Extradimensional Lenses

Definitions

A dimension, according to Oxford Languages, refers to a mode of linear extension which corresponds to a coordinate set specifying the position of a point. Different dimensions are the different coordinates required to specify a point in space. As we live in three-dimensional space, we require three coordinates to do so. Two-dimensional beings would require two, and so forth.

• It would be inaccurate to think of a 2D square as akin to a sheet of paper, After all, if one were the size of an atom, a sheet of paper would be an incredibly large rectangular prism. If the three dimensions are "up/down", "left/right", and "forward/back", then a 2D square would have zero depth. 2D planes have no concept of "up/down". Even a printed image on that piece of paper has some sort of depth within its ink. Just as you can zoom into graphs infinitely, all 3D objects always occupy a portion of all three dimensions.
• In the seventeenth century, a mathematician named Cavalieri realized that every solid body is the superposition of an infinite number of planes. Thus, dimensions are the direction on which those planes align themselves.

Example of topological equivalence.
Source: Mug and Torus morph, Lucas Vieira (Wikipedia).

Topology is a branch of mathematics concerning the properties of geometric objects that are "preserved through continuous deformations" (Wikipedia). For example, a shirt, no matter how you twist, crush, bend, stretch, or otherwise manipulate it (without opening, closing, tearing apart, stapling or gluing together any of its surface), will always have four openings. Therefore, to topologists, doughnuts are synonymous with coffee mugs — both have a single hole. (Holes are not loosely defined in topology, despite not having a big fancy name. Thus, the interior drinking portion of a mug is not considered a hole, merely a depression in the mug's surface, or a "divot".) We would say they are topologically equivalent.

• Technically, real objects are almost never actually topologically equivalent, due to complications with things like tiny holes in fabric and whatnot. Realistically, objects are not topologically equivalent, but mathematically, they are.

A manifold is any topological surface that can exist in any dimension. For example, a 1D manifold would be a line, a 2D manifold would be a plane, and a 3D manifold would be a cube. But there are other shapes as well, like the circle (1D), the torus/doughnut (2D), and the tetrahedron (3D).

Orthogonal (in this context) means perpendicular. Typically it is defined as "of or involving right angles"/"at right angles" (Oxford Languages).

Extradimensional is a word whose meaning I have not confirmed with any dictionary and which I have been using frivolously to mean "of an unfamiliar amount of dimensions".

The Mysterious Fourth Dimension

Some theorize that time is the fourth dimension; if true, it is a very simple one, since one can only move through it in one direction, at the same rate. (4D beings may be able to manipulate this property, or not follow it entirely, although us 3D beings must, but we'll get into that later.) Spatial dimensions, however, like the three we know and experience, can be traversed in all directions. (A dimension kind of is a direction.)

• This is not an arbitrary speculation; time may possibly be our gateway to understanding the fourth dimension (see "Understanding Extradimensional Shapes").
  Also possible is that time is not necessarily the fourth dimension, just an additional dimension existing alongside us. Or maybe there are no more dimensions at all, although this idea is not very plausible considering a number of mathematical and physical theories that have emerged in the last century or so.

Understanding Extradimensional Shapes

However, assuming time is not the fourth dimension, and that 4D exists as something other than a hypothetical, a fourth dimension of space is impossible to perceive or even visualize for us. So, in order to put this into perspective and give you some idea of what it must be like, we'll go a dimension lower, into two-dimensional existence.

Two-dimensional creatures do not understand spheres or cubes. They only know circles and squares. In order to perceive a sphere, such a creature (Bob) would have to view it one slice at a time in its 2D world.

Example:

Bob has a friend named Squirrel, who is featured in this depiction.
Source: Paradox of the Möbius Strip and Klein Bottle - A 4D Visualization, drew's campfire on YouTube (2022).

It seems obvious to us what a sphere is, but Bob would not be able to comprehend it unless it adapted to his 2D perspective so that he could properly view it — in other words, he cannot see the entire thing at once like we can. This is just like the nature of 4D. In order to understand all of a 4D object, we have to see it one slice at a time in our little 3D world in order to begin to understand all of it. Remember Cavalieri's observation from before? Well, these infinitely thin slices that Bob is seeing each occupy one single plane, since planes are 2D and he is also 2D.

• It's interesting to note that if time is a dimension, it's the one compensating for Bob's lack of comprehension for a third dimension. He watches each slice of the sphere appear and disappear, over time, until the whole thing has passed through his field of view and he can understand it. So, instead of space, Bob is taking advantage of time, to use as his third dimension.

Cubes are slightly more complicated than spheres, since spheres look the same no matter what angle you view it from. Bob has no concept of 3D rotation, so if a cube were to rotate along the third dimension, it would merely look as though it were shrinking and growing in odd ways, like so:

Bob's uncle Marley, utterly bamboozled by 3D objects.
Source: Simulating Biology in Other Dimensions, Curious Archive on YouTube (2023).

Although it seems obvious to us what is happening, the 2D entity is clueless. This is the case with us and 4D. It is a humbling experience, to realize we must seem like bumbling buffoons to 4D beings. Just as 2D must with 3D, we would have to see a sequence of slices in order to have some grasp of higher-dimensional shapes. Things existing in the fourth dimension have the capacity to become literally invisible to us.

Extra Note

The difference between 2D and 3D spheres: A 2D sphere is a two-dimensional manifold, as it exists only as a closed surface and, while it can be given shape/texture/etc, it still has no depth. 3D spheres have, well, interiors, and are known as balls. You can stick a pencil through a 3D sphere, but it will not go through a 2D sphere, since there is nothing inside of it and a pencil can't exist in 2D anyway. (A 2D pencil would be stuck on the sphere like a sticker, if the shape of the dimension were following the manifold.)

A 2D sphere has two "sides" (but no depth, because it is infinitely thin): the visible exterior and the interior, which is not visible unless you cut a hole in the manifold. Another way to put it is, 2D spheres aren't hollow balls floating in space. They are 2D manifolds, and there's no space around them or inside them. They exist within a singular plane. So, now that you're not confused, let's talk about embedding:

• Something important to note is that although 2D and 1D objects can be embedded in a 3D world (mathematically, not realistically), it doesn't make them 3D manifolds. They'll still only occupy one of the three accessible dimensions. However, it does not work the other way around. 4D objects can't live in a 3D world without existing only in slices. Rather, as four-dimensional beings, they will simply already be occupying four dimensions, and will appear through only three dimensions at a time. It translates in the same way to 3D beings in a 2D space.
  If four-dimensional beings were somehow shoved into a 3D world, they would not survive, much in the way that we would die if squashed into a 2D world. Thus, crossover is extremely difficult. But 3D beings may be able to live in four dimensions, even though parts of everything would be invisible at different times.
• Since the two axes of 2D space are orthogonal to each other, and the three axes of 3D space are also orthogonal, the axes of 4D space and above are orthogonal as well. This is only possible because of those extra dimensions. In 2D, there is only one unique slope B perpendicular to slope A, but the more dimensions you add, the more perpendiculars you can have. This is how we know dimensions are not placed at random angles (espeically since said dimensions are the sole thing making up the room for those angles).

Extra Note 2: Electric Boogaloo

did u know that snow is 90% air (placeholder)

P.S. If you're looking for further resources, I highly recommend watching the two videos that I made GIFs out of. They're both put together quite excellently.