First: I recommend you read the book Flatlanders. You can probably find it online.
Here's one place
edit: I didn't notice above that Matthias99 also recommended it. That's 2 of us now, plus probably anyone else who's read it will recommend it to you, since you asked this question. /end edit
I did my senior thesis in college on modelling the 4th dimension by looking at 3 dimensional shadows and intersections. Some will argue that the 4th dimension is time, but go beyond time for a moment.
Imagine you live in a two dimensional world. Time would be your 3rd dimension. You would be incapable of understanding 3-dimensional concepts such as volume. Now, imagine a 3-dimensional object passes through your 2 dimensional world. Let's say it's a sphere. You would only see the intersection of that 3 dimensional object on your "2 dimensional universe" (although, some of the physicists in your universe may have postulated that there are more than 2 dimensions, more than 3 in fact, if you're considering time to be the 3rd dimension.) The intersection of that sphere as it passes through your universe would be a circle of changing radius, and perhaps moving (depending on the angle at which the sphere struck your universe, from a 3-d perspective) You would see that 3-dimensional object as a moving 2 dimensional object. So, if you think of the 3rd dimension as time, you're all set. The object is a moving circle with changing radius. Except, remember the premise: this is a 3 dimensional object - a sphere. So, all those moments in time where you're observing particular circles -- well, all those circles exist simultaneously as one object.
Shapes such as a cube passing through 2nd dimensional space could appear as squares OR, they could appear as a number of other shapes, depending on the orientation of the cube (take a cube and slice it in some direction not normal (perpendicular) to any of the surfaces - you get triangles, and IIRC, you can even get a hexagon.
Ready for a headache?
Take a straw. Tilt it at 45 degrees and look at its intersection with a piece of paper. Oval in shape. Now, imagine a straw that's kind of oblong, such that when tipped 45 degrees, its intersection forms a circle on the paper. Imagine the straw moving vertically (not along its own axis) toward the piece of paper. It would trace out a sequence of circles moving on the paper. So, if you lived in a 2 dimensional world, what you'd observe is a circle moving in your world. BUT, we know that it's really a 3 dimensional shape and all those circles exist at the same time as separate planes through the straw.
pic to help
Now... since most of us on this forum can comprehend 3-D in our minds, this is no problem.
Think about this long enough, and you WILL have headaches
(some of my thesis was related to this)
Picture a basketball rolling down the street.
How do you know it's not a 4-dimensional object, all of those individual moments in time that you observe actually all exist at the same time some place on that 4 dimensional object - and you're simply looking at a 3 dimensional intersection with what we perceive as a 3-d universe?
The challenge of the thesis to me, was writing the software that would take a 3 dimensional cross section of a 4 dimensional object, and represent it. Are you looking at your computer screen right now? That's right... what you would be observing is a 2 dimensional snapshot of a 3 dimensional object created by the intersection of 3 dimensional space with a 4th dimensional object. Look at a picture of a cube - your mind can perceive the 3 dimensional object that its a 2 dimensional image of. NOW, train your mind, if you can, to look at a 3 dimensional image (a sculpture of something - it'd have to be 3-D) that's really a 4 dimensional object. Close one eye (to simplify my next statement) What you're perceiving as 3 dimensions is really just 2 dimensions plus time constructed together in your brain to form a 3-dimensional idea. With one eye, there's no depth perception, but with a minimal amount of motion (time), your mind deduces the distance to objects and thus the 3rd dimension.
I need some tylenol now.
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