Even though our planet is a relatively temperate place in terms of climate and geography, there are places on it that will amaze you with their level of extremes, be it the coldest place on Earth or the deepest trench in the ocean. Get ready for these 25 places to surprise you with their fantastic performance!

Hottest inhabited place - Dallol, Ethiopia

The average daily temperature here is 34.4 degrees Celsius.

The deepest cave is the Krubera-Voronya cave

It is located in Abkhazia, the depth is more than 2000 m.

Highest point - Mount Everest

The height of the mountain is 8,848 m above sea level.

The farthest point from the center of the Earth is Chimborazo, Ecuador.

The most remote island is Bouvet Island

Norwegian island in the south Atlantic Ocean located 1000 miles from Antarctica and almost 1500 miles from South Africa.

The most distant continental point is the Antarctic Pole of Inaccessibility

This is the farthest point on the continent from any ocean. And Antarctica is the most remote continent.

Flattest place - Salar de Uyuni, Bolivia

The world's largest salt marsh with an area of ​​4086 square meters. miles.

The highest navigable lake is Titicaca

The lake on the border with Bolivia is located at an altitude of 3812 m.

The lowest point on land is the shore of 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 5,000 miles long and runs through 7 countries in South America.

The deepest man-made hole - the Kola superdeep well

Its depth reaches 12,262 m.

Rainiest place - Chocó, Colombia

It receives 11,770 cm of precipitation per year.

The driest place is the Atacama Desert, Chile

The most populous landlocked country is Ethiopia.

70 million people do not have access to coastline.

Largest elevation change - Mount Thor, Canada

Height 1250 m, average angle 105 degrees.

The coldest settlement is Oymyakon, Russia

Temperatures here remain well below zero for 7 months of the year.

Windiest Place - Commonwealth Bay, Antarctica

Winds regularly exceed 240 km/h, and the average annual wind speed is 80 km/h.

Tallest Falls - Angel, Venezuela

Its height reaches 1054 m, and the water manages 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 31,820 square meters. miles.

The country with the longest coastline is Canada

The coastline stretches for 151,019 miles.

The largest gorge - Grand Canyon, USA

It is nearly 220 miles long and about a mile deep.

Largest glacier - Lambert Fisher, Antarctica

Extends over 100 miles.

Shortest River - Roe, Montana

Its length is only 61 m.

Lowest point - Challenger Deep

Located in the lower part of the Mariana Trench at a depth of 10911 m below sea level.

Coastline length

Is it measurable?
Do we have the right to give the length in textbooks?
coastline and won't we be embarrassed,
asking this figure from students?

K.S. LAZAREVICH

In geography lessons we operate with many 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. But you just have to dig deeper into any number and it ceases to be clear. Sometimes it crumbles to dust. Here are examples.
We are opening the recently published Atlas of the World, which has just gone on sale (M.: Federal State Unitary Enterprise Cartography Production Association, 2003). In the table “States and territories of the world” we find: “The capital of France is Paris (2,125.2 thousand inhabitants). If a student gives such a figure in an 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 established urban cluster, it is a ten-million-dollar city. A lot depends on how you count. This does not mean that we can accept any number in the range from 2.2 to 10 from the student as an answer; When citing this or that number, the student must understand what is behind it, what is measured and how.
A million tons of high-calorie coal and brown coal are different millions.
But it seemed like kilometers. A kilometer is also a kilometer in Africa. And what is measured in kilometers can be questioned? But it turns out that even when giving lengths in kilometers, the author of the textbook must first think. A teacher, using a textbook, must also subject a figure to critical analysis before broadcasting it to students and requiring them to memorize it. We read a textbook for the 10th grade: “Canada has three oceans, and the total length of its coastline (about 250 thousand km) is unparalleled in the world.” How was the coastline measured, what was measured, how was it measured, what was it measured with? How can you even measure a coastline?

