Can someone explain 4th dimensional objects?

geometry

I'm not sure if I should ask this in mathematics or in physics.
From what I can tell, there are only 3 dimensions: X, Y, and Z. However, I have seen a lot of things about fourth and even fifth dimensional objects. I have tried for a year or two now to wrap my mind around the concept but I never have been able to do so. Can someone please enlighten me?

Solution 1:

The thing is you're not supposed to "wrap your mind around" higher-dimensional shapes.

Instead, what happens is that we take a formalism that is made to describe 3-dimensional shapes (which we can understand more or less intuitively), and then we just see what happens when we replace all of the "$3$"s in that theory with "$4$" or "$5$" or more. The outcome of this is occasionally useful, but it's not supposed to be about things that can exist in the world -- the utility comes because the theory can be used to reason about certain problems that are not in themselves about shapes.

With a bit of experience one can gain a more or less reliable sense of when one's intuition about 3-dimensional situations gives valid results about higher dimensions, but the touchstone of that is always what one can prove in the formal model we're really speaking of.

Solution 2:

There have been people who reportedly can visualize things in four dimensions as easily as other people can in three. It's rare, however. Moreover, visualizing four dimensions may not help much when you want to solve a problem in five dimensions or more. So as Henning Makholm's answer states, to do anything really useful in higher dimensions you need a mathematical model in which you can formally work out the answers.

It is a nice mental exercise to try actually to visualize four-dimensional objects, however, so I recommend not to stop trying. One way to do this is to try to "construct" well-shaped four-dimensional objects. Consider the following pattern (from http://www.math.union.edu/~dpvc/talks/2000-11-22.funchal/cube-unfolded.html) that folds into a three-dimensional cube:

unfolded cube

The pattern itself fits in two dimensions, in a flat plane, but in order to assemble the cube you have to make parts of the pattern "pop out" from that plane so that you can join the edges that need to be joined.

By analogy, the following (from http://im-possible.info/english/articles/hypercube/) is a three-dimensional pattern from which a four-dimensional cube (known as a hypercube or tesseract) might be constructed:

enter image description here

In the assembled tesseract, cubes that are joined only at one edge in this pattern need to be joined at the faces adjacent to that edge. To do this, you have to make the cubes "pop out" of three-dimensional space. The fun part is trying to imagine where the cubes can "pop out" to.

In Robert Heinlein's story, "And He Built a Crooked House," someone builds a house in this shape and it folds up into a real tesseract with people inside. Some of the details of the story explore how the rooms become connected in the folded tesseract. There is at least one line of sight in which someone could see themselves as if they were some distance away.

Solution 3:

There are text explanations, so I will post some pictures.

We will try to build some intuition by comparing to zero-, one-, two-, three- and four-dimensional tables/arrays.

Introduction:

What's zero-dimensional array you ask? It is a single cell, as single point is a zero-dimensional space.

zero_and_one_dimension

Then there is one-dimensional array and it can be split into $n$ one-less-dimensional strucutres. Similarly a segment can be split into infinitely many points.

When we go into second dimension, we can split in different ways: either rows of columns first. Still, whatever way we do it, a 2D array is still a collection of $n$ 1D arrays, which can be split up into $n^2$ two-less-dimensional structures.

two_dimensions

This pattern holds in three dimensions, where we can split and split and split our structures. In other words, a 3D array is a collection of $n$ 2D arrays, or a collection of $n^2$ 1D arrays, etc.

three_dimensions

That is, we can think of 3D array as 1D array of 2D arrays...

vector_of_matrices

... or 2D array of 1D arrays, etc.

matrix_of_vectors

The fourth dimension:

Now, let's go up one dimension:

vector of 3D arrays

This is a 1D array of 3D array, or (1+3) = 4D structure. It's hard to imagine visually, but you could try scaling the cubes into the inside (and ignore that tiny little voice which says "such overlaps cannot happen in reality"). Another way to think about it is to mentally remove the inside of the cube and leave only the shell (6 faces), then such a thing can be scaled and combined with no overlaps. Trying with wireframes only it looks like this:

scaling cube

Or we could move it to the side:

moving cube

However, in 4D we can do yet something else, namely 2D array of 2D arrays:

matrix of matrices

This is also (2+2) = 4D structure, and you can think of it as a product of two squares (with full insides). That would also be a bit hard to imagine, so to make it simple, take squares with no insides (only the edges). To further boost our intuition, observe that product of two circles gives you a torus. The wireframe for the 4D cube will look a bit alike, although remember that these are two very different objects.

from torus to hypercube