September 1, 2006 11:48 PM   Subscribe

I can visualize complex 3D objects in my mind, and give them arbitrary rotations. Why can't I do long multiplication in my head?

Clearly the brain is capable of doing far more complex math than simple long division and long multiplication. In order to rotate a 3D model computationally, you have to do a number of math operations per element.

Since I can imagine objects I have never seen before, I can't simply be playing back memories. My brain must be generating new data. So why can't I spit out the answers to simple equations involving long numbers?
posted by b1tr0t to Science & Nature (21 answers total) 4 users marked this as a favorite
I would say human brains have evolved in an environment where you need to be able to think conciously in 3D, but not do long division.

Think of the amount of math needed to aim an easy throw of a baseball or spear, but the amount of time it takes most people to learn calculus.
posted by scodger at 12:15 AM on September 2, 2006

yeah, just because you can visualize something that is computationally difficult to model does not mean that your brain is actually performing those same computations.
posted by sophist at 1:04 AM on September 2, 2006

The two things use completely different parts of the brain. When you rotate things in your mind you don't actually do any maths, you just do it - but a lot of people can't, as that part of their brain isn't as well developed.
posted by Orange Goblin at 1:04 AM on September 2, 2006

Multiplication is an algebraic operation, essentially symbol manipulation, while visualizing and rotating objects is a geometric operation. These are essentially unrelated skills.
posted by number9dream at 1:15 AM on September 2, 2006

The easiest explanation is that the brain hasn't had as much pressure to optimize multiplication. Since one is inundated with experiencing multiple perspectives of the same visual field during waking hours (as one interacts with the environment), the brain, naturally, is comfortable with such operations.

Orange Goblin: When you rotate things in your mind you don't actually do any maths, you just do it

This gets into semantical territory, but the same is true of "math" as well. When asked to add 23 - 6, you "just do" all the intermediate steps. Calling some operations 'math' and some not is an outward categorization. Internally, all cognition's undertaken on the same substrate.
posted by Gyan at 1:37 AM on September 2, 2006 [1 favorite]

Just because you can visualize something that is computationally difficult to model does not mean that your brain is actually performing those same computations.

Quite. I can imagine catching up to to objects that are moving slower than I am - but this doesn't mean I should expect to be able to solve Xeno's paradox as a result.
posted by ed\26h at 2:10 AM on September 2, 2006

Snails can grow their shell in the golden ratio. I doubt, however, that they can compute it.
posted by dmd at 6:04 AM on September 2, 2006

If you practiced multiplying/dividing/squaring big numbers as often as you model 3d objects in your head, you'd be pretty good at it.

I read a book that lets you learn how to do these feats of mental math very quickly in your head. With some practice, you can make it look like you are just plucking the answers to stuff out of thin air.

Also, I think you're conflating two concepts in your question. You are comparing a human brain and a machine. A human brain has consciousness, and a computer doesn't. Even a machine that can spew out a complex 3d model of an object doesn't understand it like you do in a holistic sense. It's still just a bunch of bits in memory.
posted by popechunk at 8:38 AM on September 2, 2006

In grade school I was much better than average at geometry and found that I had to put little to no work into it to get good marks, but I sucked horribly at math even though I put more work into it. I couldn't understand how some of my friends were just the opposite - looking at something and figuring it out that way seemed so much easier than trying to do abstract calculations. Some people are just more visually-oriented learners.
posted by jimmythefish at 9:07 AM on September 2, 2006

If it makes you feel any better I'm the same way. I can do complicated geometric proofs in my head because I can "see" the problem... never having to write anything down.

On the other hand I can't even balance a simple chemical equation (and algebraic task) without getting really pissed and using alot of scrap paper and a calculator.
posted by JFitzpatrick at 9:46 AM on September 2, 2006

Gyan, I suspect you don't understand how 3D rotations work. I've studied them some, and they are very, very hairy.

