What Is a Manifold?

The bug can move forward or backward; there’s no way of telling whether it is living on $\mathbb{R}$ or on $S^1$, as mathematicians like to call the circle. From the perspective of our small bug, $\mathbb{R}^2$ looks like this:

In other words, the bug can move in two independent directions here, and its world will appear to be a large plane, much like the Great Plains would appear to us. Already in the $1$-dimensional example, we asked the question how a bug might be able to figure out that they are living in $S^1$ or in $\mathbb{R}$.

Source: bastian.rieck.me