Let's look at one more optical illusion which reveals important features of how our brains perform inference, and suggests how better awareness of these processes of inference can lead to improved thinking, including in our daily lives. This time around the theme is ambiguity.

The spinning dancer

Spinning Dancer

The spinning dancer is a remarkable piece of work. Do you see the dancer pivoting clockwise on her left foot? Or counterclockwise on her right? If you're at all like me, you're now seeing one or the other - but if you look at the picture for some time, you'll suddenly see the dancer spinning in the opposite direction. It may help (for reasons I'll explain below) to focus on the pivot foot or thereabouts, mentally blocking out the rest of the image. Can you pick a direction on purpose? (Part of what makes investigating the human mind fascinating hobby is finding out what others' brains can do that mine can't, and vice versa. Initially I had no control at all, but interestingly, over the course of writing this post I got much better.)

Focusing on the foot helps explain how the illusion works: the image provides no clue to distinguish between "toes front" or "heel front", it's all a dark shadow with no depth information. What we see is a foot-shaped cutout growing and shrinking due to a foreshortening effect, alternately to the left and right. This information is perfectly compatible with either direction of spin. (The rest of the image is more of the same; the entire image is functionally equivalent to a dark bar growing and shrinking in alternate directions.) The interesting question is then, why does our consciousness insist on reporting that we're seeing a perfectly unambiguous direction of motion? We're not seeing an ambiguous dancer: we're seeing one that is clearly spinning one way - until the "flip" happens, and we see her just as clearly spinning the other way.

Ambiguity, uncertainty and inference

This is reminiscent of the way our explicit models of the world have trouble dealing with quantum uncertainty - our intuition is that things must happen one way or the other, "as if the half-silvered mirror did different things on different occasions", as if the dancer actually was spinning one way then another. In the latter case at least, sober reflection tells us this can only be in the map, not in the territory - the animated GIF isn't being changed right under our noses.

"Perception is inference from incomplete information", says Jaynes - I have noted previously how this may bring insight into where our biases come from. The spinning dancer tells us something about how it feels, from the inside, to perform inference under uncertainty. That is what ambiguity is - a particular kind of uncertainty. Not the one that results from a paucity of information: the dancer illusion works because the image is quite detailed, not in spite of it. Rather, it is uncertainty that results from having too many hypotheses available, and lacking some crucial information to distinguish the correct one among them.

(Abstractly, we see that it is best to be right about something, and worst to be wrong; to be uncertain is somewere in the middle. Our visual system has a different opinion, and prefers the following ranking: right > wrong > uncertain. "Make a decision, even if it is the wrong one, we can always revise it later." Actually, many of our decision processes have that same bias; I'll revisit this topic when I post at greater length about "real options". For now, the topic I'm sticking to is ambiguity as a particular type of uncertainty.)

Spinning words

Last night I shot an elephant in my pajamas. How he got in my pajamas, I’ll never know. - Groucho Marx

It takes some effort to construct ambiguous pictures, whereas anything expressed in words seems to enjoy a head start. This is great news for people with a sense of humor: linguistic ambiguity is a constant source of merriment. In fact, though there are many competing theories of humor, it makes at least some sense to see ambiguity and its related themes of frame-crossing, reinterpretation as playing a key role in humor in general.

To some, ambiguity is much less funny. Some professions, including jurists, proponents of synthetic languages such as Lojban or Loglan, and software engineers see ambiguity as an evil to be uprooted. The most famous cartoon (excepting perhaps some Dilbert favorites posted near every cubicle) among software professionals is an extended lament about the consequences of ambiguity in human language. Ambiguity is the source of unmeasurable amounts of confusion and even hurt among people relying on the written word to communicate - an increasingly common circumstance in the Internet age. Not that oral communication is exempt; but at least in most cases it offers more effective mechanisms for error correction.

Sacrifice, duality, reframing: the powers of ambiguity

And yet, vexing as it may be to acknowledge it, instrumentally effective thinking often seems to rely on ambiguity.

In the ancient game of Go, there is a certain level of play that can only be reached by mastering the art of sacrifice. Now, in some cases this may be part of a pre-established plan: stones that you are defending will get a better position if, as a preliminary, you place a stone within enemy territory solely in preparation for a move that threatens to rescue it. In many instances, though, sacrifice involves redefining a previously valuable stone or set of stones as "sacrifice stones". What is called "light" play often involves deliberately ambiguous moves, in which you have a plan to abandon one of several stones without depending on which one; the opponent's play will determine that.

Another example involves the mathematical theme of duality, where a given kind of structure can be expressed in one of two ways which are equivalent in meaning but rely on different tools or operations. Some problems turn out to be easier to solve in a domain which is the dual of that where they were originally formulated. Unfortunately, my math having gone rusty for quite a while, I don't recall offhand any examples that I feel familiar enough with to discuss here, but to give you a hint of the flavor, consider a frequent "trick" in computing probabilities: when asked the probability of A (say, "at least two of the people in this room share a birthday") it's often easier to consider the probablity of not-A ("no two people share a birthday") and taking the complement.

