Cutting a sphere into "almost square" sections

Question

Mathematics

I recently had a question: is it possible to cut a sphere into small sections that strongly resemble squares (and the smaller the sections, the closer each of them to a square)?

I intuitively feel that “no, you can’t”, but I’m not sure. Options with a few unique non-square plots are easy to think of, but not very interesting…

Answer 1

Imagine a cube. Inflate it into a sphere. The result is a sphere, divided into 6 squares.
If you need smaller squares, then you can divide each face into 4 squares.

Answer 2

This option easily comes to mind, but I want to

the smaller the areas, the closer each of them to the square

Ideally, you also want each “square” (in quotes, because in reality it is “almost a square”) has exactly 4 neighbors, and exactly 4 “squares” converge at each node.

Answer 3

If I understand the question correctly, then yes. After all, atlases (and cartographic services) exist - just a sphere > squares (tiles). Search for "Mercator projection" ( for example )