# Map projections: what xkcd actually jokes about

Xccd is one of the most iconic web comics, and the minimalistic style of sketching as if from a mechanics textbook has become his hallmark . The author of the comic, Randall Munroe, admitted that he diligently seeks inspiration for new releases: he tries new programs and games, works on mathematical problems, and follows the news of science and technology. If he did not, then the comic would talk about how the artist sits at home at the computer.

Sometimes the release of the xkcd comic is a superficial joke in a narrow field of knowledge. Understand such humor by a specialist or at least poorly acquainted with the affected area, and the rest will only be perplexed. An example of such a release is xkcd.com/977Map Projections 2011. For a complete understanding of the issue, you only need to roughly represent the history and function of the various projections of world maps, otherwise the comic will remain a black and white picture. Even seemingly unambiguous questions have several solutions. The shape of the Earth is a geoid similar to a ball, but for ease of perception it is better to deploy it on a flat map. There are several ways to do this. Each of them will be executed with various compromises, since distortions of shape, angles or lengths are inevitable. Some projections give more distortion, others are easier to perceive, to some we are just used to.

Not all of us are looking at a map for high seas navigation. Often the choice of a method for transforming the shape of the Earth is not a matter of life and death, it is an artistic illustration. In these cases, the projection is chosen from established tastes and preferences - just like choosing clothes or a car. You can laugh about the fashion for expensive sneakers among the directors of large companies in Silicon Valley or about the popularity of understated cars. In the same way, cartographic projection preferences lend themselves to jokes, as was done in Map Projections.

We will divide the comic into several parts and consider them in the order convenient for us.

# The globe!

Yes, you are very smart.

Earth is a complex body with a relief. Even if we take it for a ball, an ellipsoid of revolution or a geoid , the Earth will still not be able to “flatten” without distortion. The lengths of rivers and roads, the area and shape of countries and continents, and angles for navigation are distorted. Any projection is a compromise between what kind of distortion is undesirable.

Therefore, any attempt to represent the Earth on a plane without distortion is doomed to failure, which is reminiscent of this answer option. Why try to do the impossible? The easiest way to answer a question about your favorite map projection is to smug smugly: “Globe!”

It is clear that this is not a projection, but only an attempt to be clever. Munro indicates this in the signature.

# Equidistant Projection (Plate-Carré)

It seems to you that this is suitable. You like how X and Y are converted to latitude and longitude. Other projections complicate things. You want me to leave you behind and give you a quiet dinner.

This is one of the oldest projections of a map of the world - probably, it was invented by Ptolemy before our era. The geoid is transformed into a cylinder and unfolds on a sheet of paper. The distances along the equator and all meridians are preserved, but the angles and area are distorted. It is convenient to use such a projection in computer geographic information systems: linear coordinates on the projection are easily converted to latitude and longitude.

Equidistant projection distorts angles and areas. Pole points on this projection stretch in a line. Due to distortion, an equidistant projection is not used in navigation.

In Munroe's view, such a projection may appeal to a busy person who is not interested in projections and related wisdom.

# Mercator

Cards are of little interest to you.

In the publication of the Atlas of 1569, the Flemish cartographer Gerard Mercator, on eighteen sheets with a total size of 202 by 124 centimeters, an equal-angled cylindrical projection was presented. The name contains its characteristics: at each point, the angles are transmitted without distortion, and the parallels and meridians maintain parallelism.

I had to pay for this with distortions of scale: the further the object moves from the equator to the pole, the larger it is. The poles themselves are transformed from points to infinitely large objects. Therefore, the map does not extend to the poles; a cut-off by latitudes of ± 80 ° –85 ° is used.

Because of this, not the largest Antarctica takes on enormous dimensions in the Mercator projection, and Africa and the countries of South and Southeast Asia seem smaller than they actually are. On the Mercator projection, Greenland looks the size of the entire African continent, although in reality its area is smaller than the area of ​​the Democratic Republic of the Congo. The True Size Of

site shows how much the Mercator projection distorts the area.

But the Mercator projection came in handy for navigating navigators and, subsequently, air traffic. If the compass needle retains its position, then the trajectory of the ship in the projection of the Mercator will be represented by a straight line. The intersection of the two roads will retain its angle in this projection. Mercator projection is widely used in navigation to this day.

