# Flat and convex maps

Like many other animals, humans are territorial. This implies having a convex perception of space. To understand it we can observe a convex mirror. We have a very precise idea of our immediate environment, so precise that we tend to magnify its size, while the mental image of the environment becomes imprecise and diffuse as we move away. A good example is found in ancient cartographies, such as the map produced in 1375 by Catalan cartographer Abraham Cresques, or the map of 1402 prepared by Korean cartographers Yi Hoe and Kwon Kun. This perception can still be seen today if we start to draw the world map from memory. You will see that what you draw best is your country, and as you move away, the shapes of the continents are increasingly imprecise and awkward.

The fact is that maps are actually two-dimensional representations of a reality that has three. Maps are flat but Earth is spherical. The German mathematician and astronomer Carl Friedrich Gauss demonstrated in the 18th century that the surface of a sphere cannot be represented as a plane without deformation. We can see this with an inflatable globe. If we cut it with a knife in two halves you will see that it is impossible to lay it flat on a table. There are always wrinkles or bubbles. The mathematical procedure for flattening a sphere is called a projection, and there are dozens of different types. The basic ones are cylindrical, conical and azimuth, and can be tangent or secant. Probably the best known is Mercator, which is a tangent cylindrical projection, its main advantage is that it preserves the perpendicularity of meridians and parallels, and therefore, it is very useful for navigation. However, it has the drawback that it greatly deforms the surface of the continents at high latitudes. At the opposite extreme we find the Gall-Peters projection, which preserves the relative surface of the continents, but greatly alters the shape. The consequence of Gauss’s discovery is that there is no optimal projection, they all have strengths and weaknesses.

Maps serve to guide us and to travel. The big transformation in location came with space technology. Placing artificial satellites into orbit revolutionized telecommunications and allowed the introduction of the global positioning system, GPS. Formed by a network of at least 24 satellites, arranged in 6 different planes, at about 20,000 km altitude, they orbit the Earth twice a day. At this distance they are subjected to a gravity 17 times lower than that of Earth, therefore, they experience Einstein’s relativity effects, which must be corrected.

We have technologically sophisticated phones or watches with GPS, but instead, the cartographic basis they use is still the Mercator projection of the year 1569, with its limitations. Since there is no optimal projection, the only way to see Earth with the continents in the correct size and shape is by looking at the globe. We use a convex map, according to a convex perception.