Another way to exploit ambiguity turns up in the domain of interpersonal relationships, in the guise of "framing" and "reframing". Some of the advice in Alicorn's recent post, for instance, involves casting around for reframes - ways of interpreting behaviour that you dislike in a person that make these behaviours tolerable or understandable instead of irritating and repulsive. In Stumbling on Happiness Daniel Gilbert argues that ambiguity is a key component of psychological resilience. A person's success in life is partially determined by their ability to redefine their values, sense of happiness, etc. on the fly, in answer to the difficulties they encounter. This is only possible if our interpretation of the world contains lots of ambiguity to start with.

Artificial ambiguity

As readers of LessWrong, "hobbyists of the mind", we can often gain insight into our human minds by framing questions about constructed minds - by forcing ourselves to confront the design space of possible minds. A corollary attitude is to be cautious about the reverse process - unreflectively projecting some perceived attribute of human minds, such as randomness or reliability, into a necessary property of artificial intelligences. Still, I have come so far in this post in large part to ponder whether ambiguity is a necessary capability of minds-in-general, rather than a human design flaw.

Douglas Hofstadter's Fluid Concepts and Creative Analogies discusses fluidity, a theme that strikes me as closely related to ambiguity (and is explicitly discussed throughout). Hofstadter is fond of "microdomains", simplified settings where human intelligence neverthelesss still easily exceeds what we can, as a rule, program machines to do. A favorite of mine is "Do this!" exercises, which inspired Robert French's Tabletop research program: two people are seated at a typical restaurant or café table, with similar (or very different) implements on each side of the table: plates, forks, knives, glasses... One person touches an item and challenges the other to "do this!".

Ambiguity arises in the Tabletop domain because exact mappings between the two sides may not exist, but "analogical" mappings often do: your wine glass maps to my water glass, for instance. The fun starts when more than one plausible analogical mapping suggests itself. Your lone wine glass maps either to my wine glass (paired with a glass of water) or to my salt shaker (the only non-plate unpaired item on my side).


I suspect that efficient cross-domain generalization requires dealing with ambiguity and analogy. Back in the human mind-space, few theories of the scientific process deal explicitly with ambiguity and analogy, Pickering's mangle being a notable exception. In some vague sense it seems to me that ambiguity provides "degrees of freedom" which are necessary to the conceptions of plans, and their flexible execution - including sacrificing, changing domains or notations, and reframing. I expect it to be a major theme of instrumental rationality, in other words, and therefore would like to delineate a more precise formulation of this vague intuition.

This post has explored some themes I'm likely to return to, or that form some kind of groundwork (for instance when I touch on "programming as a rationalist skill" later on). But more importantly, it is intended to encourage further discussions of these themes from other perspectives than mine.

What do you know about ambiguity?

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Will: "Damn, how do people make things like that?"

Nick: "I'm guessing a very high visuo-spatial intelligence."

Will: "I bet a superintelligence could create a whole universe like that."

Both, after thinking for a few seconds: "Gyahh!!"

Will: "Everything would have an infinite amount of interpretations, all of them equally correct. Everyone's beliefs would be right and everyone would be happy! Yay!"

Nick: "But whether or not everyone's interpretations were equally correct would depend on one's interpretation."

Will: "Doesn't this look like the world we live in?"

This argument for postmodernism brought to you by the Singularity Institute Visiting Fellows Program.


*grumbles about how I ought to be a shoe-in*

Scattered comments:

1) It's not that some illusions give hints as to how human cognition works; all of them do, because that's what makes it an illusion: a deviation from accuracy that is used as a usually-harmless shortcut.

2) All inference, not just perception, is based on incomplete information. MacKay says (quoting someone far before him), "To make inferences, you must make assumptions."

3) I don't think it's most accurate to say that the brain prefers wrong to uncertain. Rather, it prefers sometimes-wrong to always-uncertain. The heuristics it uses will usually be right, giving the benefit of being able to make use of greater (assumed) knowledge of a scene, at the cost of being wrong in a few rare cases. This is better than remaining non-committal, such as what would result from only using an Occamian heuristic.

4) The spinning dancer illusion doesn't work because of the detail; it works because it strips away all of the hints your brain normally uses to disambiguate a scene (such as light gradients, relative sizes, etc.)

5) Here's another great multi-stable perception illusion -- with something like 6 possible perceptions.

6) Great article! I'm also writing one about the non-obvious constraints that general intelligence must adhere to.

ETA: So I guess you guys want more of this kind of comment and less of the other?

I'm also writing one about the non-obvious constraints that general intelligence must adhere to.

