Now I'm happy to be completely wrong here, but I have a feeling that the disappearing of a 4D object in a 3D world cannot avoid using time to reveal the 4D nature of it.

Just like a in a 2D plane, to represent a 3D object and see all of it, you have to iterate though time to see it all, so parts will disappear from our perception at a given time interval.

That's if you do a "projection", like light rays from a 3D object focused onto a screen. This is more like taking a 3D slice of a 4D object. If you think of taking a 2D slice of a 3D object, you can see how objects disappear when they move out of the plane that you're drawing.

If you sliced a tesseract then presumably you'd encase it in the knife and a square, with no depth (the depth is in the 4th dimension that we're not experiencing), would appear?

Hypercubes have always been a difficult one for me to intuit, basically I have no 4th dimensional intuition. When I think of a 4D hypersphere all I can get is a simple sphere. I don't have any intuition as to whether that's wrong, it seems in a way it should be a sphere with infinite spheres on it's surface - from analogy with the tesseract - but from analogy of constructing a sphere from a circle it should be a case of rotating the sphere around itself perpendicular to the extra dimension?

It might help if you consider spheres as a surface.

3D spheres are a 2D surface wrapped into a 3D space, likewise, hyperspheres would be a 3D surface wrapped into a 4D space. There's no "infinite spheres on its surface", I think the "rotation" is a better analogy.

Take a line rotated around an orthogonal axis and you have a circle, a circle rotated around an axis orthogonal to the other two is a sphere, a sphere rotated around another orthogonal axis is a hypersphere.

Are all spheres hyperspheres? When you rotate in the 4th, or higher ordinal, dimensions don't you get the same shape?

I guess it would depend on whether you consider all circles to be spheres (or all spheres circles)?

You could take a 2D slice of a tesseract, but what most visualizations do is take a 3D slice and then project that in a more familiar way to 2D. So you could get any of these shapes: https://en.wikipedia.org/wiki/Tesseract#/media/File:Orthogon... If you 2D-slice a regular cube you could get various shapes too. https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-g...

A hypersphere has a similar description to a regular sphere or a circle, which is that every point a fixed distance from the center is part of the sphere. So I'd say it's a little boring. No matter what 3D section you take, it's just a larger or smaller sphere. It gets smaller if you take a slice off to the side, just like taking a slice of a sphere gives you a smaller circle near the side.

A 3-sphere (hypersphere embedded in 4 dimensional space) is a sphere that can be incrementally rotated in 6 orthogonal directions. :-)

I think the difference is projection versus cross-section. A video game is a 3D world projected on a 2D screen. The game is not a cross-section of the 3D world. In this game, you play within a cross-section of the fourth dimension, which you can move along a slider, sort of like a CT scan.

That having been said, I don't have any clue what a 4D projection into 3D space would look like. I suspect we would have much more difficulty understanding it.

Well look at the video for projection. But you raise an interesting point, perhaps we could encode the 4the dimension in color or the stereoscopic channel (with VR gear) and remove this channel from the 3D part as we can do 3D without stereo.