Flat in this case doesn’t mean literally 2 dimensional like a piece of paper. What we mean by flat here is not curved. In other words, the universe obeys Euclidean geometry. This is the geometry we’re all most familiar with in our daily lives, i.e., parallel lines never meet and the sum of angles in a triangle is always 180 degrees.
Something that’s not flat is curved. Think of the surface of a sphere. On the surface of a sphere, parallel lines will always meet and the sum of angles in a triangle is not 180 degrees. In this analogy, the surface of a sphere is a 2D representation of what curved 3D space would be like.
Our measurements point to our universe being flat or extremely close to flat.
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