Why Cant We Visualize More Than Three Dimensions?
Physicists and mathematicians who think about higher-dimensional spaces are, if they allow their interest to somehow become public knowledge, inevitably asked: “How can you visualize more than three dimensions of space?” There are at least three correct answers: (1) You can’t. (2) You don’t have to; manipulating abstract symbols is enough to help you figure things out. (3) There are tricks to help you pseudo-visualize higher-dimensional objects by cleverly projecting them into three dimensions; see here and here.
But really, why can’t we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?
I can think of a couple of explanations why it’s so hard, with different ramifications. One would be simply that our imaginations aren’t good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it’s like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it’s like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.
The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from “tuning a dimensionality knob” on our imaginations. The truth is certainly more complicated than that (and I’m not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us.
Maybe this could be a new tantric discipline: visualization in higher dimensions. Or maybe the Maharishi already offers a course?
Everyone knows what it’s like to experience the hallucinations that accompany certain kinds of drug use (among other mind-altering contexts) — if not from direct experience, at least from depictions in movies and literature. We’ve seen the colorful, swirling patterns, or the illusory tunnels stretching before us. It turns out that hallucinations are by no means random; there are certain recurrent patterns reported by people who experience them. These patterns were studied by Heinrich Kluever in the 1920’s and 30’s, and classified into four different structures: spirals, spokes, honeycombs, and cobwebs.
Subsequent work has suggested more complicated hybrid forms, such as that portrayed here, but the basic types seem to be robust.
Here’s the good part: the appearance of these particular hallucinations can be explained by physics! I know this because I’m sitting on the thesis committees for two students, Tanya Baker and Michael Buice, working with Jack Cowan of the math department. Cowan is a pioneer in mathematical neurophysics, developing sophisticated physical models of the behavior of neurons in the brain. He is a co-originator of the Wilson-Cowan equations to model neural behavior, and his students and collaborators have been thinking recently about hallucinations and other emergent properties of the brain.
The cerebral cortex of the human brain is basically a thin crumpled sheet, about three millimeters thick and one square meter in area. Naturally, physicists are going to think of it as a two-dimensional problem. The visual field, as observed by our eyes, maps smoothly (but with distortion) onto an area called V1, the primary visual cortex. (I’m glossing over details, including the fact that we have two eyes and our brain has two hemispheres. This is because I don’t really know what I’m talking about, and will try to limit what I say to stuff in which I am confident.) So if we see two parallel lines, it activates a set of neurons that describe two non-intersecting curves in the physical layout of our visual cortex. How do we know the map from the visual field to the cortex? Well, it involves monkeys, or sometimes cats, and electrodes, and noble sacrifices in the name of science. We won’t dwell on the details.
The brain is complicated, so we begin by making approximations. There are enough neurons that we don’t worry so much about the discreteness of cells, but model the cortex as a smooth plane. At each point is a neuron, which can be either be activated or deactivated, or any value in between. The complication comes in when we consider the stimuli to which the neurons respond. These include not just the color and brightness of the point in the visual field to which it corresponds, but also interactions with other neurons, near and far.
These interactions allow the neurons to be sensitive to nonlocal features of the image, such as spatial or temporal frequencies in the brightness pattern, or the presence of correlated orientations within the image. This last capability makes us sensitive to the existence of straight lines — so much so that our brains fill them in when they aren’t even there, such as in the triangle illusion illustrated at right. The space of features of visual stimuli to which neurons are sensitive is not only high-dimensional, it can even be topologically nontrivial.
So as physicists (or applied mathematicians) we want to come up with a mathematical model describing the state of the neurons as a function of the input stimuli and the state of all the other neurons. We end up with an equation of the schematic form
da(x,φ,t)/dt = -a(x,φ,t) + I(x,φ,t) + ∫ dx’ dφ’ f(x-x’,φ-φ’)a(x’,φ’,t)
Here, a(x,φ,t) represents the state of the neuron — either activated, deactivated, or in between. The variable x is the position on the cortex, t is the time, and φ represents all of the things to which the neurons can be sensitive — brightness, spatial frequencies, color, orientation, and so forth. On the right-hand side, the first term -a(x,φ,t) is just minus the current state of the neuron, which makes an unstimulated neuron decay back to the deactivated state. I(x,φ,t) represents the direct stimuli received from the optic nerve, whatever they may be. The interesting part is the final term, an integral representing the interaction with other neurons in the visual cortex. The function f(x,φ) tells us how sensitive the neuron is to the states of other neurons a certain distance away, which can be different for different features of the visual field (frequencies, orientations, etc.). This function typically has a “Mexican Hat” form — it is positive for short distances, negative for intermediate scales, and goes to zero far away.
Okay, the previous paragraph may or may not have made any sense to you. But here is the punchline: patterns of hallucinations reflect normal modes of the neurons in the visual cortex. By “normal modes” we mean the characteristic patterns of vibration, just as for a violin string or the head of a drum. The idea is that a drug such as LSD can alter the ground state of the visual cortex, so that it becomes excited even in the absence of stimuli. In particular, certain oscillating patterns can appear spontaneously. Generally these would take the form of different configurations of straight lines in the cortex itself; however, due to the distortion in the map from our visual field to the brain, these appear to us as spirals, tunnels, and so on. Indeed, Cowan and collaborators have shown that these normal modes can successfully account for all of the basic forms of hallucination classified by Kluever decades ago.
So, the next time you have a near-death experience, and see a tunnel stretching before you leading to a beckoning light, it’s not Jesus calling you into the afterlife. It’s just some characteristic jiggling of the neurons in your weakened brain. Which, to my mind, is much more interesting.