Scientific Intuition or How to Imagine 4th Dimension

I have been approached over the years by many people asking me questions like “How do you imagine X?” I always tried my best to explain to them that some things just cannot be imagined, that you have to accept them as they are. Recently, I started thinking about this a bit more. I was preparing a general talk on the topic of “The Strangeness of Quantum Mechanics” and decided that I would also try to explain some things about science in general and the intuition of scientists. So I conducted a survey asking my scientist friends (and also friends that were at some point in the past involved in science) to answer two questions:[1]

  1. What do you do to imagine and how do you imagine or visualize n-dimensional space (especially 4-dimensional)?
  2. How do you imagine or visualize wave-particle duality? In particular, in what way do you visualize what happens in the double-slit experiment (with single photons or electrons)? Do you imagine a particle, a wave, or something else, or nothing?

I discuss the answers that were given below. Since I cannot expect all readers to understand every concept I quizzed my friends on (wave-particle duality being the most likely to be alien to the average reader), I shall briefly introduce them.


Physicists always talk about dimensions, and they rarely stop at three. Sometimes they say that space-time is 4-dimensional, and sometimes they say that the universe has 11 dimensions (or some other arbitrary number). But very often, on a more abstract level not connected to the “physical” space we’re used to, they use n-dimensional space (where n can easily be of order 1023). Some even talk about infinite-dimensional spaces (I know some that will engage in lengthy discussions about the intricacies of infinite-dimensional spaces, yet are quite lost when the discussion turns into finite dimensions). Finally, the layman tend to just look at these scientists and think how strange we are when we imagine such nonsense. This is how we – scientists – are usually perceived.

Let me say that we are not strange (not all of us at least), and I hope I can make this clear in the next few lines. With my first question I knew what to expect for an answer and I was not disappointed. The answer is simple: you simply do not imagine larger dimensional spaces, because you cannot. We live in a three-dimensional space and this is what we experience on a daily basis; we do not know anything else. This is exactly why we know how to imagine 3 dimensions (and smaller integer-valued dimensions, which are easily embedded in our 3D space) but we do not know where to put the fourth one. Even though we cannot visualize the fourth dimension (although some respondents had tried) it is an easy task to replace our intuition with known concepts. One can easily think of the fourth dimension as time, or color. However, this seems to be, in general, as far as we can push this idea.

So, what do we do with higher dimensions? We use analogy: in the same way one goes from one dimension to two and from two dimensions to three by adding just one more “identifier” (spatial position in this case) one can add time to get fourth, color to get fifth. But what’s more you do not have to imagine it – once you know the rules, you just accept them. In other words we just need to look at the underlying mathematical formalism and keep in mind the rules that are applied. A nice analogy to understanding the way we imagine things was given to me by Andrej Gendiar. After using a formalism a large number of times one is not getting any better at visualizing things, but you do acquire an intuition. As Andrej wrote: ” Now, I can see no difference among 3D, 4D, and infinite-D, because I do not see any difference between two integer numbers +4 and -4 (the latter one being absolutely unrealistic).” There is one more thing scientists do when they work in higher dimensional spaces and need to get intuition. They project the problem either into 3D or 2D space, which our brains are accustomed to. In the same way you can assess the properties of a hanging object from its shadows on the wall, one assesses what happens in higher dimensions from what happens in the lower-dimensional case.


A slightly different approach is to use what happens in a smaller dimension to acquire intuition about what happens in the higher dimension, just like you can gain some intuition about cubes by studying squares. In the end no one visualizes higher dimensions and all one can do is build some intuition about it from considering what happens in smaller dimensions.

Wave-particle duality

While the first question was just to confirm my suspicions, the second question was more interesting to me. There is not much one can do to imagine larger dimensions, but I was quite curious how others try to imagine the concept of wave-particle duality.

Before we consider the given answers, what do I mean by wave-particle duality? When one lets a small particle such as a photon, electron or atom (anything that could be considered “quantum” in nature) go through a double slit and then measures where it hits the wall located past the slits one finds that two quite contradictory explanations are at hand: One is that the particle will hit the wall at a point-like location (like a particle, right?), but on the other hand the probabilities describing possible positions the particle could hit the wall seem to follow the rules for a wave – the probability of hitting the wall at some location is proportional to the height of the wave that would arrive at that location. There is certainly more to be said on the topic (see e.g. Wikipedia) but this blog is not about the effect. All we need to remember is that before hitting the wall, our photon/electron/atom behaves like a wave and then after the collision acts as if it is a particle. This is a strange phenomenon that is certainly counter-intuitive and has been with us from the onset of quantum theory. In the (classical) world around us, the waves are always waves and particles are always particles, and so there is this desire to ask: how can you imagine/visualise this wave-particle duality?

As in the previous case, I was not really surprised by the answers, although I had secretly hoped that somebody would have some amazing way of imagining it. The answers I got were quite closely related to the fact that the effect is statistical, meaning that the supposed “wave” is just a theoretical prediction of the (amplitude of the) probability that the particle is located on a given place at a given time. What people did was they visualized a wave, sometimes composed of small balls, or a “ship on the wave” that represented the particle.

It turned out that all these visualizations were, however, only a crutch. As is the case with imagining higher dimensions, what’s more important is the underlying mathematical structure. Your intuition is very limited and comes only from the observation of the classical world around us, which has no respect for “quantum laws”. That is why at one point or another our inherent intuition fails. We now have a firm grasp of the laws according to which quantum theory works and while their consequences might be counter-intuitive, we have to accept them. In the case of wave-particle duality, we just have to accept that the wave – as a mathematical tool – determines the probability that the particle hits the wall at a given location, and there is not much more to it.

As you can see, there is nothing in these cases that scientists are better at imagining than anyone else. The things we talk about that cannot be imagined are always based on a solid mathematical background that we respect. There is, however one point worth mentioning: while it is true that we cannot imagine these things, when one works with them a lot, one does acquire some intuition about these things, and in the end it is not the intuition that leads to knowledge, but knowledge that leads to intuition. In the same way that a cook, through hours of testing and tasting, acquires an intuition about which ingredients work well together, scientists, by working through the mathematics, acquire some intuition about how things work.

[1] The experiment was conducted on 38 participants. They answered the questions willingly and were allowed to answer anonymously. No animals or scientists were harmed during the experiment.

Many thanks to Tom Bullock for having the patience to go through the text and for his corrections.

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