There certainly are ways to ‘see’ more than 3D.
If a hypersphere were to intersect my 3D world right in front of me. I would see a point grow into a sphere, get bigger, then start shrinking back down to a point, and then disappear.
Now, I can assign direction names for travel in the 4th dimension, perhaps call them fourwards and quatwards. Then when the sphere appeared again and was growing before me, I can call to the hypersphere, ‘Stop!’ and it would remain in the same size.
Then I’d say, “Which direction are you going?”
“Fourwards,” the hypersphere would reply.
“Go three inches quatwards, please,” I would say.
“Sure.” And the sphere would shrink a bit.
Then I’d think, “Now, if I were able to go three inches quatwards, I’d see it grow back into the size it was when I said ‘stop.’”
Thus, I’d be putting together in my mind, the total shape of the hypersphere, even if I couldn’t see it all at once.
It’s like watching a merry-go-round. You don’t see all the horses at once, but you get an intuitive feel of where all the horses are at any one time. You can, in your mind, open up the merry-go-round and see all the horses lined up in order in a straight line. You can’t see all the sides at the same time with actual sight, but you get an intuitive sight of the entire structure.
There are other tricks to help the mind ‘see’ more than the three spatial dimensions.
When I used to play the ancient video game, Asteroids; I used to be quite good ‘visualizing’ how I can shoot off one side of the screen to hit something on the other side of the scree (a wrap-around screen world). Though I couldn’t see it with actual sight, I intuitively could see it.
Skeet shooting is 4D sight, if you count time as a fourth dimension (which you can, thanks to Einstein). You’re targeting in 3D, but have to antipate future placement in the 3D world in time. That’s seeing in 4D.
Peace.