The Third Dimension is Not Simple

Ever since Einstein got us thinking about the fourth dimension and string theorists got us worried about ten and eleven dimensions, we have not really given serious thought to the mundane old third dimension. Several things, ranging from the emerging three-dimensional Internet over at Second Life, to the delightful modern religion of Parkour and the Nintendo Wii controller, have made me think seriously about the third dimension in recent weeks. It isn’t just badly-developed characters in movies and books that are two dimensional — you and I are as well, in fundamental ways.

The third dimension is not a conceptually simple extension of two dimensions. It is also not as familiar a place as you might think, once you attempt to actually engage all three dimensions at once. So here is a socio-cultural-mathematical-cognitive meditation on the third dimension. I’ll start with two anecdotes, segue into some thoughts on spatial cognition, take a detour into some math (which you can skip without much loss if you like), discuss 3d art forms like dance and Parkour, and conclude with some observations on the 3d Internet which is emerging. Take a deep breath.

Two Anecdotes

Two anecdotes should illustrate what I mean when I say the third dimension is not a familiar place for humans. The first is from when I was an enthusiastic swimmer as a kid. Occasionally I’d go swimming rather late, after dark. One such evening, I was the only person in the deep end of the pool, and I decided to do the underwater equivalent of spinning around on a swivel chair, to make myself dizzy for fun. So I took a deep breath and turned as many underwater somersaults as I could, trying to change my spin axis in random ways as I did so. Result: I got massively disoriented several feet under water. I couldn’t tell which way was up, and for a few seconds, in a blind panic, I was actually swimming away from the surface after I ran out of breath. For those few seconds, I had a powerful sense of the third dimension — it is a scary place.

The second incident is more recent. While a postdoc at Cornell, I lived near the Finger Lakes trail, and one afternoon, I took off on a hike armed with a topographic map — the sort with contours. I somehow managed to wander off the trail and get hopelessly lost, and stupidly convinced that the trail continued on the other side of a deep gorge, with steep sides. I scrambled down one slope and began walking upstream alongside the narrow creek at the bottom, looking for a place to cross. In a few minutes the stream ended, and I’d walked into some sort of un-nameable topographic mess that looked nothing like the map I was holding. It was noon, and I had no sense of North. The map looked like it was gullies and little canyons all over. Suddenly I realized I’d have to navigate in full three dimensions with a bad two-dimensional aid. Vastly harder than navigating in an essentially two-dimensional place like a city, which involves a trivial bird’s-eye-to-worm’s-eye translation around well-behaved cuboidal things like buildings. During the few minutes of clambering up an impossibly convoluted slope, the full complexity of three dimensions hit me. At the time I was working on vanilla two-dimensional robotic navigation algorithms, and I remember a very powerful, sinking feeling of “my research is rubbish and nothing like the real world!”

So what is full 3d really about?

Our Two-Dimensional Minds

I won’t belabor this point, but our minds are essentially two-dimensional in the way they process visual sensory input. Stereoscopic vision gives us some sense of depth, but very loosely, our firmware, I suspect, is limited to some sort of projective geometry in two dimensions. The details of the visual cortex all seem to be about two-dimensional processing. (our auditory system seems more legitimately three dimensional, since it is associated with the semi-circular canals, but I have no insight to offer there at the moment.)

Almost all the time, visually, our attention is focused on one of two planes: the vertical plane in front of us, with “up” defined by gravity, or the horizontal plane (organized with a left/right/forward/backward relative frame of orientation). Call these plane V and plane H (the third, “sideways” plane doesn’t seem to get much action, since we turn around to make it V if we need to process something oriented that way). With one major exception, all human motion and activity involves information processing in one of these planes. Reading is a plane V activity, most walking is a plane H activity. Walking down a long corridor is plane V.

The major exception is tactile manipulation (a subject studied with neat elegance by CMU Roboticist Matt Mason in his under-appreciated Mechanics of Robotic Manipulation) with our famous opposable thumbs. I am convinced though, that the processing going on during tactile manipulation is fundamentally not spatial processing at all. When you solve Rubik’s cube (if you can, that is; I can’t), I suspect you are invoking a very different brain capacity that runs on finite/countable group theory and algebraic topology rather than the geometry of the three-dimensional continuum. This is why expert Rubik’s cube solvers can solve the puzzle blindfolded. They aren’t thinking spatially in a meaningful sense.