Irregular curves on a map can be measured using a curvimeter - the wheel of this device is rolled along the curve, carefully recording each curve. However, the tortuosity of the coastline is often so great that it is impossible to follow it with a curvimeter. You have to walk along the curve with a measuring compass. 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 over a small segment, but the relative error is more or less preserved.
Let's, for the sake of an example, try to measure the length of the coastline of the Chukotka Autonomous Okrug. Let's take a map from the School Atlas on the Geography of Russia (scale 1: 22,000,000) and walk the entire Chukchi coast with a two-millimeter compass step (44 km). The result will be 4300 km (98 compass steps). Let's make the same measurement using the scale map
1: 7,500,000. Here we will 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 more extensive.
Let's do one more experiment. The length of the coastline of the Leningrad region. on the map
1: 22,000,000 - 300 km, according to the map 1: 2,500,000 - 555 km, and according to the topographic map
1: 500,000 - 670 km. At the same time, the length of the coastline of the Vyborg Bay alone (where the shores are especially indented with bays and coves), measured on a topographic map, is 338 km, while according to the school atlas - 65 km (a difference of more than
5 times!).
Thus, there is a natural increase in the length of the measured coastline with increasing scale. The reason is not only that the two-millimeter step of the compass corresponds to an increasingly smaller value on the ground, but mainly because 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 shore of the Leningrad region. Only Vyborg Bay, Neva Bay and small bends of the southern coast of the Gulf of Finland are visible. On a map of scale 1: 2,500,000, the outlines of the Vyborg Bay are already quite complex, and in the south the Koporskaya and Luga bays are clearly visible. On the half-million-year-old map, there are many other small bays within the Vyborg Bay, some of which have their own names (Baltiets Bay, Klyuchevskaya Bay), and only South coast The Gulf of Finland looks little changed compared to the previous scale; the coastline there is much less rugged.

How to determine the exact length of the coastline?
The English meteorologist Richardson set himself this goal, choosing his home island, Great Britain, as a testing ground. He came to the conclusion that the length of the coastline increases with increasing scale of the map by which this length is measured (Fig. 2). Is there a limit to this increase? Hardly. The length of the coastline is increased by every small sand spit jutting 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, yet in reality all these irregularities in the coastline exist.

There are many examples of how the use of mathematical methods can make geographical research more convincing and 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. The English name for this concept is fractal, but in Russian it has not yet been completely settled and is found in three versions: fractal(genitive and instrumental cases will be fractal, fractal), fractal in the masculine gender ( fractal, fractal) And fractal in the feminine gender ( fractals, fractal); behind Lately seems to be leaning 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 the figure usually called the Koch snowflake, although this name is incorrect: this snowflake was built at the beginning of the twentieth century. Helga von Koch, and her last name should not be declined.
Let's take an equilateral triangle. Let's divide each side into three equal parts and construct an equilateral triangle on the middle segment of each side. The result is a regular six-pointed star, a figure with six convex angles and six incoming ones. Let's divide each of its sides (and there are 12 of these sides) into three equal parts and again construct an equilateral triangle on the middle segment of each side. The result will be a figure with 48 sides, with 18 convex and 30 recurrent angles. Repeating this operation an infinite number of times (this can be done, of course, only mentally), we will obtain 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 is, three equal segments of this side are replaced by four equal ones and therefore the length of each side (and therefore the entire perimeter) increases by 4/3 times, and any number greater than one to a power equal to infinity (and we do the construction an infinite number of times) tends to infinity.

The border of the snowflake will be something like a wide, shaggy line, filling the entire border area of ​​​​this figure. The concepts of “broad 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), acquired citizenship rights with the development of the theory of fractals. It is believed that a 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 volume, three coordinates are needed. And the theory of fractals introduces the concept of fractional dimension: the line has not become two-dimensional, but has ceased to be one-dimensional. This is quite 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 if you look at it, squatting, then standing up at full height, then climbing a mountain, then taking off on an airplane or spaceship, we will not so much understand as we will feel what a complex system this line represents; For her, one characteristic is definitely not enough - length.
And the theory of fractals, born from geographical research, itself comes to the aid of geography. A method for studying relief as a fractal has not yet been developed, but definitely has promise. Looking at the relief in 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 already appear. Even larger - and you can see the hummocks and wind ripples on the sand. But this is not the limit: there are individual pebbles and 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 giving the length of the coastline, you should always indicate on what scale maps it was measured and in what way. And be sure to stipulate at the same time, whether the coastline of the islands was taken into account or not. Without indicating the scale of the maps and whether islands are included or not, any data on the length of the coastline becomes meaningless. Unfortunately, even in sources that claim to be completely reliable, one can find terrible absurdities. For example, the famous CIA website “The World Factbook”. Here, coastline data is given for each country and ocean, but the measurement method is not specified. As a result, the coastline of Canada turns out to be more than 200 thousand km, the Arctic Ocean - 45.4 thousand km, the Atlantic Ocean - 111.9 thousand km (the data is given - don't think wrong of it! - to the nearest kilometer). Canada was considered taking into account the islands, that is certain; How the oceans were considered is unknown, but the coastlines of two of the three oceans that surround Canada add up to less than the coastline of Canada alone. For Norway the figure is 21,925 km and the 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.” The sum totals exactly the indicated total length of the coastline. But why the shores of the fjords are not part of the coastline of the mainland, why the length of the jagged edges is added to the length of the coastline of the mainland, which islands are considered large - we can only guess about all this. 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”.

