Since land has features at all levels, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there are no obvious limits on the size of the smallest features, and hence no well-defined land perimeter has been recorded. Various approximations exist under certain minimum size assumptions.
An example of a paradox is the well-known UK coast. If the UK coastline is measured using fractal units of 100 km (62 miles) in length, then the length of the coastline is about 2,800 km (1,700 miles). With a unit of 50 km (31 mi), the total length is about 3,400 km (2,100 mi), about 600 km (370 mi) longer.
Mathematical aspects
The basic concept of length comes from Euclidean distance. In a familiar euclidean geometry, the straight line represents shortest distance between two points; this line has only one finite length. The geodesic length on the surface of a sphere, called the great circle length, is measured from the surface of a curve that exists in a plane containing the endpoints of the path and the center of the sphere. The length of the main curve is more complex but can also be calculated. When measuring with a ruler, one can approximate the lengths of a curve by adding the sum of straight lines connecting the points:
Using several straight lines close to the length of the curve will produce a low estimate. More and more use short lines will produce a sum of lengths that approximates the true length of the curve. The exact value of this length can be established using calculus, a branch of mathematics that allows you to calculate infinitely small distances. The following animation illustrates this example:
However, not all curves can be measured in this way. By definition, a curve is considered a fractal, with complex changes in the scale of measurement. Considering that the smooth curve gets closer and closer to the same value as the measurement accuracy increases, the measured value of the fractals can change significantly.
Length " true fractal" always tends to infinity. However, this figure is based on the idea that space can be subdivided to indeterminacy, i.e. be unlimited. This is a fantasy that underlies Euclidean geometry and serves as a useful model in everyday measurements, almost certainly does not reflect the changing realities of "space" and "distance" at the atomic level.Coastlines are different from mathematical fractals, they are formed from numerous small details that create models only statistically.
For practical reasons, you can use the dimension with an appropriate choice of the minimum size of the ordinal unit. If the coastline is measured in kilometers, then small variations are much smaller than one kilometer and can be easily ignored. To measure coastline in centimeters, tiny changes in size must be considered. The use of different measurement techniques for different units also destroys the conventional belief that blocks can be converted by simple multiplication. Extreme coastline cases include the fjord paradox of the heavy coast of Norway, Chile, and the Pacific coast of North America.
Shortly before 1951, Lewis Fry Richardson, in a study of the possible effect of frontier length on the likelihood of war, noted that the Portuguese reported their measured frontier with Spain as 987 km long, but Spain reported it as 1214 km. This was the beginning of the coastline problem, which is mathematically difficult to measure due to the irregularity of the line itself. The predominant method for estimating the length of a boundary (or coastline) has been to overlay N number of equal segments of length ℓ with delimiters on a map or aerial photographs. Each end of a segment must be on a boundary. By investigating discrepancies in bound estimates, Richardson discovered what is now called the Richardson effect: the sum of the segments is inversely proportional to the total length of the segments. Essentially, the shorter the ruler, the larger the measured border; Spanish and Portuguese geographers simply measured the boundary with different lengths of rulers. As a result, Richardson was struck by the fact that, under certain circumstances, when the length of the ruler ℓ goes to zero, the length of the coastline also goes to infinity. Richardson believes that based on Geometry of Euclid, the coastline will fit to a fixed length, how to make similar estimates of the correct geometric shapes. For example, the perimeter of a regular polygon inscribed in a circle approaches the circle as the number of sides increases (and the length of one side decreases). In geometric measure theory, a curve as smooth as a circle, which can be approached by small straight segments with a certain limit, is called a rectifiable curve.
More than a decade after Richardson completed his work, Benoit Mandelbrot developed a new area of mathematics - fractal geometry to describe just such non-rectifiable complexes in nature in the form of an endless coastline. Own definition new figure serving as the basis for his research: I came up with a fractal from the Latin adjective " fragmented' to create irregular patterns. So it makes sense... that, in addition to "fragmented"... broken should also mean "irregular".
