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Make a ring in your mind, have it spin about one of its axis.
Make another ring inside the first, have it spin on a different axis than the first.
Make a third ring, put it inside the first two, have it spin on a different axis than the second.
That is as far as I can go, it's hard, but I can make it look right if I concentrate, and it has to be in one particular orientation. I absolutely can not get a 4th ring into the mix, the details get fuzzy and I can't get it to move right. How far can you go?
I tried visualising 4 dimensions once or twice, but it doesn't really work out for me.
So I have no way to get that 4th axis for the 4th ring.
It's extremely easy upto the 2 rings and I am also able to change the axes while moving them.
The moment I add a 3rd ring, the difficulty seems to abruptly increase, making me take quite a while to imagine it and even the slightest diversion, making me forget the simulation.
I just realised that in your explanation, you didn't explicitly require orthogonal axes. Now it makes more sense to try the 4th ring. Perhaps I will try doing that when I am about to sleep, after doing the 3 rings for a while and see if I am able to do so.
There is a way to cheat yourself out of this, by first making the animation and watching it on a screen and then internalising it. I will keep that for later, after knowing how well I can do without it.
I also feel the need to say that while in my above description, I am able to do something similar to visualisation, that doesn't really help me in drawing stuff. Even when I try using a 3D CAD program, that way of visualisation ends up making it harder for me to make it. Instead, I need to force myself to reduce the extra information of touch and feel and taste, to make it easier for me to model it and it takes further effort to make a 2D projection.