# Mathematics down to Earth

A map is an image build up by a mathematical process called cartographic projection, a tool that allows us to move from the Earth – a “sphere” – to a flat surface. The Mercator, Gall-Peters, Eckert VI projections are just some examples of these tools created in the course of history. Why are there so many different cartographic projections? The exhibition develops following a description of the main cartographic projections, narrating their origins and their mathematical properties. Different necessities during the history gave rise to different cartographic projections. In some cases they became obsolete and in other cases they are still used today.

The tour of the exhibition will become a voyage along geography, history and mathematics. The creation of cartographic projections has been, and continues to be, of fundamental relevance in the evolution of the modern society.

## Elements of the exhibition

The main element of the tour is the collection of maps, each one together with a description from the point of view of history and from the point of view of mathematics. The six maps are the Mercator projection, the Gall-Peters projection, the equidistant azimuthal projection, the sinusoidal projection, the Lambert conformal conic projection and the gnomonic projection. We choose them among the great number of possible other examples because of their relevant meaning in history and of their peculiar geometric properties. Many other maps are presented in the exhibition with a more concise description.

Four interactive experiences and some videos enrich the exhibition giving the public the possibility to learn by playing with the concepts.

Exhibit “Comparison of cartographic projections”

This software allows the public to interact via a touchscreen with different cartographic projections. The guests can draw loxodromic and geodesic lines, or draw free hand and see the results on the earth and on a map to compare them.

Earth Race

This software allows two visitors to play the role of aviators that have to make a race. Two airplanes move at a constant speed and, if the cloche is keeped free, the route turns out to be a geodesic line. The planes fly on a map that can be choosen at the beginning of the game (among the maps presented in the exhibition).

The winning strategy then is to understand how is the shape of the geodesic lines on the map in order to choose the most efficient way to get to the goal.

Fuller’s puzzle

Here the public can try to piece together three puzzle, obtained from three Fuller’s maps (the net of an octahedron, of a cube and of an icosahedron, respectively).

Videos

These videos help to give some interesting descriptions of the mathematical background of the exhibition.

• What is a cartographic projection?
• Conformal maps, area-preserving maps
• Geodesic lines
• Loxodromic lines
• Gauss’s Theorema Egregium