The key property of a fractal is self-similarity, that is, the same general configuration appears at any scale. The coastline is perceived as bays alternating with capes. In a hypothetical situation, a given coast has this self-similarity property, no matter how much any small stretch of coast appears in magnification, a similar pattern of smaller bays and capes is superimposed on larger bays and capes, down to a grain of sand. At the same time, the scale of the coastline instantly changes into a potentially infinitely long thread with a random arrangement of bays and headlands formed from small objects. Under such conditions (as opposed to smooth curves) Mandelbrot argues, "the length of the coastline turns out to be an elusive concept that slips between the fingers of those who want to understand it. "There are different kinds of fractals. The coastline with the specified parameters is in the "first category of fractals, namely the curves with fractal dimension greater than 1." This last statement is Mandelbrot's extension of Richardson's thought.
The Mandelbrot Statement of the Richardson Effect:
where L, the length of the coastline, a function of the unit of measurement, ε, is approximated by the expression. F is a constant and D is the Richardson parameter. He didn't give theoretical explanation, but Mandelbrot defined D with a non-integer form Hausdorff dimensions, later - fractal dimension. Rearranging the right side of the expression, we get:
where Fε-D should be the number of units of ε required to obtain L. fractal dimension- the number of fractal dimensions used to approximate the fractal: 0 for a point, 1 for a line, 2 for an area. D in the expression is between 1 and 2, usually less than 1.5 for the coast. The broken coastline dimension does not extend in one direction and does not represent an area, but is intermediate. This can be interpreted as thick lines or stripes 2ε wide. More broken coastlines have a larger D, and therefore a larger L, for the same ε. Mandelbrot showed that D does not depend on ε.
Source : http://en.wikipedia.org/wiki/Coast#Coastline_problem
http://en.wikipedia.org/wiki/Coastline_paradox
Translation: Dmitry Shakhov
The modern political map of the world has its own records for the area of countries, location, length of borders, population, time of occurrence, political structure, number of nationalities, peculiarities of the location of capitals, etc.
Everyone educated person, student, student must know a certain part
The largest state
The largest state in terms of area is the Federation. Its area is 17.0754 million sq. km. With the accession to Russia in 2014, the area of the country increased by 26 thousand square kilometers. The area of Russia is 11.5% of the entire world's land surface.
The following places behind Russia in terms of area are occupied by: Canada (9.976 million sq. km), USA (9.3726 million sq. km), Brazil (8.512 million sq. km).
Russia is the coldest country
Spain is located in Europe and its Canary Islands in Africa.
Portugal in Europe and the Madeira Islands in Africa.
Yemen is located in Asia, its Socotra Islands are located in Africa.
Countries located in several parts of the world
On the political map of the world there are countries that, without taking into account their possessions, lie simultaneously in several parts of the world.
Most of Russia lies in the eastern hemisphere, while the extreme northeast of the country lies in the western hemisphere. Many countries are located simultaneously in the eastern and western hemispheres: England, Algeria, Mali, Burkino-Faso, Ghana, Fiji,.
There are countries that are simultaneously located in the northern and southern hemisphere: Indonesia, in Africa (Equatorial Guinea, Sao Tome and Principe, Liberia, Somalia).
An island nation of 16 islands, Kiribati, located in the Pacific Ocean, is located in four parts of the world. This country is located in the Gilbert Islands. The name is given by the name of one of the travelers who visited the islands. The name was given by the Russian traveler I. Kruzenshtern. This state appeared on the world map in 1977.
The state occupies an entire continent
It occupies a whole continent with an area of 7.7 million square meters. km. It can accommodate 33 Great Britain.
largest island nation
Indonesia is the largest island nation in terms of area. Its area is 1.904 million sq. km. It stretches from north to south for 2000 km. and from west to east for 5000 km. It is a country of 13,000 islands. Kalimantan is the 3rd largest island in the world. The island of Sumatra is equal in area to Sweden. Java is 4 times larger than Belgium and equal to the area of Greece.
The name of the country was given in 1884 by the German traveler, geographer and ethnographer A. Bastian. He proposed calling the people of the country Indonesians, combining the word "India" and the Greek "nesos" - islands, that is, the inhabitants of island India. because the Indian cultural influence is clearly visible in the culture of the main peoples.
A country where the form of government changed frequently in a short time
In the Central African Republic, the form of government has changed four times in 30 years.
