Understanding Four Dimensional Space

Lots of people have tried to explain this. I think I might be able to do it better than others I have seen.

What I really wanted was to understand what shapes would be like in such a world, and what it would be like for things to move around.

First, we need to know a few things...
Things like:

What is a "dimension"?
A dimension (in general) is anything you can measure. Not just length and height, but also temperature, pressure, time, and whatever else.

Dimensions are generally separate. Air pressure is not the same as time. Four degrees Celsius is not equal to some length or distance.

Three separate spatial dimensions means we can mark three completely different directions/axes to measure. They are "completely different" because if something is taller, it isn't necessarily longer. If something is longer, it isn't necessarily wider. We can call one direction "height", another direction "length", another "width". Or we could use other names ("depth", "thickness", "distance in direction 1", "distance in direction 2", etc.).

We can show you a picture of a fourth dimensional space
It just won't "feel" fourth dimensional to you. Because your brain's visual system is trained to process everything as three dimensional. Bummer. But I'll try to help us understand it anyways!

Difficulty with words in the fourth dimension
Most of our words are made for our 3d world. There's a piece of paper, a ball, a box, and so on. Some words are ok in lower dimensions, there are 2d shapes like squares and triangles and circles and so on, and 1d lines, and longer lines and shorter lines and broken lines and dotted lines.

But what about 4d? Well, there are some words, like tesseract/hypercube and hypershpere. But what about the lower dimensions as they exist in 4d space? Can I just say "cube in 4d" (thin in one dimension, but not the other three), or will that be confused for a tesseract/hypercube (equally wide in all 4 dimensions)? Can you even tell what I'm trying to say right now? And what about arbitrary shapes? It takes a bit of extra effort to communicate, due to the lack of names for all these shapes.

Viewing "lower dimensions" from higher dimensions
In our 3d world, we can't really view lower dimensions. But we can see pictures thin enough in one of their dimensions, flat enough, that it's kind of like a two dimensional thing. Make it thin in another direction, and you suddenly have a line, which is kind of one dimensional. Shorten the line down to a dot, and it's kind of like a zero dimensional thing.

What would this be like in a 4d world? Very similar. If something was thin enough in one direction, it would kind of be like a three dimensional thing! Thin in two directions, it would be like a two dimensional thing. Thin in three directions, it would be like a one dimensional thing. Thin in four directions, it would be like a zero dimensional thing.

Looking at things from different angles
"2d" picture/painting in 3d world. View from different angles. From the side, it isn't really visible. It's flat.

"3d" "picture/painting" in 4d world. View from different angles. From the side, it isn't really visible. It's "flat" (in one of its dimensions).

You can look at a stained glass window (or one of those window paintings sometimes in shop windows) from one side or the other side. Each side is like a mirror image of the other. Walking around the window, your view of it makes it seem to squish, until it is flat, and then enlarge again into the full reversed image. Similarly in a 4d world, a thin "3d" "picture/painting" could be viewed from other side, giving a mirror image. Walking around this object, your view of it makes it seem to squish, until it is flat, and then enlarge again into the full reversed version of the object.

Building things out of "lower dimensional" things
In our 3d world, if you stack a bunch of thin papers (thin in one dimension, but not the other two) on top of each other, you start to get a box shape (thick in all 3 dimensions).

In a 4d world, you could stack a bunch of "thin boxes" (thin in one dimension, but not the other three) on top of each other, and you would get a 4d shape (thick in all 4 dimensions).

Some Youtube Videos, their Pros and Cons
There are a lot of videos on youtube attempting to explain and visualize four dimensional space. Yet none of them seem perfect. Here I'll go through some of them, and point out their strengths and weaknesses.

Drawing the 4th, 5th, 6th, and 7th dimension, by Physics Videos by Eugene Khutoryansky
Pros: Cons:
 * provides a visualization which "fills up " like water, or adding to a stack of papers, similar to what I said in my section above about "building things out of "lower dimensional things"".
 * correctly points out that you can display any number of dimensions on a flat screen
 * visualizes the removal of one dimension of a four dimensional object, bringing it back down to three dimensions
 * different colors for each surface, not just wireframe
 * instructs you to "double the points" to go from one dimension to another, which is unnecessary.  All you need is one additional point for each additional dimension.  They should have said that you need to double the points only if you want all the lines to be connected at 90 degrees to each other.
 * could do a better job of explaining what is going on, and how it generalizes to different shapes

4th Dimension explained by Matthew Dina
Pros: Cons:
 * neat blackboard trick to get you thinking about the dimensions you are seeing
 * correctly shows it is only necessary to add one additional point for each additional dimension
 * fails to show an axis for four dimensions, incorrectly claiming it is impossible
 * a bit too much wireframe and such, made with "3d" image software probably
 * a bit weird, the intro and the reference to "reptiles"