Inspired by the sphere, we can study more complicated surfaces, like this bagel shape (called a "torus") using multiple 2D coordinate systems.
If we want to visit somewhere on the bagel where a coordinate system doesn't reach, we can make a new coordinate system by clicking the button above.
Try moving all the way around the surface of this torus. If you experiment, you can see that it's possible to return to where you started, only by moving in straight lines. But returning to where you started is impossible in normal Euclidean space. We've learned something interesting about the torus-shape just from these 2D coordinate systems.

And this strategy doesn't just work on toruses and spheres - it works on even incredibly complicated surfaces, like this one! The key idea: To study complicated spaces, which we might not know how to work with, use tons of Cartesian coordinate systems, which we do know how to work with.
To make this idea precise, mathematicians invented the word "manifold". If you're interested in the technical details, a manifold is the data of these three parts:

• A space you want to study, like this mammoth.
• A ton of Cartesian coordinate systems, called "charts", which together cover all of the space in part #1.
• A ton of "transition functions" which tell you how to change from one coordinate system in part #2 to another.

Our mammoth is a space we want to study, and you've covered it in tons of Cartesian coordinate systems. So this mammoth is a manifold! Next