On December 1, 1958, the former French colony was proclaimed an autonomous state within the French Community. Currently, the republic celebrates December 1 as the Day of the Proclamation of the Republic.
On August 13, 1960, it was proclaimed independent state within the French Community. Its first president, D. Dako, was elected.
January 1, 1966 in the Central African Republic occurred coup d'état and the military came to power. Colonel J.B. Bokassa became president. The country became known as the Central African Empire.
On September 20, 1979, with French military assistance, another coup took place. D. Dako came to power again.
The oldest state and republic in the world
Most ancient state and the oldest republic in the world - San Marino. The Republic has been in existence since 301 AD. The name San Marino officially appeared in documents from the 10th century. The country is located in the northeast of the Apennine Peninsula. Its area is 61 sq. km. The population is 24.3 thousand people. This amazing country is visited annually by 3 million tourists.
The oldest federal state
The oldest federal state is Switzerland, more precisely the Swiss Confederation. It was formed on August 1, 1291 from the Alpine cantons (Uri, Unterwalden and Schwyz). These cantons entered into an "alliance for all eternity" among themselves. Later, neighboring lands were added to the union of the three cantons. At the Congress of Vienna in 1814-1815. the exact boundaries of the state were established. In 1848, a constitution was adopted, which stated that the country began to be considered federal.
The youngest state
The youngest state is Eritrea, which was officially proclaimed on May 14, 1993. Until that time, it was under the sovereignty of Ethiopia for 40 years. This country is located on the Red Sea coast of Northeast Africa. The area of the country is 125 thousand sq. km. Population No. 6 mln. The capital, Asmere, is home to 400,000 people. The name of the country comes from the Greek "eritos", which means red. Until now, they argue about the origin of the name of the country. Maybe it came from the name of the sea, maybe from the color of the soil.
Continent with the most borders
This continent is Africa. There are 108 borders.
The longest border between countries
The longest border between Canada and the United States. Its length is 8963 km. taking into account the length of the borders between the state and Canada (2547 km).
Longest continuous land border
This is the border between Russia and Kazakhstan. Its length is 7200 km. The long land border between Argentina and. Its length is 5255 km.
The shortest border
The shortest land border is at the Vatican. Its length is only 4.07 km. The length of the borders between Spain and Gibraltar is even shorter. The length of the border is 1.53 km.
The country with the most big number of land borders
It is such a country. It borders 15 countries.
Russia borders on 14 countries, Brazil on 10, Germany and the Democratic Republic of the Congo on 9.
Country with the most maritime borders
Indonesia has the largest number of maritime borders.
It borders on 19 countries.
longest maritime border
The longest sea border is between Canada and. Its length is 2697 km.
State with the longest coastline
Canada has the longest coastline. The total length of the coast is 96009 km. On the mainland coast, the length of the coast is 28,737 km, and on numerous islands, their length is 67,272 km. Canada's coastline is four times the size of the US.
Sovereign country with the shortest coastline
The shortest coastline of the seaside state is Monaco. It is only 5.61 km long. Monaco is located on the northern coast of the Ligurian Sea between France and Italy. Monaco is a principality. Monaco lives off tourism. gambling business, construction of residences. Here S.P. Diaghilev created the Russian ballet in 1911. Monaco is home to the headquarters of the International Hydrographic Organization and a famous oceanarium.
The oldest capital
most ancient capital is a city in Syria. It has been around since 2500 BC.
In 10 - 8 centuries. BC. the city was the center of the Damascus state. The name Damascus from the Semitic means "useful", "business".
The youngest capital
The youngest capital of the world in 1997 was the city of Akmola (Astana) in Kazakhstan.
This is the new capital of Kazakhstan. By the presidential decree in 1998, it was renamed and became known as Astana. Translated from Kazakh means "capital". The city is located on the banks of the Ishim River.
The most populous capital in the world is considered to be the capital of Japan. About 26 million people lived within this urban agglomeration at the end of the 20th century. It became the capital in 1869. Tokyo means "capital" in Japanese. It is part of the Tokaido metropolitan area.
