visualizing_the_fourth_dimension

You have all heard the common wisdom: it is not possible for humans to visualize a fourth dimension of space, without resorting to over-simplistic "flatland" analogies.

I am here to tell you: the common wisdom is wrong. I can visualize the fourth dimension. And in the following exercise, I will show you how you can too.

In this exercise, we will consider a simple hypercube of four dimensions.

First, we must name the cube's dimensions, and the units with which they are measured.
We will call the first dimension "length".
We will call the second dimension "width".
We will call the third dimension "height".
And we will call the fourth dimension "depth".
We will call the 2-space of the hypercube's base "area".
We will call the 3-space of the hypercube's body "volume".
And we will call the 4-space of the entire hypercube "fullness".

The common unit of length, width, height, and depth will be the "meter".
The unit of area will be the "square meter".
The unit of volume will be the "cubic meter".
The unit of fullness will be the "quartic meter".

THE GOAL OF THE EXERCISE IS TO RENAME THE QUARTIC METER.
Specificly, we must give the quartic meter a name which is intuitive, which allows us to visualize it as a property of ordinary objects, and which is mathematically sound.

First, observe that, because (length)*(width)=(area), it necessarily follows that (area)/(length)=(width). Thus, the meter of width can be rewritten as the "square meter per meter of length". This is abbreviated as mm2, which obviously simplifies to m. (In other words, width can be measured using a horizontal line of squares.)

Next, observe that, because (area)*(height)=(volume), it necessarily follows that (volume)/(area)=(height). Thus, the meter of height can be rewritten as the "cubic meter per square meter of area". This is abbreviated as m2m3, which, again, simplifies to m. (In other words, height can be measured using a vertical stack of cubes.)

Now, you may be wondering where I'm going with this. Well, here's the reveal.

Because (volume)*(depth)=(fullness), it necessarily follows that (fullness)/(volume)=(depth). Thus, the meter of depth can be rewritten as the "quartic meter per cubic meter of volume". This is abbreviated as m3m4, which, just as before, simplifies to m.

Why do we care? Well, recall that THE GOAL OF THE EXERCISE IS TO RENAME THE QUARTIC METER.

What fits in the blank?
"_________ per cubic meter."

The only intuitive answer...
... is "kilograms".

And so, we arrive at the conclusion of our exercise: mass may be thought of as a form of hypervolume; and density may be thought of as a linear spatial dimension. One might say, for example, "Water has a density of one meter," and be perfectly correct in saying so, if, by convention, the quartic meter were defined as 1000 kilograms.

All four-dimensional geometry flows from this revelation. A sphere of even density is a hypercylinder; a sphere whose density increases towards the center is either a hypersphere or a hypercone. A cube whose density increases towards the center is a hyperpyramid. A sphere of zero mass is a hyperdisc. A singularity of any mass is a hyper-line.

The fourth dimension no longer seems so mystifying; does it?

Reaction 1: I must admit, it's pretty clever. It's not physically correct (at least not for non-elementary particles), but clever. If it were physically correct, then you should be able to rotate over mass+space, like special relativity is a rotation over time+space - there should be observers to whom mass appears as distance, and distance in one dimension as mass. Instead, if you do that, m^2=E^2-p^2 and s^2=t^2-x^2.

All you've basically done is intuitively derive that parameters are isomorphic to dimensions. You can do the same for other parameters, and the collection of all these parameter dimensions is called phase space.

Explanation 1: Observers would see density as distance, not mass. (And vice-versa.) Thus, black holes would be points to some observers, and lines to others.

I'm not suggesting that there exists a fourth dimension of space, and that that dimension is density; I'm suggesting that density may be treated, mathematically, as hypervolume perpendicular to 3-space, and that it therefore represents an ideal didactic analogy for the fourth dimension, far superior to the "flatland" analogy of Feynman, Sagan, and others. (Though, the flatland analogy is not wholly without merit, and should certainly not be abandoned; it should merely be supplemented with this exercise.)

Reaction 2: OP, what you're imagining isn't a spatial dimension at all. It is a function of three variables. Density isn't the only function of (x,y,z), so is pressure, temperature, electric fields, etc. A function of (x,y,z) can't even map to a fourth dimension (unlike time).

I typically think of more than three dimensions as just a free parameter. So in a problem with six dimensions/variables, the best we could hope for is visualizing a 3D plot of three of those dimensions, and watching how it changes as we vary the other variables.

Reason 2: Take a sphere.
f(x y)=r2−x2−y212
z=r2−x2−y212
Therefore, z=f(x y). Every spatial dimension can be expressed as a function of every other spatial dimension.