The thing with 3 dimensional objects is that you get a lot of those in real life. All it takes to show a blind person what 3 dimensional looks like is to pick up a keyboard, show its different sides, and describe that "This is the height, this is the width, and this is the depth."

What? How would you think of three dimensional objects without ever being able to feel them? (I say as a someone that's got reasonable vision, but surely sound and especially touch is much better at spatial positioning that mere faulty "stereoscopic" vision -- there's a reason why you might want to practice martial arts with a blindfold...).

As a counterpoint it's routine in topological mathematics to work with 8 or 12 or 11 dimensional objects.

Good point! But is this done without any reference to 3 dimensional objects, or does it take the principle and not make any assumptions, or describe it in such a way that the concept is explained without any aspects related to sight?

Genuinely curious...

It is ... difficult for me to describe just how dangerous it is to try to use 3 dimensional intuitions in a higher dimensional space. As the number of dimensions increases the amount of "space" there is for things to get weird get much more complicated; for example any knot that exists in 3 dimensions can be untied trivially in a 4 dimensional space.

Suffice to say if you try to use your ordinary everyday life intuitions doing higher dimensional mathematics you will continuously embarrass yourself until you learn not to. Some folks develop the necessary adjustments and have higher dimensional intuitions; I do not and have to do the math anywhere higher than around 4.

You could take a linear algebra approach and explain dimension as the number of possible independent directions of movement. This could be experienced through touch. Start with a pencil, then a piece of paper, then 3-space, and so on.

> I'm genuinely curious how you would think of three dimensional objects without ever being able to see them... [...] > without reference to anything you can see with your eyes, how would you describe space?

But the thing is: I wouldn't have to. People without sight live in space too. What mental models of the universe anyone has inside their head may vary wildly, but at the end of the day a torus is still torus-shaped when the lights are out, and West is to your right when you turn North while upside down.

well, you can feel depth and surfaces with your hands, i don't think that is a problem.