The highest capital city in the world
The highest mountain capital in the world is the city of La Paz (Bolivia). This city is located in the Andes Mountains in the Bolivian Highlands at an altitude of 3400 meters.
The southernmost capital in the world
Wellington, New Zealand is the southernmost capital in the world. The city is located in the south of the North Island. The population of the city is 150 thousand people. The city was founded in 1839.
oldest ruling dynasty
The oldest ruling dynasty in Japan. The 125th Emperor Akihito, who was born on December 23, 1933, descended from the first Emperor, Jimmu Tennō.
The country of Canada is one of the countries with the largest territory in the world, ranking second after Russia. The territory of Canada is 9,984,670 km², while the population of the country in 2016 was 36,048,521 people. But the density of the country is only 3.5 people per km2, which is one of the lowest in the world. Canada is also famous for having the longest coastline in the world - 243,791 km! Canada is located on the mainland North America, in its northern part. It has a land border only with the United States, and has sea borders with Denmark (Greenland) and France (Saint Pierre and Miquelon).
Canada is washed by the Arctic Ocean in the north, the Pacific Ocean in the west, and the Atlantic Ocean in the east. The length of Canada from north to south of the country is 4600 km, and from west to east of the country - 7700 km.
The capital of Canada is Ottawa. The monetary unit is the Canadian dollar. The current monarch of Canada is Elizabeth II.
Canada is a constitutional monarchy with a parliamentary system. It was founded in 1534 by J. Cartier. The country consists of 3 territories and 10 provinces. There are two in the country official languages- English and French.
Flag of Canada:
Today, this country is an industrially and technologically advanced state. Canada has a diversified economy based on trade and natural resources Canada is rich in.
Relief of Canada
The central part of the country is occupied by plains. It is possible to distinguish the Hudson Bay Lowland, which is marked by a flat relief, the Laurentian Upland, which is characterized by a hilly relief, and the central plains. In the west of the country is the Cordillera mountain system. The highest point is Mount Logan. mountain system, whose height reaches 5959 m above sea level. In the northeast of the country there is a strip of mountains up to 2000 m high, and in the southeast the region of the Appalachian highlands.
Climate of Canada
The climate of Canada is quite diverse, due to its large territory. In total, Canada has three types of climatic zones - Arctic, Subarctic and temperate. The temperature is very different in the north and south of the country. In winter, the difference in average temperatures in the south and north reaches almost 30 units, and in summer it is slightly less.
For example, the average maximum temperature in the north in winter reaches -28 degrees Celsius, and in the south of the country -0.4 degrees Celsius. In summer, the average maximum temperature in the north reaches 6 degrees Celsius, and in the south of the country 29 degrees Celsius. At the same time, in the summer in the south of the country the temperature can rise to 35-40 degrees Celsius, and in the north of the country it can drop to -45-60 degrees Celsius with strong icy winds.
Canada's climate is quite harsh. These are long, snowy winters that last up to 8 months a year and short summers. At the same time, in winter in the south of the country the sun shines 8 hours a day, and in the north it does not shine at all. Since icy winds from the north and warm winds blowing from the USA occur on the territory of the country, quite a lot of precipitation falls over Canada. a large number of precipitation.
Inland waters of Canada
Canada occupies one of the first places in terms of the number of lakes. About 10% of Canada is covered by water. On its territory are the Great Lakes (Ontario, Superior, Erie, Huron), as well as smaller lakes and numerous rivers throughout the country. The most important river in Canada is the navigable St. Lawrence River, which connects the Great Lakes to the Basin Atlantic Ocean. Thanks to Canada's climate, all of its lakes and rivers are covered with ice for 5 to 9 months a year.
Flora of Canada
The vegetation in the country varies from deciduous and mixed forests in the south of the country to tundra, taiga, which in the north of the country turn into arctic deserts. Of the forests in Canada, coniferous forests predominate. In the forests, you can most often find plants such as: black spruce, pine, white spruce, thuja, larch, oak, beech, chestnut, alder, birch, willow, cedar, fir, strawberry tree, elm and many other plants.