If you don’t get the mathematical speculation, this might help. Twiddling your thumbs or solving Rubik’s cube is a discrete/digital type of activity like counting, not an activity involving a sense of continuous space . There is a subtlety here though — all our two-dimensional visual processing is also discrete in non-trivial ways beyond our neural hardware being digital (for example our visual systems detect parallel lines with a special layer of neurons). But at the level of awareness, we sense our two-dimensional thinking to involve the two-dimensional continuum. So in a sense this discreteness of 2d is merely a sort of sampling.

So how do we process three dimensions when we have to? I think we can only do it in two ways: we can train ourselves to think (with low intuition) about three dimensions, with mathematics as our crutch, or we can learn limited, specialized sensori-motor three dimensional skills.

Thinking about 3d with a mathematical crutch

(You can skip this section with no loss of continuity if the mathematical references are obscure to you)

There are differences between two and three dimensions that are purely mathematical and topological, which makes three dimensions a fundamentally tougher space to think about (for instance,knots and knot theory only meaningfully exist in three dimensions). But I am interested in limitations that arise from the way we think.

We have mathematics to help us even when we are thinking of “continuum geometry” aspects of three-dimensional space, rather than finite/countable group-theoretic or algebraic topological aspects (which, as I said before, seem to be more innate in our circuitry). Quick example: solving the Rubik’s cube involves thinking about turns along only 3 orthonormal axes that are integer multiples of 90 degrees. Most human artifacts also limit themselves to such finite/countable group-theoretic limits (for example, agile aircraft maneuvering can be understood in terms of what are called maneuver automata, introduced by Emilio Frazzoli, that make three dimensional rotations “countable” in a sense). Not much of what we do in 3d is in “rich” continuous 3d. Dropping a spinning top in turbulant water is the sort of thing I mean when I say “rich 3d.” Or the instantaneous snapshot structure of a stormy ocean surface.

Continuous three-dimensional structure is something you can start to wrap your mind around using the differential geometry of curves (the Frenet-Serret equations). I recently played briefly with a problem involving these equations (think roller-coaster design). Perhaps I am stupid, but I managed to get no sense at all for how to work them at an intuitive level, even after writing some rather neat visualization code. I was reduced to groping with the actual symbolic math, which I never like to do for long. That tells me I’ll never be a mathematician, since I don’t trust symbolic manipulation to take me where my visualization abilities fail.

You don’t need to get to differential geometry. Even your more basic three-dimensional SO(3) rotational matrices can be tricky to understand (I don’t find quaternions helpful at all). Once you probe the subtleties of these creatures in the context of sufficiently complex three-dimensional phenomenology (like say, dog fighting in fighter aircraft) you will soon lose all intuition. By contrast, analogous two-dimensional problems, like planar robotic navigation, retain intuitiveness almost everywhere. I have never found a two-dimensional result that I understood mathematically but not intuitively. In three dimensions though, most of my understanding is purely technical, and I have no sense of how the stuff works. There is a lot to explore here, including spinning tops, nutation and precession, continuous (Lie) groups and the like. These are subjects I am exploring slowly and painfully at a conceptual/philosophical level, and I may blog about them as I understand more. But for now, I’ll leave you with the thought that in three dimensions, mathematics is a necessary crutch for most of us, unlike in two dimensions, where intuition easily keeps pace with mathematics. I’d like to meet someone who has true three dimensional intuition. I don’t.

Dancing in Three Dimensions

A more familiar way to get beyond our two-dimensional limitations is through training of our bodies and sensori-motor mental capabilities. Dance is the most familiar example. I have often thought, observing good dancers, that they seem to actually exploit all three dimensions in a rich way. That is what distinguishes them from us pointy-finger clowns.

But a much more dramatic example has surfaced in recent years. This is the French sport/religion of Parkour and its relative, freerunning. Two recent movies have featured the practice, the James Bond flick Casino Royale and the fourth installment of the Die Hard series, Live Free or Die Hard. The New York Times recently did a feature on this movement. There is some mildly interesting Zen-like philosophy and aesthetics the practitioners bring to their art, but it is the art itself that is simply stunning. If you get a kick out of the action scenes in the Spider-man series, you will be enthralled by live Parkour and freerunning (a Web search should yield plenty of videos if you are curious).

