How many times harder is to think in 3D as compared with 2D?

Question

I'm a chemist, currently going through a course about molecular symmetry and group theory applied to Chemistry. This subject is very demanding in terms of visualization in 3D space. To really grasp the subject, one must be able to see, for example, how the orbitals around a given nucleus transform when subjected to the different symmetry operations in space.

I'm stumped by the sheer difficulty I have at it, specially given I deem myself good at reasoning with geometric concepts in the 2D plane. Add just one dimension and now it feels like to be in a quagmire.

That makes me wonder, can you put hard numbers in this increased overhead when going from 2D to 3D? How much more processing power is necessary to reason about 3D space? A linear scaling can be surely ruled out as, at least for me, 3D doesn't feel like just 50% more difficult. It feels like a several fold increase in complexity. Could be the case it scales like area or volume, as a power of the number of dimensions, say, n³? What about a generalization to higher dimensions? How much more difficult a 4D being would have reasoning about 4D as compared with 3D?

I would not be surprised if grasping something like a 4D were very difficult or even impossible for us, as we don't experience 4D spatial dimensions. But we do live in 3D space, and yet thinking in 3D can be very hard.

Answer 1

This answer is based on my experience.

Usually for me three-dimensional things are easier to imagine than two-dimensional those. This is because to imagine the first I have just to reproduce a slightly modified image from my daily life experience. The latter is mostly three-dimensional. In particular, I feel solid things more real than plain those.

This situation is typical. Nicholas Bourbaki wrote that “the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition, which is not the popular sense-intuition, but rather a kind of direct divination (ahead of all reasoning) of the normal behavior, which he seems to have the right to expect of mathematical beings, with whom a long acquintance has made him as familiar as with the beings of the real world”.

So I think that for people like me visualization depends not on a processing power (especially on quantifiable one) but on experience. Also it can require some kind of psychical energy. For instance, for me can be hard to provide a detailed visualization when I am tired.

Hard cases can require to detailed study the object first, in order to “saturate” it in mind and then sometimes an image can be forced to flash in mind.

Probably a visualization skill can also be developed by performing visualization exercises, which can be found, for instance, in some parapsychological books.

A real visualization is hard for me. I mean, for instance, to imagine an object not as some related spatial intuition but a concrete image, for instance, a small red ball, which is seen as in real life or dream.

On visualization of high-dimensional things see my recent answer here.

Answer 2

It is often with examples that one can approach a subject.

I give here 5 counter-examples where 3D can be a "blessing" for understanding a 2D situation...

A) First, an answer of mine some time ago about Monge and Desargues theorems.

B) A different situation where a family of curves can be more globally interpreted as a set of level curves on a surface. Consider for example the following representation of a double family of hyperbolas situated "on" the parabolic hyperboloid with equation $z=x^2-y^2$:

Remark: The horizontal sections are hyperbolas whereas the vertical sections are parabolas.

C) Still with a level curve interpretation, but with a physical situation: the fact that sound waves emitted from one of the foci of an elliptical mirror are reflected on the mirror in such a way that they are eventually re-focalized into the other focus. You will find a description in this answer of mine (again !): begin by the last figure and consider the figure before this one as a oblique "lifting" in 3D of the 2D situation: just an issue of (shifted) level curves...

Edit (Nov 16, 2020)

D) Let me present here, still another case, with some common features with example C). Consider the first figure below:

It describes (in a classical way) a conical curve by a point (a focus), a line (its directrix) and a number $e$ (its eccentricity) as the set of points $M$ such that the distances' ratio (from the point to the focus and the shortest distance from the point to the directrix) is constant and equal to $e:

$$MF/MH = e\tag{1}$$

This process encompasses the three types of conic curves : ellipse if $e<1$, parabola if $e=1$, hyperbola if $e>1$.

Could this picture be seen in a different manner ? Yes, in the following way (second figure): we can transform in a certain vertical plane our different types of curves in a unified type, i.e., circles, moreover concentric... with a well placed light source $S$:

In fact, afterwards, one can think to the different curves of the first figure as the limits of the light projected on the ground by a torchlight having an adjustable conical beam width...

Remarks:

The "mathematical machinery" behind the transformation seen in the second figure is "projective geometry" which has become a recognized branch of geometry in the 19th century, whereas the definition of conics we just saw using foci, directrix and eccentricity has $2300$ years...

2. Squaring relationship (1) gives a 2nd degree equation common to all conics:

$$x^2+y^2=e^2(1-x)^2.$$

E) See this recent answer as well...