and extra note.
the 4-D can be interpredted in many particular types of problems
1) the Euclidean way (a hypercube, hypersphere) in other words your regular geometic extensions of the circle , the square, and so on.
2) as the coordinate system for imaginary equations such as (X+ i)2=0, so you let x=real, y=i, z=-real and w=-i
3) x=lenght, y=widht, z=depth and d=time
4)as defined by hamilton, he define the properties that would make a 4-D system be a field . in other words it has an identity, it has inverses except for 0., the associative property, the distributibe property and ..... property hold. and so on . but be careful here the definition of inverses is not like in the real numbers (it's more like a hyperbolic coordinate system)
so you see that these are specific usage of a 4-D system, but trying to see 4-D objects in general has nothing to do with these specific definitions of the 4tth dimension in general.
a good way see the fourth dimension is to draw a cube in 3-D but instead of using dotted lines to connect the sides that you do not see, use solid lines. and stare at it . What do you see?
you should be able to see a box sometimes you see the bottom sometimes you see the inside, guess what that flipping of the box occurs in 4-D. not possible in 3-D unless you move the box up and down with your own hands, but if that movement happens just by staring at it . that can only happen if the flips in 4-D. easy example of seeing the power of 4-D.. ENJOY