The modern political map of the world has its own records for the area of ​​countries, location, length of borders, population, time of origin, political system, the number of nationalities, the location of capitals, etc.

Every educated person, pupil, student should 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 its accession to Russia in 2014, the country’s area increased by 26 thousand sq. km. Russia's area makes up 11.5% of the world's total land surface.

The following places behind Russia in 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 is in Europe, and the Madera Islands are 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 hemispheres: 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 corners 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 an entire continent with an area of ​​7.7 million square meters. km. Its territory can accommodate 33 Great Britain.

Largest island state

Indonesia is the largest island state by 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. This is a country of 13,000 islands. The island of Kalimantan is the 3rd largest 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 Indian cultural influence is clearly visible in the culture of the main peoples.

A country where the form of government has 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 declared an autonomous state within the French Community. Currently, the republic celebrates December 1 as the Day of Proclamation of the Republic.

On August 13, 1960, it was proclaimed an independent state within the framework of the French Community. Its first president, D. Dako, was elected.

On January 1, 1966, a coup d'état took place in the Central African Republic 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, a coup took place again with French military assistance. D. Dako came to power again.

The most ancient state and republic in the world

The most ancient state and the oldest republic in the world is San Marino. The Republic has existed 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.

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 a “union for eternity” among themselves. Later, neighboring lands were annexed 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 this time, it had been under Ethiopian sovereignty 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 million people. The capital, Asmer, is home to 400 thousand people. The name of the country comes from the Greek “erithos”, which means red. There is still debate about the origin of the country's name. 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 there.

Longest border between countries

The longest border between Canada and the USA. 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.

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

Such a country is. It borders 15 countries.

Russia borders on 14 countries, Brazil on 10, Germany and Democratic Republic Congo from 9.

Country with the largest number of maritime borders

Indonesia has the largest number of maritime borders.

It borders on 19 countries.

Longest maritime border

The longest maritime 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 96,009 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 larger than that of the United States.

Sovereign country with the shortest coastline

The shortest coastline of a coastal state is Monaco. It is only 5.61 km long. Monaco is located on the northern coast of the Ligurian Sea, a 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 aquarium.

The most ancient capital

Most ancient capital is a city in Syria. It has existed since approximately 2500 BC.

In the 10th – 8th centuries. BC. the city was the center of the Damascus state. The name Damascus from Semitic means “useful”, “business”.

The youngest capital

The youngest capital of the world in 1997 was the city of Akmola (Astana) in Kazakhstan.

This new capital of Kazakhstan was renamed by presidential decree in 1998 and began to be called Astana. Translated from Kazakh it means “capital”. The city is located on the banks of the Ishim River.

The capital of Japan is considered the most populous capital in the world. At the end of the 20th century, about 26 million people lived within this urban agglomeration. It became the capital in 1869. Tokyo means "capital" in Japanese. It is part of the Tokaido metropolis.

The highest capital in the world

The highest capital in the world is the city of La Paz (Bolivia). This city is located in the Andes Mountains on the Bolivian Highlands at an altitude of 3400 meters.

The southernmost capital of the world

The southernmost capital in the world is Wellington (New Zealand). 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, descends from the first Emperor Jimmu Tenno.

The country of Canada is one of the countries with the largest territory in the world, ranking second after Russia. Canada's territory is 9,984,670 km², while the country's population in 2016 was 36,048,521 people. But the country’s density 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 whole world - 243,791 km! Canada is located on the North American continent, 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 in the north by the Arctic Ocean, in the west of the country by the Pacific Ocean, and in the east Canada is washed by the Atlantic Ocean. 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 currency is the Canadian dollar. The current monarch of Canada is Elizabeth II.

Canada is a constitutional monarchy with a parliamentary system. It was founded back 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 developed state. Canada has a diversified economy that is based on trade and natural resources, which Canada is rich in.

Relief of Canada

The central part of the country is occupied by plains. We can distinguish the Hudson Bay Lowland, which is characterized by flat terrain, the Laurentian Upland, which is characterized by hilly terrain, and the central plains. In the west of the country is the Cordillera mountain system. Most high point is Mount Logan this 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 Hills.

Climate of Canada

Canada's climate is quite varied, due to its large territory. In total, Canada has three types of climate zones - Arctic, Subarctic and temperate. The temperature in the north and south of the country is very different. 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. Moreover, in winter in the south of the country the sun shines 8 hours a day, but in the north it does not shine at all. Since the country experiences icy winds from the north and warm winds blowing from the United States, a fairly large amount of precipitation falls over Canada.

Canadian inland waters

Canada ranks one of the first places in the number of lakes. About 10% of Canada's area is covered by water. Its territory contains 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 Atlantic Ocean basin. Thanks to Canada's climate, all its lakes and rivers are covered with ice from 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 and taiga, which turn into arctic deserts in the north of the country. Coniferous forests predominate among the forests in Canada. In 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.