Ah, I'm looking forward to that. I've been wondering for some time now about what invariants can be expected in the design space of general minds.

it prefers sometimes-wrong to always-uncertain

I'm referring to the trope that "indecision is worse than a bad decison", which enjoys some currency despite being correct only under very constrained conditions. I think I first came across it in one of Heinlein's novels, where he mentions it as a heuristic of military command.

it works because it strips away all of the hints

That too: and the reconstructed motion relies, it seems to me, on the details which make it a convicing picture of a spinning dancer, as opposed to an abstract dark figure. However...

another great multi-stable perception illusion

Wow. Neat. Especially neat is how the cues work; once you've had the perception you no longer need them and can bring flip almost at will. There are some intriguing aspects, like I can't keep on perceiving the helix if I look away from the center.

"To make inferences, you must make assumptions"

"When you assUme, you make an ass of U and me" - another great quote about the role ambiguity can play in human communication and interpersonal relations that I wanted to work into the post but couldn't find a spot for. ;)

Ah, I'm looking forward to that. I've been wondering for some time now about what invariants can be expected in the design space of general minds.

Thanks, but it's not so much about what to expect of minds in general, but rather, it's a list of criteria you'll need to meet if you want to design a good, intelligent, cross-domain-optimizing mind. However, the sets do overlap, and you can probably use it the way you describe, to some extent, just as long as you don't expect everything to always hold.

That too: and the reconstructed motion relies, it seems to me, on the details which make it a convicing picture of a spinning dancer, as opposed to an abstract dark figure.

Okay, fair point then. (Still, did they have to make the, um, chest area so detailed at the ... um, endpoint...?)

(Btw, I noticed after clicking on your name that you can add a location and website to your handle, which I just completed.)

you can add a location and website to your handle

Yes: I noticed because of the work I've done recently on the integrated Anti-Kibitzer, but otherwise I suspect it will go unnoticed. Wondering if the Tricycle folks are planning a brief top-level post to advertise the changes.

I have also added my info - I would definitely approve of such a post.

Happen to be on FICS? I just realized that there are probably a lot of chess players on Less Wrong and it wouldn't be hard to make a Google group if there were enough frequent users.

I mostly play postal-like chess on Gameknot as "packbat", but I apparently have an account as "RobinZ". I've never used it.

Cool, didn't know what you looked like until now.

Nice improvement on not having profiles. It still seems oddly limiting... no room to list an email. I'd like to see a Hacker News style empty box.

I agree with most of your points, but this stood out:

All inference, not just perception, is based on incomplete information.

There's a very important difference between (i) perceiving the black shape as a dancer spinning on her right foot and (ii) deducing that if Socrates is a man, and all men are mortal then Socrates is mortal.

In case (i) the 'premises' (i.e. the animation) didn't contain sufficient information to determine that the conclusion was correct. In case (ii) the premises are sufficient.

No matter how hard I try, I can't see the dancer on her right foot. Its not possible!


Can you easily make a version that has the coloured cues on the upper part of her body going in opposite directions from those on the lower part? It would be interesting to see whether that suffices to make the un-cued version look as if the two parts of her body are rotating in opposite directions.

[EDITED to add: oh, wait,you didn't actually make it, did you? So I should have said: "Would anyone like to make ..."; it seems like it should be not-too-painful with a decent image editing program.]

If you cover L and look at a spot between C and R, and then cover R and look at a spot between C and L, and then cover L and look at a spot between C and R, you can make her look like she's swinging her leg left and right in front of her. =)

Also, doesn't the fact that the shadow/reflection of her moving foot is visible on the right-left suggest that her leg is further away at this point, and thus she's going anti-clockwise?

First thing I've seen that can make me see the dancer on her right foot. Weird thing is, if I look at the right side of that image, I can instantly and easily see her on her right foot, but my brain immediately defaults to seeing her on her left once I look away.

Edit - 'my' brain, not 'by' brain.

That kind of helps but it seems... I dunno, a little "artificial". I don't see it as though it was actually that way, and if I stop concentrating Its still on the left foot. Come to think of it, I never even saw it on the right foot. ERRR!

Initially, I could only see her on her right foot. After looking at Alicorn's link, I can see it either way, but only by looking at the left/right image, then looking at the center... I can't force myself to change how I see it just by looking at the un-hinted version. And whenever I look away for a few seconds and then come back to it, my brain always defaults to seeing her on her right foot.

I wonder if there's any correlation between which way people see this and any other aspects of personality, cognition etc... maybe right/left handedness?

I have to rely on the side images to switch interpretations as well. I find my ineptness at controlling my interpretation both amusing and disturbing/frustrating.

I'm right handed, and see her on her left foot- always. Also, when I look at it, it looks like the camera is locked, and the dancer is moving. Do you have to switch how you view the camera to see it the other way?