Moreover, the Mercator projection well preserves the general outlines of objects. A variant of such a projection is used by all the basic mapping services: Google Maps, Bing Maps, OpenStreetMap and so on. The difference of the “Web Mercator” is that it is cut off at latitudes of ± 85.0511 ° to make the map perfectly square.

The problem of cartographic projection is solved, but not completely. Yes, walking along the sea along the Mercator is very convenient, but it will not be possible to depict two small sections around the poles. Squares and lengths are distorted. Such a projection may appeal only to those who are indifferent to the problems of cartography, the signature hints.

# Gall - Peters Projection

I hate you.

The layman is accustomed to the Mercator projection, which brazenly distorts the real sizes of countries near the equator. Europe is almost the largest of South America, although in fact it is almost half the size. Greenland is actually 14 times smaller than Africa, although they look about the same for Mercator. The states of Europe and North America look larger than near-equator countries, which belittles the importance of the latter. At least that's what politically motivated criticism of Mercator's projection sounds like.

In 1855, a Scottish priest, James Gall, described a projection called the "Gall Orthographic Projection." In this projection, the area of ​​objects is correctly displayed due to the distortion of their shapes. At that time, the projection had no political implications; it was a byproduct of astronomical observations.

In 1967, the German Arno Peters created an identical projection, and in 1973, during the political climate of the search for social justice, presented it as a new invention. Peters projection was demonstrated as a method of combating imperialism and Eurocentrism. Europe is large and central in the Mercator projection, the author pointed out, and Africa and the countries close to the equator are of the right size on the "new" projection.

The very name "Gall - Peters projection "first appeared in a 1986 publication by Arthur Robinson. The projection is mentioned, for example, in the television series West Wing , which talks about the work of the fictional administration of the US president. The plot implies that cartographers suggested changing the cards in schools to "more socially fair."

The problem of the Gall - Peters projection consists in the initial assumption that the Mercator projection sets as its main or secondary goal the raising of the significance of European states due to their enlargement on the world map. In fact, the Mercator projection is only convenient for marine navigation, nor does it distort the local forms of objects.

The Gall - Peters projection preserves areas due to the distortion of angles, distances and the shape of objects. The distortions are significant, although Peters denied this. Despite claims to the contrary , the Peters projection does not represent the world in the form in which it actually looks.

The funny thing is that this map, on the contrary, depicts poor countries at the equator with distortions. The fact is that on this projection at 45 degrees north and south latitude the shape of the objects does not change relative to real ones. The farther from these two lines, the greater the distortion of the shape. One of the two lines without distortion enters Europe - that is, the area of ​​the world from the influence of which the Gall - Peters projection is so eager to rid us. Thus, Africa is most distorted by Peters, although the wealthy “imperialist countries” (Europe, USA, Japan, Australia) almost do not change their real form.

For comparison: in the Berman projection, the line runs at 30 degrees without distortion, in the Lambert’s isometric cylindrical projection - at the equator.

Distortion ellipses on the Gall - Peters

projection The map in the Gall - Peters projection is rectangular, which is a bit strange for projection in the 20th century. Before the projection of Peters, projections with preservation of scale already existed, although the map was presented as something unprecedented before. Finally, if we assume that the larger the country, the more powerful it is perceived, then Antarctica will be the most important in the Mercator projection.

To summarize, the Gall - Peters projection finds a problem that does not exist, cannot solve it, but admits many new distortions. Nevertheless, she has supporters who sincerely believe that the Mercator projection should be a thing of the past. For example, this projection is used by UNESCO, some British schools and some schools in the state of Massachusetts.

Munroe is not limited to a short remark that he hates those who have this projection loved. Also in the alt text of the comic, he hid a message with a joke in the style of Horatio Kane : “What? You think I don’t like the Peters map because the challenge to my cultural assumptions makes me uncomfortable? And you don’t think that you don’t :: put on glasses :: project? ”

# Hobo - Dyer Projection

You want to avoid cultural imperialism, but you heard bad things about Gall - Peters. You are non-conflict and buy organic products. You use the newly invented set of gender-neutral pronouns and think that the world needs a revolution in self-awareness. Three names are encoded

in the name of the Hobo-Dyer projection : Howard Bronstein and Bob Abramas, the managers who ordered the projection from ODT Maps, and the cartographer Mick Dyer, who completed the order. It was created in 2002 as a more pleasing to the eye version of Gall - Peters.

The map still distorts the shape of the countries. At 37.5 ° north and south latitude, there is no distortion.