The 3d Internet: Second Life vs. Bumptop

It is hard to explain the premise of the 3d Internet. Let’s clear some conceptual confusion first. The current Web isn’t a two-dimensional Web. It is a symbolic Web whose basic metaphor is the document, not two-dimensional visual space. The real two-dimensional artifact in our computing lives is the two-dimensional desktop metaphor. The natural descendant of the symbolic Web is the Semantic Web. The natural descendant of the two-dimensional desktop is something like the three-dimensional Bumptop.

The 3d Internet only makes evolutionary and conceptual sense if you first assume the convergence of the Desktop and the Web (and therefore the blending of the document and desktop metaphors). Okay, having cleared that up, and assuming Web 2.0 and Google Docs culminate in the desktop-Webpage convergence, that sets the cognitive stage for a true 3d Internet. At the moment, this has been conceptualized in only one way and realized only in one way: a photo-realistic 3d world based on 3d games, the leading example of which is Second Life. (It is not yet quite clear what Google Earth is, in this framework).

You have to experience it to understand where it is taking us. Simply put, what happens if you take the game out of massively multiplayer online role playing games (MMPORGs)? You get a 3d virtual world where you can build stuff, fly around and really stretch your mind in unusual ways. There are social aspects that are interesting in some ways that I may talk about at some point, but it is the foundational spatiality and underlying metaphors of the concept that intrigue me. If you haven’t yet checked it out, go log on to Second Life. Here is a picture of my avatar in Second Life:

Venkat in Second Life

The most curious experience I had in Second Life was building. To build stuff in Second Life you manipulate objects (in translation and rotation modes). Though the interface is extremely well-crafted, it is still extremely unintuitive and feels conceptually and metaphorically wrong. I suspect that is because it is based on an underlying continuum mathematics as a metaphoric base: you do things by moving/rotating things continuously along or around axes. At the same time you have to manipulate your viewpoint and zoom level to get the right look at what you are trying to do (without your avatar getting in the way — the animation doesn’t actually show the avatar doing tactile manipulation, more like magical hand waving).

I suspect a truly effective three dimensional interface will somehow draw on algebraic topological or group-theoretic concepts, not 3d geometry and rotations. CAD software, and to a lesser extent, Google Sketchup, do this to some extent, but not enough(the underlying “machine shop operations” metaphor that drives these has limits). Perhaps 3d graphic arts software is where the great idea lives right now (things like Maya or the open source Blender). I haven’t used these.
But at a more fundamental level, I suspect the 3d Internet, when it arrives, will take on a much more metaphoric character than Second Life. Bumptop shows us some of the possibilities, as do the neat interface of Windows Vista and Tom Cruise’s cool hand-waving display in Minority Report.

But the true high concept of the 3d Internet still eludes us.


I’ll stop there and say nothing about the Wii controller since I haven’t experienced it. If you’ve experienced it, any philosophical thoughts there?

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About Venkatesh Rao

Venkat is the founder and editor-in-chief of ribbonfarm. Follow him on Twitter


  1. tubelite says:

    Interesting. In a slightly different sense, I think urban creatures are aware of only one dimension. We’ve been trained to follow roads. We know very well how long to go along one road, when to turn left and right and so forth. But when a road ends, you think “Well, the world ends here”. There’s a nice bit of cognitive dissonance when you see beyond the end of the world in Google Earth.

    This is also why it’s so easy to conceal things by putting 8-feet walls around them. After sometime, passers-by automatically tune it out. Like blinkered horses, they stare straight ahead – following the twisting line of the one-dimensional road – everything else is Somebody Else’s Problem.

    I was very surprised to find that along the road to my work place, which I’ve only been going to every day for 4 years, are big military warehouses and tanks, invisible behind high walls. Thank you, Google Earth. You’ve introduced me to the second dimension.

    I guess urban dwellers would also have trouble estimating distances and (especially) areas in open countryside. And most people have a very poor idea of volumes: would you credit the notion that everyone on earth can be squeezed into a cube only a mile on each side? Or – for the metrically inclined – that all the gold ever mined would fit in a cube 20 meters a side?

