases of the coastline paradox include the fjord
-heavy coastlines of Norway
, Chile
and the Pacific Northwest
of North America
. From the southern tip of Vancouver Island
northwards to the southern tip of the Alaska Panhandle
, the convolutions of the coastline of the Canadian province of British Columbia
make it over 10% of the entire Canadian coastline—25725 km (15,984.8 mi) vs 243042 km (151,019.7 mi) over a linear distance of only 965 km (599.6 mi), including the maze of islands of the Arctic archipelago.
Counterintuitive
The word "counterintuitive" literally means counter to intuition, and so it essentially means that something does not seem right or correct.A counterintuitive proposition is one that does not seem likely to be true when assessed using intuition or gut feelings...
observation that the coastline of a landmass
Landmass
A landmass is a contiguous area of land surrounded by ocean. Although it may be most often written as one word to distinguish it from the usage "land mass"—the measure of land area—it is also used as two words.Landmasses include:*supercontinents...
does not have a well-defined length. This results from the fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
-like properties of coastlines. It was first observed by Lewis Fry Richardson
Lewis Fry Richardson
Lewis Fry Richardson, FRS was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...
.
More concretely, the length of the coastline depends on the method used to measure it. 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 limit to the size of the smallest feature that should not be measured around, and hence no single well-defined perimeter to the landmass. Various approximations exist when specific assumptions are made about minimum feature size.
For practical considerations, an appropriate choice of minimum feature size is on the order of the units being used to measure. If a coastline is measured in kilometers, then small variations much smaller than one kilometer are easily ignored. To measure the coastline in centimeters, tiny variations of the size of centimeters must be considered. However, at scales on the order of centimeters various arbitrary and non-fractal assumptions must be made, such as where an estuary
Estuary
An estuary is a partly enclosed coastal body of water with one or more rivers or streams flowing into it, and with a free connection to the open sea....
joins the sea, or where in a broad tidal flat the coastline measurements ought to be taken.
Extreme cases of the coastline paradox include the fjord
Fjord
Geologically, a fjord is a long, narrow inlet with steep sides or cliffs, created in a valley carved by glacial activity.-Formation:A fjord is formed when a glacier cuts a U-shaped valley by abrasion of the surrounding bedrock. Glacial melting is accompanied by rebound of Earth's crust as the ice...
-heavy coastlines of Norway
Norway
Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...
, Chile
Chile
Chile ,officially the Republic of Chile , is a country in South America occupying a long, narrow coastal strip between the Andes mountains to the east and the Pacific Ocean to the west. It borders Peru to the north, Bolivia to the northeast, Argentina to the east, and the Drake Passage in the far...
and the Pacific Northwest
Pacific Northwest
The Pacific Northwest is a region in northwestern North America, bounded by the Pacific Ocean to the west and, loosely, by the Rocky Mountains on the east. Definitions of the region vary and there is no commonly agreed upon boundary, even among Pacific Northwesterners. A common concept of the...
of North America
North America
North America is a continent wholly within the Northern Hemisphere and almost wholly within the Western Hemisphere. It is also considered a northern subcontinent of the Americas...
. From the southern tip of Vancouver Island
Vancouver Island
Vancouver Island is a large island in British Columbia, Canada. It is one of several North American locations named after George Vancouver, the British Royal Navy officer who explored the Pacific Northwest coast of North America between 1791 and 1794...
northwards to the southern tip of the Alaska Panhandle
Alaska Panhandle
Southeast Alaska, sometimes referred to as the Alaska Panhandle, is the southeastern portion of the U.S. state of Alaska, which lies west of the northern half of the Canadian province of British Columbia. The majority of Southeast Alaska's area is part of the Tongass National Forest, the United...
, the convolutions of the coastline of the Canadian province of British Columbia
British Columbia Coast
The British Columbia Coast or BC Coast is Canada's western continental coastline on the Pacific Ocean. The usage is synonymous with the term West Coast of Canada....
make it over 10% of the entire Canadian coastline—25725 km (15,984.8 mi) vs 243042 km (151,019.7 mi) over a linear distance of only 965 km (599.6 mi), including the maze of islands of the Arctic archipelago.
See also
- How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional DimensionHow Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2...
- Fractal dimensionFractal dimensionIn fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
- Paradox of the HeapSorites paradoxThe sorites paradox is a paradox that arises from vague predicates. The paradox of the heap is an example of this paradox which arises when one considers a heap of sand, from which grains are individually removed...