What is beyond a volume in Geometry?

Question

I found this nice explanation on Spatial Geometry inside my encyclopedia:

A moving point is a line, a moving line is a surface, a moving surface is a volume.

I am aware of String Theory and the 10th theoretical dimension, but for example, the 4th dimension, spacetime, is not something that would follow volume (depth) as far as I suppose, hence String Theory has nothing to do with Spatial Geometry. Correct me if I'm wrong.

My question is, in advanced Topology or Geometry is there anything beyond a volume? And if so, how would it be represented? Is a moving volume even something or just...more volume? Maybe Topology is not what I'm thinking it is and I'm saying something senseless?

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$To paraphrase Ian Stewart, mathematicians don't say "the fourth dimension", but "a fourth dimension". Empatically "yes", there is something "beyond a volume". Moreover, one doesn't need to go further than linear algebra (early undergraduate-level) to encounter this material formally.

To a modern mathematician, the prototypical "$n$-dimensional space" is Cartesian space $\Reals^{n}$, the set of ordered $n$-tuples of real numbers, equipped with context-dependent extra structure that may include:

• Coordinate-wise addition and scalar multipication, turning $\Reals^{n}$ into the Cartesian vector space.

• The dot product or its associated norm, in which case one speaks on $n$-dimensional Euclidean space.

• The collection of open sets defined by the Euclidean metric, and the consequent notions of continuous and smooth functions. A topological space modeled on the smooth structure of Cartesian $n$-dimensional space is a smooth $n$-manifold.

A smooth $n$-manifold itself can be (and often is) equipped with the additional structure of a Riemannian metric. Geometric quantities such as length (of tangent vectors, or of smooth curves), angle (between two tangent vectors at a point), and area (of smooth surfaces) make sense in a Riemannian manifold.

In general relativity, one instead equips spacetime with a Lorentzian metric, similar to a Riemannian metric, but "with one timelike direction" at each point.

At risk of speaking for an entire profession, most geometers define a curve to be a (smooth) $1$-manifold, and a surface to be a (smooth) $2$-manifold. A "moving surface" suggests a solid region of $\Reals^{3}$, a very special type of $3$-manifold; the term "volume" is therefore potentially ambiguous, and in my experience not commonly used. In the sense I think you're asking, however, the progression "curve, surface, volume" continues with (smooth) manifolds of dimension four, five, etc. Clearly there is no upper bound on the dimension, no end to this sequence.

Answer 2

It is true that in Relativity one conceives of space and time as being unified into one 4-dimensional framework. Imagining a fourth dimension as being time, over which objects in three-dimensional space are evolving, can also be a helpful way of getting a handle on higher-dimensional reasoning.

However, it does not make much sense to say that time is "the" fourth dimension. After all, if time is the fourth dimension, then what is, say, the 3rd dimension? What are the second and first dimensions? Where do they point to?

Furthermore, the spacetime we deal with in relativity is really just one example of a 4-dimensional geometry. In string theory, it is often posited that there are 10 spatial dimensions, as you mention, and that the "extra dimensions" that we don't see are "tightly curled up", so microscopically, that we don't observe them in our day-to-day interactions. However, there are many formulations of string theory, and some assume that there are actually 26 dimensions; others that there are 11. The bottom line is that none of these theories have any experimental grounding and are all likely to remain deeply theoretical for a long time. It's also possible that none of them are true.

But we don't have to look only to theoretical physics to find examples of higher-dimensional geometry, and more importantly, we don't have to suppose that some number of dimensions would have to "exist in space" in order to deal with that level of higher dimensional geometry. Often, in mathematics we encounter abstract spaces with a large number of dimensions, which consist of a set of points that have some sort of spatial structure on them, which does not come from "existing in space". A key concept in classical mechanics and in engineering is that of a "configuration space". Namely, we have some system, which we could imagine for example to be a robotic arm with some number of joints, and we want to imagine the ways that this system can evolve, or in the case of the robotic arm, how this arm can rotate and move about. Perhaps this arm has some tip at the end which interacts with whatever product it operates on. The configuration space of this arm then consists of all the states that the arm can be in; literally, all its possible configurations.

Now, it's possible that the tip of this arm can be at the same spot while the arm is in two different configurations. In any case, we consider these two configurations to be two distinct "points" in the configuration space. Indeed, it requires some expenditure of work in order to make the arm transition from configuration $A$ to configuration $B$, even if these two configurations determine the same point at which the tip rests. Further, if we wish to move the tip from some point $a$ to $b$, it is possible that the arm cannot take a direct route between the two points (perhaps the joints have moved as far as they can go in the direction pointing directly towards the destination point), and so the arm has to twist around in some manner before the tip can be made to rest at point $b$. This illustrates that the arm really has to move about within its space of configurations, and finding a path between two points within the space of configurations can be a highly nontrivial mathematical problem for the programmer of said robotic arm.

This is all to show that higher-dimensional geometry is actually eminently applicable to a wide range of scientific and engineering disciplines. The number of dimensions of a configuration space can be very large, and go up for each degree of freedom of the system in question (for example, with the number of joints in our robotic arm. Further applications include, for example, solution sets to partial differential equations, which crop up everywhere in science and engineering. And there are higher-dimensional analogues of length, area, and volume, which can all be described as hyper-volume. We can also describe higher-dimensional shapes, such as a tesseract.

Visualizing all these notions is exceedingly difficult, and in some sense it may be impossible to get a full intuitive picture of what's going on. For this reason people often rely on intuitive crutches to understand it, such as pretending that "the fourth dimension is time". These often only serve to perpetuate misconceptions and inhibit real understanding about what higher-dimensional geometry truly is, and it really frustrates me to see educational sources continue to perpetuate these inaccurate ideas, either out of laziness or from the teacher's own failure to really understand what's being taught. At any rate, I hope that this exposition helps to clear up your understanding, as well as the resources that Andrew D. Hwang has pointed you to.