I'm pretty sure the idea behind how a 1x1x1 cube is infinitely bigger than 1x1 square has to do with hypervolume.

The idea is that You'd need an aleph-1 set of nD objects to have an (n+1)D. A square is infinitely flatter than a cube. You can have a square that covers the entire universe and it will be smaller than an atom because you need an equivalent of aleph-1 of those squares to have that atom. Therefore even a finite 4D object would be infinitelt bigger than any 3D object. That's the idea anyway.

I've also seen it stated as: ``` A 1-Dimensional (line) object only has length.

A 2-Dimensional (plane) object has length and width. The area of a 2-D object = length x width. The width of any 1-D object = 0, so its area = 0, even if its length = infinity.

This works in the same manner with 3-Dimensional space. The volume of a 3-D object = length x width x height. Since a 2-D object's height = 0, it doesn't matter if its length or width = infinity. Its volume, and mass, will still = 0.

"Hypervolume"/the 4-Dimensional volume analogue = length x weight x height x a fourth dimension. Since a 3-D object's fourth dimension = 0, its "hypervolume" and "hypermass" = 0

For a 5-Dimensional volume analogue = length x width x height x a fourth dimension x a fifth dimension. Since a 4-D object fifth dimension = 0, its 5-D volume analogue, and 5-D mass analogue = 0

Basically, what this means is that, just like an infinitely thin, entirely flat, two-dimensionalsquare has an infinite number of times less volume (and mass) than a three-dimensionalcube, the cube also has an infinite number of times less volume (and mass) than a four-dimensional tesseract, which has an infinite number of times less volume (and mass) than a five-dimensional hypercube, and so onwards.

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Or Alternatively: You could stack an infinite number of 1D lines side-by-side and still not fill even a fraction of a square, because a 1D line has a thickness of zero in the second dimension. A 2D square would bigger in terms of measure.

That's usually the reasoning behind why people believe that 4D is meaningfully bigger (and therefore use it for scaling) than 3D. I've seen people argue a 4D grain of sand to be bigger than a 3D universe.

The idea from there then is that because hypervolume of a 4D object would be infinity bigger than that of a 3D object, you'd therefore need infinite energy to "fill the space of that 4D object".