Wildlife of Canada

In the south of the country animal world the most diverse, and in the north the scarcest. The country is home to deer, elk, sheep, goats, arctic fox, hare, chikari squirrel, chipmunks, jerboas, porcupines, American flying squirrel, beaver, raccoon, wolf, fox, bears and many other animals. 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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Since land has features at all levels, from hundreds of kilometers in size down to tiny fractions of a millimeter and below, there are no obvious limits on the size of the smallest features, and therefore no well-defined land perimeter is fixed. 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 a fractal unit of 100 km (62 mi) in length, then the coastline is approximately 2,800 km (1,700 mi) long. With a unit of 50 km (31 mi), the total length is about 3,400 km (2,100 mi), approximately 600 km (370 mi) longer.

Mathematical aspects

The basic concept of length comes from Euclidean distance. In a friend Euclidean geometry, a straight line represents the shortest distance between two points; this line has only one finite length. The geodesic length on the surface of a sphere, called the great length of the circle, is measured along the surface of a curve that exists in a plane containing the end points 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, a person can approximate the lengths of a curve by adding the sum of the straight lines connecting the points:

Using several straight lines to approximate the length of the curve will produce a low estimate. Using shorter and shorter lines will produce a sum of lengths that approaches the true length of the curve. The exact value of this length can be determined using calculus, a branch of mathematics that allows you to calculate infinitesimal distances. The following animation illustrates this example:

However, not all curves can be measured in this way. By definition, a curve with complex changes in the measurement scale is considered fractal. Given that a smooth curve moves closer and closer to the same value as measurement precision increases, the measured value of 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 the point of indeterminacy, i.e., to 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 patterns only statistically.

For practical reasons, you can use the measurement with the 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 usual confidence that blocks can be converted by simple multiplication. Extreme coastline cases include the fjord paradox of the heavy coasts of Norway, Chile, and the Pacific coast of North America.

Shortly before 1951, Lewis Fry Richardson, in a study of the possible influence of border length on the likelihood of war, noted that the Portuguese presented their measured border with Spain as 987 km long, but Spain reported it as 1,214 km. This was the beginning of the shoreline problem, which is mathematically difficult to measure due to the irregularity of the line itself. The predominant method of estimating the length of a boundary (or coastline) was to superimpose N numbers of equal segments of length ℓ with delimiters on a map or aerial photographs. Each end of the segment must be on a boundary. By investigating discrepancies in boundary estimation, Richardson discovered what is now called the Richardson effect: the sum of segments is inversely proportional to the total length of the segments. Essentially, the shorter the ruler, the larger the measured boundary; Spanish and Portuguese geographers simply measured the border using different lengths of rulers. As a result, Richardson was struck by the fact that, under certain circumstances, when the length of the ruler ℓ tends to zero, the length of the coastline also tends to infinity. Richardson believes that based on Euclid's geometry, the coastline will approach a fixed length, how to make similar estimates of regular 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 smooth curve such as a circle, to which small straight segments can be approximated with a certain limit, is called a rectifiable curve.

More than ten years after Richardson completed his work, Benoit Mandelbrot developed a new area of ​​mathematics - fractal geometry to describe precisely such non-rectifiable complexes in nature in the form of an endless coastline. Own definition of a new figure serving as the basis for his research: I came up with a fractal from the Latin adjective “ fragmented» to create irregular fragments. 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 coastline has this property of self-similarity, no matter how much any small section of coastline appears enlarged, a similar pattern of smaller bays and headlands superimposed on larger bays and headlands, down to the 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 capes formed from small objects. In such conditions (as opposed to smooth curves), Mandelbrot states, "the length of a coastline is 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 curves with fractal dimension greater than 1." This last statement represents Mandelbrot's expansion of Richardson's thought.

Mandelbrot Richardson Effect Statement:

where L, the length of the coastline, is a function of the unit of measurement, ε, and is approximated by Eq. F is a constant and D is the Richardson parameter. He did not give a theoretical explanation, but Mandelbrot defined D with a non-integer form Hausdorff dimensions, later - fractal dimension. Regrouping the right side of the expression we get:

where Fε-D must be the number of ε units needed to obtain L. Fractal dimension- number of fractal dimensions used to approximate a fractal: 0 for a point, 1 for a line, 2 for an area. D in the expression is between 1 and 2, for the coast it is usually less than 1.5. The broken dimension of the coast does not extend in one direction and does not represent an area, but is intermediate. This can be interpreted as thick lines or stripes with a width of 2ε. More broken coastlines have larger D and therefore 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