Animal world of Canada
In the south of the country animal world the most diverse, and in the north the most scarce. Deer, elk, rams, goats, arctic fox, hare, chikari squirrel, chipmunks, jerboas, porcupines, American flying squirrel, beaver, striped raccoon, wolf, fox, bears and many other representatives of animals are found on the territory of the country. There are also many migratory and game birds. Rivers and lakes are rich in fish. But the list of reptiles and amphibians is not so numerous.
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Although our planet is a relatively temperate place in terms of climate and geographic data, there are places on it that will amaze you with their level of extreme, whether it be the coldest place on Earth or the deepest trench in the ocean. Get ready to be blown away by these 25 locations!
Hottest inhabited place - Dallol, Ethiopia
The average daily temperature here is 34.4 degrees Celsius.
The deepest cave is Krubera-Voronya cave
It is located in Abkhazia, the depth is more than 2000 m.
The highest point is Mount Everest
The height of the mountain is 8,848 m above sea level.
Farthest point from the center of the Earth - Chimborazo, Ecuador
The most remote island is Bouvet Island
Norwegian Island in the South Atlantic Ocean is located 1,000 miles from Antarctica and almost 1,500 miles from South Africa.
The most distant continental point is the Antarctic Pole of Inaccessibility
It is the furthest point on the continent from any ocean. And Antarctica is the most remote continent.
The flattest place - Uyuni Salt Flats, Bolivia
The world's largest salt marsh with an area of 4086 sq. miles.
The highest navigable lake - Titicaca
The lake on the border with Bolivia is located at an altitude of 3812 m.
The lowest point on land is the Dead Sea
This point separates Jordan and the West Bank at 418 m below sea level.
Longest mountain range - Andes, South America
The ridge is 5000 miles long and passes through 7 countries of South America.
The deepest man-made hole - Kola Superdeep Well
Its depth reaches 12,262 m.
The rainiest place is Choco, Colombia
It receives 11,770 cm of precipitation per year.
The driest place in the Atacama Desert, Chile
Most populous landlocked country - Ethiopia
70 million people do not have access to the coastline.
The largest vertical drop - Mount Thor, Canada
Height 1250 m, average angle 105 degrees.
The coldest inhabited locality - Oymyakon, Russia
For 7 months of the year, temperatures here are well below zero.
Windiest place - Commonwealth Bay, Antarctica
Winds regularly exceed 240 km/h, and the average annual wind speed is 80 km/h.
Tallest waterfall - Angel Falls, Venezuela
Its height reaches 1054 m, and the water has time to evaporate before it reaches the ground.
Highest mountain pass - Marsimik La, India
Located at an altitude of 5582 m.
The largest freshwater lake is Lake Superior
Its area is 31820 sq. miles.
Country with the longest coastline - Canada
The coastline stretches for 151,019 miles.
The largest gorge - the Grand Canyon, USA
It is almost 220 miles long and about a mile deep.
The largest glacier - Lambert-Fischer, Antarctica
Extends over 100 miles.
Shortest River - Row, Montana
Its length is only 61 m.
Lowest point - Challenger Deep
It is located in the lower part of the Mariana Trench at a depth of 10911 m below sea level.
coastline length
Can it be measured?
Do we have the right to give in textbooks the length
coastline and will we not be embarrassed,
asking this figure from the students?
K.S. LAZAREVICH
In geography lessons, we operate with a variety of statistical indicators. Most of them look very simple and clear: so many millions of people, so many millions of tons of coal, so many kilometers. But that's if you don't think about it. And one has only to dig any figure deeper - and it ceases to be clear. Sometimes it crumbles to dust. Here are examples.
We open the recently published and just went on sale Atlas of the World (M.: FSUE Production Mapping Association "Cartography", 2003.). In the table “States and territories of the world” we find: “The capital of France is Paris (2,125.2 thousand inhabitants). If the student gives such a number on the exam, will the examiner be satisfied? After all, Paris is one of the largest centers in Europe and no less than St. Petersburg. But there is no mistake in the given figure: this is Paris within the administrative boundaries of the city of Paris. And within the boundaries of a really developed urban cluster - this is a ten millionth. A lot depends on how you count. This does not mean that we can accept from the student as an answer any number in the range from 2.2 to 10; giving this or that number, the student must understand what is behind it, what is measured and how.
