Ok I see, it's a 2D shape in 3D space, you actually need the 3th dimension to contain this shape... At least I find that there's s distinction between a 2D shape you can draw on a 2D screen (like a filled rectangle or a disk), and a shape that's a 2D surface itself but requires 3-dimensional space to sit in (like a torus/donut or a non-filled 2-sphere)

So I guess the 126-dimensional shape actually also is in 127-dimensional space then

But the article says "Over the years, mathematicians found that the twisted shapes exist in dimensions 2, 6, 14, 30 and 62.".

To me "Exists in dimension 2" sounds like a shape in 2D space, not in 3D space, but apparently that's not what they mean and the way I understand this language is wrong

> So I guess the 126-dimensional shape actually also is in 127-dimensional space then

Sometimes you need more dimensions to embed the manifold. For a 2-dimencional object, the most famous example is the Klein bottle https://en.wikipedia.org/wiki/Klein_bottle You can construct one of them in 3-dimmension only if you cheat. Yhey look nice and you can buy a few cheating-versions. But you can embed the Klein bottle in 4-dimensions (without cheating).

For the manifold in the article, I'm not sure how many additional dimensions you need. Perhaps 127 (n+1) is enough or perhaps you need 252 (2n) or perhaps something in between. You can always embed an n-dimensional manifold in the 2n space, but that is the worst case. https://en.wikipedia.org/wiki/Whitney_embedding_theorem