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.
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.
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?