An old question considered

I missed watching this video put out by Sabine Hossenfelder some months ago:

It deals with, although too briefly, the question of whether it is possible to consider the 4 dimensional spacetime universe as being embedded in a higher dimensional universe.   I mean, the idea of extra dimensional objects (or beings) being able to pass through our lowly 3 (spatial dimension) universe was very popular for a while in 20th century science fiction, but I don't think it ever got a mention much in real physics, and I never understood why.   (And yes, I know that string theory was about compacted extra dimensions, but that's different.)

She said (with no further explanation, and starting at about the 6 min 30 second mark) that yes, you could consider our universe to be embedded in higher dimensions and to be expanding into them, but it (generally) takes 10 dimensions to make this work, and as they are understood (or "constructed"?) to be non observable, it is not scientific to think they are real.

Well, now I need to know why it takes 10 dimensions... 

Update:   OK, so according to this explanation in AEON, the 10 dimensions that Sabine mentions is about the extra compacted dimensions that are relevant to string theory - but as I said before, I didn't think compacted dimensions were relevant to the idea of our universe being embedded in extra dimensions that it can expand into.    Anyhow, here is the explanation:

If moving into four dimensions helps to explain gravity, then might thinking in five dimensions have any scientific advantage? Why not give it a go? a young Polish mathematician named Theodor Kaluza asked in 1919, thinking that if Einstein had absorbed gravity into spacetime, then perhaps a further dimension might similarly account for the force of electromagnetism as an artifact of spacetime’s geometry. So Kaluza added another dimension to Einstein’s equations, and to his delight found that in five dimensions both forces fell out nicely as artifacts of the geometric model.

The mathematics fit like magic, but the problem in this case was that the additional dimension didn’t seem to correlate with any particular physical quality. In general relativity, the fourth dimension was time; in Kaluza’s theory, it wasn’t anything you could point to, see, or feel: it was just there in the mathematics. Even Einstein balked at such an ethereal innovation. What is it? he asked. Where is it?

In 1926, the Swedish physicist Oskar Klein answered this question in a way that reads like something straight out of Wonderland. Imagine, he said, you are an ant living on a long, very thin length of hose. You could run along the hose backward and forward without ever being aware of the tiny circle-dimension under your feet. Only your ant-physicists with their powerful ant-microscopes can see this tiny dimension. According to Klein, every point in our four-dimensional spacetime has a little extra circle of space like this that’s too tiny for us to see. Since it is many orders of magnitude smaller than an atom, it’s no wonder we’ve missed it so far. Only physicists with super-powerful particle accelerators can hope to see down to such a minuscule scale.

Once physicists got over their initial shock, they became enchanted by Klein’s idea, and during the 1940s the theory was elaborated in great mathematical detail and set into a quantum context. Unfortunately, the infinitesimal scale of the new dimension made it impossible to imagine how it could be experimentally verified...

It goes on to explain that the idea got revived in the 1960's to help explain the weak and strong nuclear forces:

Kaluza’s and Klein’s ideas bubbled back into awareness, and theorists gradually began to wonder if the two subatomic forces could also be described in terms of spacetime geometry.

It turns out that in order to encompass both of these two forces, we have to add another five dimensions to our mathematical description. There’s no a priori reason it should be five; and, again, none of these additional dimensions relates directly to our sensory experience. They are just there in the mathematics. So this gets us to the 10 dimensions of string theory. Here there are the four large-scale dimensions of spacetime (described by general relativity), plus an extra six ‘compact’ dimensions (one for electromagnetism and five for the nuclear forces), all curled up in some fiendishly complex, scrunched-up, geometric structure.

And there's more explanation that Witten came up with 11 dimensions:

There are many versions of string-theory equations describing 10-dimensional space, but in the 1990s the mathematician Edward Witten, at the Institute for Advanced Study in Princeton (Einstein’s old haunt), showed that things could be somewhat simplified if we took an 11-dimensional perspective. He called his new theory M-Theory, and enigmatically declined to say what the ‘M’ stood for. Usually it is said to be ‘membrane’, but ‘matrix’, ‘master’, ‘mystery’ and ‘monster’ have also been proposed.

So, what about the type of extra "large" dimensions that was the subject of Flatland and science fiction?   Well, it might be there, as AEON explains:

In 1999, Lisa Randall (the first woman to get tenure at Harvard as a theoretical physicist) and Raman Sundrum (an Indian-American particle theorist) proposed that there might be an additional dimension on the cosmological scale, the scale described by general relativity. According to their ‘brane’ theory – ‘brane’ being short for ‘membrane’ – what we normally call our Universe might be embedded in a vastly bigger five-dimensional space, a kind of super-universe. Within this super-space, ours might be just one of a whole array of co-existing universes, each a separate 4D bubble within a wider arena of 5D space.

OK, that's more like it.

Update:   I realised on the weekend (because Youtube pointed it out to me) that Sabine Hossenfelder had done earlier videos on the extra dimensions idea, explaining the stuff that appeared in the AEON article above.   I didn't previously realise that the idea of compacted extra dimension had been around for so long, with string theory really just reviving it.