We know that it is not possible to draw the net of a sphere like cylinders, cones, or polyhedra. So, how we can represent a 3D sphere on a 2D paper? Our world is pretty much a spherical shape too, then, how come we have so many different world maps?
When we peel an orange, we cannot flatten the peels entirely on a two-dimensional surface (at least without deforming them!).
Mathematician Hannah Fry explains and demonstrates the orange peel idea in the video of A Strange Map Projection. That's why the all the world maps are somewhat distorted.
You may try to draw a rough world map on the orange and then peel it.
Like you can peel an orange in different ways, there are different map projections for our spherical world. You can try the interactive by Mathigon.org to see a few of these projections and how they distort the actual size and the locations of the continents.
Many Mathematicians tried to converge the sphere to different polyhedra to draw it net, therefore the map of the world.
For instance, Buckminster Fuller designed his map using triangles since he uses an icosahedron (A Platonic Solid with 20 triangular faces) as the main shape of our world.
This projection style is called Dymaxion (Fuller) projection.
One of the most famous polymaths of human history, Leonardo Da Vinci, used eight congruent Reuleaux Triangles* as the net of the sphere.
*A Reuleaux triangle is a shape with constant width like a circle formed by the intersection of three identical circles.
To see the collection of all known notebooks of Da Vinci, please visit discoveringdavinci.com
The notebook that we are looking for is “Codex Atlanticus”, and it is original pages are in this beautiful library in Milano;
You may visit the Ambrosiana Library virtually using Google Arts and Culture.
The better news is that we can find this 1119 page - notebook online and categorized as algebra, geometry, physics, natural sciences, etc ...
Another reason that this library is a sacred place for the mathematicians is, it also has the original copy of “Divina proportione” by the Italian Mathematician Luca Pacioli.
Divina Proportione is about the mathematical and artistic proportion, golden ratio and its applications. While Pacioli was writing this book, Da Vinci was taking mathematics lessons from him. In the book, Da Vinci illustrated two views for the solid shapes;
a solid view and a skeleton view where he removed the faces to better reveal the complete structure of the polyhedron.
These sketches provide a complete view of the number of sides, faces and the vertices of the polyhedra. In terms of functionality, they serve the same purpose as the 2D Nets of the solids. Let's have a look how did Da Vinci sketch the sphere for Pacioli's book;
...and as we have started, we end up with an orange in our search of the net of a sphere.