A million tons of high-calorific coal and brown coal - different millions.
But here, it would seem, kilometers. A kilometer is also a kilometer in Africa. And what really measured in kilometers can be questioned? But it turns out that even giving lengths in kilometers, the author of the textbook must first think. The teacher, using the textbook, must also subject the figure to critical analysis before broadcasting it to the students and requiring it to be memorized. Reading a 10th grade textbook: "Canada has three oceans and its total coastline (about 250,000 km) is unparalleled in the world." How was the coastline measured, what was measured, how was it measured, how was it measured? How can you even measure a coastline?
Irregular curves on the map can be measured using a curvimeter - the wheel of this device is rolled along the curve, carefully writing out each meander. However, the tortuosity of the coastline is often so great that it is impossible to pass along it with a curvimeter. You have to walk along the curve with a compass-meter. The most comfortable step length is 2 mm. On different scales, this step corresponds, of course, to different distances, such a measurement will never give an exact length, since each step straightens the curve on a small segment, but relative error more or less preserved.
Let's, for the sake of an example, try to measure the length of the coastline of the Chukotka Autonomous Region. Let's take a map from the School Atlas on the Geography of Russia (scale 1: 22,000,000) and with a two-millimeter step of a compass (44 km) we will walk the entire Chukchi coast. The result will be 4300 km (98 compass steps). Let's make the same measurement on the scale map
1: 7,500,000. Here we already count 345 two-millimeter (15 km) steps, that is
5,200 km. It is logical to assume that if an even larger scale map is used in the measurements, the measured coastline will become even longer.
Let's set up another experiment. The length of the coastline of the Leningrad region. on the map
1: 22,000,000 - 300 km, according to map 1: 2,500,000 - 555 km, and according to topographic map
1: 500,000 - 670 km. At the same time, the length of the coastline alone Vyborg Bay(where the shores are especially indented by bays and coves), measured according to the topographic map, is 338 km, while according to the school atlas - 65 km (the difference is more than
Five times!).
Thus, there is a regular increase in the length of the measured coastline with the enlargement of the scale. The reason is not only that the two millimeter step of the compass corresponds to an ever smaller value on the ground, but mainly that the line itself, even if it is very accurately measured and converted in accordance with the scale in kilometers, actually becomes longer (Fig. 1) . On the map of Russia near the coast of the Leningrad region. only Vyborg Bay, Neva Bay and small bends of the southern coast of the Gulf of Finland are guessed. On a 1:2,500,000 scale map, the outlines of the Vyborg Bay are already quite complex, and the Koporskaya and Luga bays are clearly visible in the south. There are many other small bays on the half-million map within the Vyborg Bay, some of which have proper names(Baltiets Bay, Klyuchevskaya Bay), and only the southern coast of the Gulf of Finland looks little changed compared to the previous scale, where the coast is much less indented.
How to determine the exact length of the coastline?
This goal was set by the English meteorologist Richardson, choosing his native island, Great Britain, as a test site. He came to the conclusion that the length of the coastline increases with an increase in the scale of the map, according to which this length is measured (Fig. 2). Is there a limit to such an increase? Hardly. The length of the coastline is increased by every small sandy spit that goes into the sea, every hollow that creates a tiny bay, every pebble that flows around the water. Even on the largest-scale map they are not visible, meanwhile, in reality, all these roughnesses of the coastline exist.
Many examples are given of how the use of mathematical methods makes it possible to make geographical research more convincing, more reliable. Here the opposite happened: geographical research - the study of the length of the coastline - contributed to the emergence of a new mathematical concept. English name of this concept is fractal, in Russian it has not yet finally settled down and is found in three versions: fractal(parent and instrumental will fractal, fractal),
fractal in masculine ( fractal, fractal) and fractal in the feminine ( fractals, fractal); behind recent times seems to lean towards fractal.
A fractal is a line, each fragment of which becomes infinitely more complex, the length of each fragment and the entire line is constantly increasing. An example is a figure usually called the Koch snowflake, although the name is incorrect: she built this snowflake at the beginning of the 20th century. Helga von Koch, and her name should not be declined.