Hobo - Dyer projection ellipses

Munroe plays up the history of projection, pointing out that her lover may turn out to be a less radical and more non-conflict supporter of social justice and environmental concern. Such a person buys organic products, that is, those grown with minimal use of synthetic fertilizers, pesticides and herbicides. He uses gender-neutral pronouns to not offend anyone.

Inverted world map in the projection of Hobo - Dyer

# Pierce projection

You think that when we look at a map, in fact, we see ourselves. After watching the movie “Beginning” for the first time, you sat in the cinema silently for six hours. What scares you is that there is a skeleton inside each of those around you. Once you really looked at your hands.

The planet in Pierce's projection is expanded so that an infinite pattern can be made from the projection. It correctly transmits angles in many places except in areas where the equator and meridians suddenly change directions. The equator in Pierce's projection looks like a square, although the real equator has no sharp angles. Pierce himself claimed that the distorted areas on the map take up less space than that of Mercator.

Pattern from the Earth in the projection of Pierce

Maps in Pierce's projection have no meaningful practical applications. But the projection algorithm itself was useful for representing 360-degree images on the plane - Pierce successfully turned the sphere into a rectangle consisting of two squares.

Distortion ellipses on Pierce's projection

The plot of the movie “Beginning” uses a complex nested structure, so the viewer may need several views to fully understand the movie. With the right mentality, it may seem strange to have a skeleton in the body or the complexity of a human hand. Munro probably hints that people who tend to be fixed on small details like the endless pattern of projections of the Earth and the squareness of the equator.

# Van der Grinten projection

You are a simple person. You like the Mercator projection; you just want it to not be square. The earth is not square, it is round. You like circles. Today will be a great day!

In 1922, the National Cartographic Society of the United States adopted the Van der Grinten projection as a standard world map . In this status, it lasted until 1988.

This projection is neither isometric nor equiangular. She projects the surface of the Earth into an arbitrary figure - a circle. It retains the outlines of the continents and countries that are familiar from the Mercator projection, slightly removing distortions. At the same time, the distortion at the poles is enormous - Antarctica seems to be larger than the rest of the land.

Van der Grinten projection distortion ellipses

Munroe presents a fan of such a projection in the form of a frankly stupid and naive person. The sphere and the circle have little in common, but for the lover of Van der Grinten this is one and the same. In fact, this man is not far from the projection of Mercator, hints Munroe.

# Robinson Projection

You have a comfortable pair of running shoes that you wear everywhere. You like coffee and the Beatles. You think Robinson's projection is the best without question.

Robinson's work succeeded Van der Grinten as a projection of the standard world map of the National Cartographic Society of the United States in 1988. A decade later, in 1998, the projection was changed. It is possible that the author of the comic strip himself (Munroe was born in 1984) taught geography with a map of such a projection.

Robinson Projection- This is also neither an equiangular nor an equal projection, but a compromise. Meridians are gently bent, and the poles are pulled into strings. Therefore, the distortions at the poles are huge, but even with a small departure towards the equator, the errors are not so significant. This is largely an artistic picture of the world: the projection is given by a table of values ​​in increments of 5 degrees, and not by a formula. The remaining values ​​are obtained by approximation.

Robinson projection distortion ellipses.

Munroe hints that this projection is familiar and pleasant: he associates her fan with soft and comfortable things. Musical tastes of people are recorded in adolescence . Perhaps Munroe points out that such a person grew up in the era of the Beatles' popularity, that is, he is a man in his fifties and sixty who loves a quiet life.

# Winkel Triple Projection

The National Cartographic Society adopted the triple projection of Winkel in 1998, but you adored it long before this recognition. You worry that this is the end, and you are considering switching to Kavraisky. Once you left the party in disgust when the guest came in with shoes with his fingers. The name of your favorite music genre begins with "post-".

The Winkel triple projection was published in 1921, and in 1998 the National Cartographic Society adopted it as a standard. This is the arithmetic mean between equal- spaced projection and Aitof's projection - so the cartographer Oswald Winkel in his compromise projection tried to reduce all three types of distortion.

A popular comic stereotype claims that if a certain product gains popularity, hipsters immediately refuse to use it. Munroe points out that the fan of the Winkel triple projection belongs to this subculture. Such a person will refuse it in favor of the projection of the Soviet geodesist and cartographer Kavraisky , which is not used in the West , which is generally similar to the triple projection of Winkel.

