what Manifolds are and what they’re used to understand / interpret / model?

Question

what Manifolds are and what they’re used to understand / interpret / model?

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Answer 1

When you hear manifold, think surface.

Now the word surface generally refers to 2D manifolds. Meaning, if you pick a point on a surface, you only need two numbers to describe where with is. The surface of a sphere, a 2D plane, a saddle, a bow. These are surfaces. These are 2D manifolds.

Manifolds needn’t be 2D. The circumference of a circle is a 1D manifold. The boundary of a hypersphere is a 3D manifold.