Let's take equilateral triangle. Let us divide each of its sides into three equal parts and construct an equilateral triangle on the middle segment of each side. You get a regular six-pointed star, a figure with six convex corners and six incoming ones. We divide each of its sides (and these sides are 12) into three equal parts and on the middle segment of each side we again construct an equilateral triangle. You will get a figure already with 48 sides, with 18 convex and 30 incoming angles. Repeating this operation an infinite number of times (this can be done, of course, only mentally), we will get a figure whose area is constantly increasing, but more and more slowly, gradually approaching a certain limit (Fig. 3). The perimeter of this figure increases indefinitely, since every time we build a new equilateral triangle on the side of the figure, no matter how small it may be, three equal segments of this side are replaced by four of the same and therefore the length of each side (and therefore the entire perimeter) increases by 4/3 times, and any number more than one to a power equal to infinity (and we do the construction an infinite number of times), tends to infinity.
Rice. 3 Snowflake Koch -different stages of construction |
The border of the snowflake will be something like a wide, shaggy line that fills the entire border area of this figure. The concepts of "wide line", "thick surface", seemingly absurd from the point of view of classical mathematics (the line there has no width, and the surface has no thickness), with the development of the theory of fractals, they acquired the rights of citizenship. It is believed that the line is one-dimensional, it has only a length, the position of a point on it is determined by one coordinate; the surface is two-dimensional, it has an area, the position of a point on it is determined by two coordinates; the body is three-dimensional, it has a volume, three coordinates are already needed. And the theory of fractals introduces the concept of fractional dimension: the line has not become two-dimensional, but has already ceased to be one-dimensional. It is rather difficult for an unprepared person to understand (you can’t sneeze one and a half times), but if we remember how the coastline behaves - not only on the map, but also in nature, how it changes when you look at it, squatting down, then straightening up to our full height, then climbing a mountain, then taking off on an airplane or a spaceship, we will not only understand, but feel what complex system represents this line; for her, one characteristic is definitely not enough - length.
And the theory of fractals, born from geographical research, is itself coming to the aid of geography. The method of studying relief as a fractal has not yet been developed, but definitely has prospects. Looking at the terrain general view, drawing it on a small-scale map, we see mountain ranges, plateaus, deep valleys. On an average scale, hills, small valleys, and ravines are already looming. Even larger - and bumps are visible, wind ripples on the sand. But this is not the limit: there are separate pebbles, grains of sand. In practical terms, all this is important because you need to learn how to correctly select objects for depiction on maps of different scales; one of the main mistakes of map compilers is the discrepancy between the content of the map and its scale, the map is either underloaded or overloaded.
But what to do with the length of the coastline? Refuse to measure it because it is immeasurable?
No, this is not an option. Simply, when citing the length of the coastline, one should always indicate on what scale maps it was measured, in what way. And be sure to specify, the coastline of the islands was taken into account or not. Without indicating the scale of the maps and whether islands are taken into account or not, any data on the length of the coastline lose their meaning. Unfortunately, even in sources that claim to be purely solid, one can come across terrible absurdities. For example, the famous CIA website "The World Factbook". Here, for each country and ocean, coastline data is given, but the method of measurement is not indicated. As a result, the coastline of Canada is more than 200 thousand km, the Arctic Ocean - 45.4 thousand km, the Atlantic - 111.9 thousand km (the data are given - do not think bad! - accurate to a kilometer). Canada was considered taking into account the islands, this is undoubtedly; how the oceans were counted is not known, but the coastlines of two of the three oceans surrounding Canada are, in sum, smaller than the coastlines of Canada alone. For Norway, a figure of 21,925 km is given and a note is given: "Mainland 3419 km, large islands 2413 km, long fjords, numerous small islands and small bends [literally translated notches] coastline 16,093 km. In total, just the indicated total length of the coastline is obtained. But that's why the coast of the fjords is not part of the coastline of the mainland, why the length of the notches is added to the length of the coastline of the mainland, which islands are considered large - all this can only be guessed at. Absolutely indisputable data in this table are given only for Andorra, Austria, Botswana, Hungary, Swaziland and similar countries that do not have access to the sea - it is written: "0 km".
