Well, I can elaborate, but I'm not sure how helpful it will be. "No one can be told what the matrix is" and that sort of thing. The basic idea is that it's the equivalent of the line rising out of the paper in two-dimensions, but in three dimensions instead. But that's not telling someone who has tried and failed anything they don't know, I'm sure.

If you really want to be able to visualize higher-order spaces, my advice would be to work with them, do math and computer programming in higher-order spaces, and use that to build up physical intuitions of how things work in higher-order spaces. Once you have the physical intuitions it's easier for your brain to map them to something meaningful. Of course if your reason for wanting to be able to visualize 4D-space is because you want to use the visualization to give you physical intuitions about it that will be useful in math or computer programming, this is an ass-backward way of approaching the problem.

I believe wnoise was making a joke-- one that I thought was moderately funny.

Doing specific rotations by breaking it into steps is possible. Rotations by 90 degrees through the higher dimensions is doable with some effort -- it's just coordinate swapping after all. You can make checks that you got it right. Once you have this mastered, you can compose it with rotations that don't touch the higher dimensions. Then compose again with one of these 90 degree rotations, and you have an effective rotation through the higher dimensions.

(Understanding the commutation relations for rotation helps in this breakdown, of course. If you can then go on to understanding how the infinitesimal rotations work, you've got the whole thing down.)

Adding multiple temporal dimensions effectively how I do it, so one more shouldn't be a problem*. I visualize a 3 dimensional object in an space with a reference point that can move in n perpendicular directions. As the point of reference moves through the space, the object's shape and size change.

Example: to visualize a 5-dimensional sphere, I first visualize a 3 dimensional sphere that can move along a 1 dimensional line. As the point of reference reaches the three-dimensional sphere, a point appears, and this point grows into a full sized sphere at the middle, then shrinks back down to a point. I then add another degree of freedom perpendicular to the first line, and repeat the procedure.

Rotations are still very hard for me to do, and become increasingly difficult with 5 or more dimensions. I think this is due to a very limited amount of short-term memory. As for my technique, I think it piggybacks on the ability to imagine multiple timelines simultaneously. So, alas, it's a matter of repurposing existing abilities, not constructing entirely new ones.

*up to 7: 3 of space, 3 of observer-space, and 1 of time

There's a pretty serious gap between the idea of a person evolved to visualize four dimensions and it being capable of thoughts I cannot think. This might be defensible, but if so only in the context of certain thoughts, something like qualitative ones. But the original quote was inferring from the fact that not everyone can see all the colors to the idea that there are thoughts we cannot think. If 'colors I can't see' are the only kinds of things we can defend as thoughts that I cannot think, then the original quote is trivial.

So even if you can defend 4d visualizations as thoughts I cannot think, you'd have to extend your argument to something else.

But I have a question in return: how would the belief that there are thoughts you cannot think modify your anticipations? What would that look like?