I assure you with all my heart that the brain is not doing quaternion math on a matrix of points in space. Whatever mechanism it uses for rotation, that ain't it. It's not Euler rotations either. That's how most people THINK about rotations, but thinking and mentally visualizing them are very different things. The brain doesn't end up with axis-lock due to complications from the Euler model. (quaternions avoid axis lock, but ye gods are they hard to work with.)

16-7 is something you can simply memorize; we're perfectly good at memorizing facts. You are, in essence, saying that 3D visualization is the same thing as memory, and it just isn't. You also seem to be claiming that the brain 'does math' to do rotations, and it most certainly does not.
posted by Malor at 10:25 AM on September 2, 2006

The brain seems to percieve and manipulate numbers and letters in a fundamentally different way than it does solid objects. People can have dyslexia of varying degrees, but have you ever known anyone who turned right instead of left, walking into a wall instead of through a doorway, because they "saw" the doorway being on the other side? It just doesn't happen.

Anecdotally, I was completely illiterate in my dreams until about the age of 20, although I have no dyslexia or other trouble reading in waking life. Whenever I dreamed of encountering a sign or any kind of written material, the letters just jumped around and performed a big happy dance, making it impossible for me to read. I began to be able to read a word or two at a time, and then phrases occasionally as time went on, but I still get the happy dance occasionally to this day. I don't think I've ever dreamed of a solid object that wouldn't sit still long enough for me to get a good look at it (I probably will now, lol). Anyway, like I said, the brain seems to process numbers and letters in a very different way than it does solid objects.
posted by Marla Singer at 10:36 AM on September 2, 2006

On top of everything else everyone's said, long multiplication generally involves many symbols. Rotating (e.g.) a chair in your head basically only involves one symbol, the chair, and your brain is well-developed enough to treat the entire complex shape of a chair as one whole three-dimensional manipulatable symbol in your mind. Rotating mentally is very easy once you are thoroughly familiar with the entirety of a shape. You're not really generating new information.

Something like 324 x 932, though, involves unfamiliar combinations of symbols. Humans can generally juggle about 7 symbols in their short-term memory, and you've already got seven there. 3,2,4,x,9,3,2. Once you start trying to figure out how to multiply them, you have to start juggling the results, too, and as you know long multiplication involves summing the results of a series of short multiplications; two three-digit numbers involve 9 multiplications.

It's not a very efficient way to do anything—it's sort of like picturing two shapes, each made of three chairs in various rotations, and then figuring out what a new shape would look like by combining each of the component chairs using some weird geometrical operations. (i.e., it's hard.)

Just because we have a short notation for conveying the concept of multiplying two large numbers doesn't mean that it's anywhere near as easy as writing it down. The complexity of math is why we have these computer things.

(Having said that, there are people that can do it in their head, but they generally practice a lot, every day, and are really excited about it. I'm pretty sure the average person doesn't have any real incentive to do that. There are some books you can get, though, if you're really interested.)
posted by blacklite at 12:16 PM on September 2, 2006

To add a bit to the good answers so far, one must also remember that rotation is a dynamic process, which means it occurs over time. There are two major ways of performing dynamic computations: discrete and continuous.

The discrete method breaks the process down into a sequence of snapshots, and solves each snapshot completely. When done quickly enough, discrete methods produce admirable facsimiles. All modern digital technologies depend on discrete dynamics (CD-Audio, polygonal games, etc.) Note that math must be explicitly done for each discrete step of the process.

There is another way, though. Sticking with the audio reproduction analogy, consider a record player. The dynamics are literally continuous. As the record physically rotates, the needle bumps around, which is then used by an amplifier to produce the music. No math is performed, explicitly or implicitly. There are no steps. The system continuously evolves, producing the desired result as a side-effect. One might say the universe itself does the math for you.

The brain probably does rotations in a continuous way. There might be some set of neurons dedicated to manipulating shapes, probably connected intimately to our visual centers. Thus, we can understand rotations when we witness them, and also mentally perform them without any physical muse.