# Dimaxion

Do you like Isaac Asimov, XML, and toe shoes. You think that Segways just have a bad reputation. You have 3D glasses in which you look at rotating models of 3D glasses better. Your layout is Dvorak.

Why be sure to suffer with a ball? Our planet can be approximately represented as a polyhedron, and its development can be made a cartographic projection. Buckminster Fuller published this idea in Life magazine in 1943.

In this scanning has several advantages. Distortion of the size and shape of objects is less than on the projection of Mercator or Gall - Peters. On such a map it is convenient to illustrate the scientific theory of human migration.

Dimaxion projection distortion ellipses

However, you need to get used to several new concepts. There is no “top” or “bottom” on the map. Fuller argued that the universe has "north" and "south" are absent - there is only "inside" and "outside." In the Dimaxion projection, it is difficult to find the coordinates of objects and indicate the cardinal points. It’s hard to find travel paths; they look unpredictable.

No matter how minimal the distortions are, maps of the Earth are needed to move around the world. Despite the originality and beauty of the idea, the Fuller projection did not receive any practical application.

In the signature, Munroe collected similar cases when outwardly interesting inventions in practice did not have a significant distribution, remaining something niche. The list is not the most popular XML markup language, unusual shoes with fingers, expensive and therefore unpopular Segway mobile scooters. For 3D glasses (that is, early helmets of virtual reality) Munroe hints at the lack of content and applications - this is true for the early 2010s. Dvorak’s layout is designed to speed up typing with respect to typewritten QWERTY, but requires complete retraining and does not demonstrate such significant advantages in practice.

# Waterman Butterfly

Seriously? Do you know waterman And what about the 1909 Cahill map on which it is based ... do you have a reproduction in the frame of the house ?! Wow ... listen, forget about these questions. What are your plans for the evening ?!

Waterman's projection is similar to Dimaxion: the geoid is approximately similar to a polyhedron, in this case, an octahedron. Then the polyhedron is deployed on a plane. As in previous cases, the shape of objects and their true size are not distorted so much. But on such a map it is difficult to indicate the cardinal points and build a path.

Waterman published his “butterfly” in 1996. As Munro recalls, it was based on the work of Cahill in 1909. Architect Cahill also crushed Antarctica into segments, but he did it physically: he created and patentedfolding rubber globe, and then flattened it.

As in the case of Dimaxion, this projection has not received significant practical application. Only a big fan of cartography can know about it. Munroe jokes that he has something to chat with a man who calls the “Waterman Butterfly” his favorite projection.

# Hood Projection

They say that transferring the Earth to a two-dimensional plane is like flattening an orange peel, which sounds simple to you. You like simple solutions. Do you think that we would have less problems, if we would choose normal people in the Congress instead of politicians. Do you think that the airlines should just buy food from restaurants near the terminals and serve it on board. You change the oil in your car, but secretly wonder if you need to.

In 1923, John Goode developed his pseudo-cylindrical isometric replacement for the Mercator projection. Hood's projection was designed to correctly convey the area of ​​objects. The projection has several gaps across the oceans and Antarctica that are designed to reduce distortion.

On some variants of the Hood projectionobjects are repeated. You can perform such a projection with emphasis on the ocean.

The main drawback of the Hood projection is the huge gaps in the oceans that do not allow us to visually assess the distance between the continents. Hood's projection is designed to preserve the area and does this for the continents. But the oceans in it look bigger than they really are.

Indeed, this projection has a comparison with the peel of an orange. Munroe considers the desire to turn the globe into a peel an overly simple and uninformed approach to a difficult problem, such as cartography. He cites parliamentary elections as an example: “normal” people would hardly be able to break into the strictly expressed US ruling class. And in the end, this would hardly solve the country's problems. In addition, who determines the "normality" of a person?

The perception of taste strongly depends on the composition of the air and pressure, and a regular restaurant cannot simultaneously prepare the quantities of food that are needed for 100-300 people. Also, food must be packaged in a special way. If you make a restaurant take into account all these requirements and change its technology, it will turn into a food supplier for aircraft passengers.

The car manufacturer indicates when to change the oil. Our fan of Hood projection sometimes wonders if there is a collusion between car manufacturers and car oil sellers.

All these are traps that you can fall into if you underestimate the problem being solved. Munroe points out that turning a geoid onto a plane is not as simple as it seems to a fan of the Hood projection.

Based on materials Brilliant the Maps , discusson Reddit and Explain xkcd .