  2. I pondered this ‘down to 1 dimension’ line of thinking for a while, but decided it was not true in general, but only in the very specific situation of driving. In the US, quite often nowadays, I am so completely guided by Google Maps or Mapquest, that I lose all sense of local orientation besides my position on a graph (vertices+edges graph, not the x-y thing). That makes me a 1-d creature in that context. In local driving, I find I get most disoriented in cities with curvy grids. Manhattan grids allow you to retain a strong sense of 2d location while navigating a graph.

    I haven’t yet had an Aha! moment with Google Earth though. Maybe because it has cached rather than real time satellite imagery, so I’ve never seen my own car in my parking lot or something startling like that.

    I once used your population example (though with area rather than volume) by asking a grad class on complex systems that I was teaching, to guess and then actually estimate the space occupied by all humanity standing in a square grid. Turns out to be around the size of Chicago, which surprised most of the students :). Been meaning to write a silly Web calculator that works with such bizarre scale-calibration units as “swimming pools” for volume and “atoms in the universe” for counting. Will do a post on that sometime. Saw the gold example in a book evangelizing gold investing. Wonder what the cube would look like for the most common metal (Aluminum, right?)

    Serendipitous aside: the Economist has a piece on 3D as of today:
    Economist on 3d. There is a clear trend around 3D; it just hasn’t acquired coherence yet.

  3. Our Two-Dimensional Minds:
    Although most of my activity is being increasingly limited to one or two dimensions, I do believe our bodies are wired to lead rich 3-D lives and I do think we do so in certain arenas. Here are a couple of examples.
    1) Tactile Manipulation: I agree with your example of the Rubik’s cube, but here is one anecdote I’ll offer from an experience at a pottery class. We are asked to make a face on clay and I paid special attention to the nose of the face I was making. I had been through a nasal injury a few days back and was very intimately aware of my overall nasal structure from generally probing to see if there was pain in various areas of the nose. The experience of molding the clay to emulate the contours of the nose I would call as rich-3D activity. I’ll post a photograph of the very handsome nose I sculpted sometime.
    2) Even if you think sculpting is inherently dealing with 2D surfaces with some surface bending thrown into the mix, here is one activity that should top the charts or be right next to “dropping a spinning top in turbulent water”…Eating !! Imagine the kind of manipulations that your tongue does on a morsel of food, deciphering its orientations, adjusting to a changing structure. I think our tongues (and overall mouth system) are intimately aware of the limited 3D space of our mouth (and in a very rich 3D way). Its also a conscious 3D manipulation, though you do not explicitly think about it.

    btw, one of the google ads on this page points to exploring a 4D world. It is You’ll definitely need all your mathematical crutches for that one ;) yes…I do click on some of the google ads, so now you can buy me a cappuccino sometime :)

    Thanks for introducing me to Parkour. That should definitely lead to some entertaining youtube sessions. It reminds of a lot of my childhood days when we used to run around on construction sites / half built walls, etc. with the objective being nothing more than enjoying the feeling of crossing obstacles. Never imagined at that time it would get formalized like a religion !!

  4. I am not supposed to click on my own ads, per Google’s terms, so I clicked on your link. Interesting; I got totally disoriented.

    I’ll have to think about your eating example. Seems complex, but I am not sure it is not also group theoretic like rubik’s cube (or better, doing stunts with a soccer ball on your arms and legs like pros can). The pottery example, I would definitely argue, is 2d (though not flat 2d).

    Plus, some people may have more developed 3d senses than others, like with those amazing people who visualize prime numbers without having to compute.

  5. Gillette says:

    I think Rubik cube or Kapsio’s tongue example is less 3D and more surfacial (2D subset of 3D space). If the cube had colors on the insides, it’d be 3D. Perhaps our deficiencies in dealing with 3D are purely due to us learning on a 2D medium since being a kid. Perhaps if you teach kids on a 3D blackboard say, and make them draw pictures in 3D, even sculpt etc, they would find it easier. Unlike you, I think we have got the infrastructure and it’d be like learning a foreign language, damn easy when you’re a kid, extremely difficult when your brain cells set.

  6. Aaron Davies says:

    Speaking of games reminds me of the notoriously difficult Descent series from the mid-nineties–it was much closer to true 3D than the then-new “first-person” games, and as a result was essentially impossible for many people. The modern equivalent is probably a couple specific aerial battles in World of Warcraft (Occulus and Eye of Eternity), which are among the least favorite bits of the game for most players.