I would bet that all human imagination is similar to this. That is, our imagination is just our normal perceptual systems operating detached from the external world.
posted by clord at 5:48 PM on September 2, 2006

If you're asking about yourself in particular, this AskMe question of mine might be useful.

If you're asking about humanity in general, I'm going to invoke the evolutionary argument again. An excellent book on this topic is How the Mind Works by Steven Pinker. Pinker gives evidence that the brain has actually changed its use over time, and that it used to be an organ that helped with seeing rather than thinking. Since we're using an organ that's optimized for seeing, any mental task that requires us to visualize will be relatively easier than one that requires us to taste, smell, or hear in our heads.
posted by lunchbox at 6:57 PM on September 2, 2006

Malor: I assure you with all my heart that the brain is not doing quaternion math on a matrix of points in space. Whatever mechanism it uses for rotation, that ain't it.

Leaving aside your proof for this contention, that's irrelevant and I don't see what's that got to do with anything I wrote.

16-7 is something you can simply memorize

a)How many permutations will you memorize?
b)You seem to have missed my point. All cognition, that which we classify as 'math' or 'not math' takes place on the same substrate using the same neural operators. The distinction is made by humans based on social & cultural factors.
c)I don't know where I claimed that (declarative) memory comes prominently into the picture. Humans have to move in space; they have to plan their movements so that goals can be accomplished, so visual manipulation is a very basic needed skill. Figuring out 23 * 43 hasn't been so urgent a need. The fact that in the way we consciously approach math, the latter is much more simpler than the former is tangential.
posted by Gyan at 8:00 PM on September 2, 2006

Interesting answers.
posted by b1tr0t at 10:16 PM on September 2, 2006

All cognition, that which we classify as 'math' or 'not math' takes place on the same substrate using the same neural operators.

Not true..

Characterizing the specific differences is an open question, I guess, but differences are observable.

However, I'm not completely convinced by Malor's point either...

Here is a helicopter 30deg away from axis lock about the pitch axis. It is rolling between + and - 10deg. Notice how all three joints in the gimbal mount are rotating, the roll joint between + and - 20deg (twice as large as the helicopter's motion).

Two camera angles of the same motion. From my M.A.Sc. thesis.

I don't think people normally use Euler angles to visualize rotations - people think in principal axis rotations, which are approximately quaternion rotations.

I created that animation by first considering what would happen near gimbal lock (same as axis lock). Visualizing the helicopter making its rotation, and then wondering how the other objects (links, in robotics talk) would move. Clearly, some links would have to move a lot to allow the helicopter to move just a little - that is a mathematical scale operation, although it is approximate. Finally back to visualizing the rotation of the links..

It really is a grey matter..
posted by Chuckles at 12:39 AM on September 3, 2006

Chuckles: Not true..

Any part of the brain acts as it does, because of its connections to the the rest of the cortex, i.e. its place within the architectural hierarchy. The brain is an integrated processing system. There have been cases where lesions of brain regions have led to no functional deficits because the functions get reimplemented on the remaining substrate (may not happen because of the age of the brain and the nature of the lesion). The occipital lobe isn't "meant" to be the locus for visual processing, but that's what results from the overall architecture.
posted by Gyan at 5:40 AM on September 3, 2006

Okay, but.. Allowing for repurposing in the neocortex, it is still only one portion of the brain.

Has functional equivalence actually been shown? I can make pliers do the job of a wrench, but the results aren't as good. Sometimes I will even come up with a new use for a certain tool.. Complete uniformity is limiting.
posted by Chuckles at 11:09 AM on September 3, 2006

This is getting more & more tangential, but I'm not arguing for (super)plasticity in the mature brain. Just pointing out that function is context-sensitive, so it's misleading to say that parietal lobe does this or the occipital lobe does that. Relatively, they may be the loci of peak activity during imaging of this or that task, but that not indicate caged modularization.
posted by Gyan at 11:39 AM on September 3, 2006

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