Hmmmm, I'm right handed too.

So far, I've always been seeing the camera as fixed - it hadn't occurred to me to think of it otherwise... you mean you're picturing the dancer standing still with the camera rotating around her?

No, but I thought it might be a factor in seeing her on her right foot. Maybe its just me, but I don't even see how it could be physically possible for her to be on her right foot if the cameras locked. This is frusterating.

But that is a good point, we should do a survey and see what kind of people see it which way. Maybe its just random, but I'm guessing theres a reason people see it differently. It reminds me of those people with motion sickness. As I understand it, theres nothing different about their brain or visual system, its just the way they grew up and their neurons were trained. When the image is shaky, their mind can't compensate and it confuses them. And there are different degrees of it to.

Maybe its just me, but I don't even see how it could be physically possible for her to be on her right foot

That's exactly what I thought as well.... except the other way around.


ETA: ok, I just looked at it again, for the first time in several hours, and now I see her on her left foot, and can't make myself see her on her right foot (opposite of before).

This keeps getting freakier and freakier. I feel like my brain is defective.

Take a coffee cup (with a side handle). Hold it level in front of your eyes, and rotate it slowly around the vertical axis (in either direction).

You can tell the way it's rotating because you see the handle coming into view in front of the cup on one side, and disappearing behind the cup on the other. Inferring the direction of spin from that is easy.

However, if the cup was covered in a magic paint which absorbed all visible light, you would no longer be able to make that distinction. What would you see? You'd see the handle shrinking and growing, alternately to the left and right. This would be ambiguous as to the direction of spin.

The dancer is the same, only more detailed, and because your brain infers a whole 3-D anatomy to go with the inferred direction of motion, it's hard to let go of the conviction that this made-up story about the direction is the "right" story.

Your brain is working perfectly fine, it just wants things to be one way or the other, rather than uncertain.

Yeah, at an intellectual level I understand all that... but it doesn't make it feel any more intuitive.

And I'm still curious as to whether there's any significance to which way a person sees it. And why my preferred direction changed after being away from it for awhile (I'm still stuck on left, if anyone's keeping score)

any significance to which way a person sees it

Susceptibility to priming effects? My intuition says handedness should have little to do with it, and the initial direction perceived by someone not familiar with the illusion would be evenly distributed, BUT people would differ more characteristically in their propensity to see it both ways.

I'm having a hard time trying to figure out how to explain the phenomenon I just witnessed. I went back to that link feeling that I might have missed something and I'd give it another try. In the first image she is clearly spinning on her left foot, the same way I always see it. I then look at the second image. Same thing. On the third image, I thought I was seeing it the way I always do, but when I looked at which foot was on the ground, it was her right! Since the foot in the middle never moves except for up and down, I wondered how it could possibly be the same image. So I compared it to the one in the middle to see if it really was the same, and both rotate exactly the same! Every pixel is the same (except the ones with color.) How could this be possible! So I looked back at the first and compared it to the one in the middle. Same thing, every pixel where it should be, and her left foot was clearly the one moving up and down. How could this be! I thought about it and came up with the idea that if the backside was similar enough the the front, you could switch between which side you saw as front/back and left/right would chang accordingly. But this is not the case, they are far to different. I am completely dumbfounded to the physics behind this. Looking at it again, I can focus on the spot between the first and the second and watch them move the same, then switch to focusing on the spot between the second and third, making her suddenly stop and turn the other direction. By switching back and forth really fast, she quickly wobbles back and forth. I begging to wonder if magic is possible.

The backside isn't just similar to the front, they are both exactly the same shade of black.

This might help you understand what's going on: if you lie on the ground and have someone trace your outline, you can't tell from the outline whether you were face up or face down.

Is there a pausible version of this? If I could freeze the individual frames, I might get it. I refuse to believe that her backside is that similiar, because just look at all the detail. Also, the part that reallly bugs me is not her direction of spin, but the fact I can't even tell which way is right or left.

The detail is all in the outline (the part inside the outline is all black). From just the outline, you cannot tell

  • whether her foot is a little closer to you than it was a second ago or a little farther (so your brain picks one)
  • or whether she's facing towards you or away from you (so your brain picks one).

The Wikipedia entry (link in OP) has a frame-by-frame version. (ETA: or you could download an animated GIF editor, and yourself produce from the ambiguous original two unambiguous versions; it really doesn't take much in the way of extra cues to convince your brain.)

Also "versions of the image with an additional visual cue" (click "show").


I found that briefly very disturbing -- it seemed that the mere presence of an "L" or "R" on the vertical leg was sufficient to get my brain to see it one way or the other, even though I'd given no conscious thought to the question of which leg was which.

But of course it isn't actually the choice of letter that makes the difference. (I haven't said what is in case anyone else is in the same boat as me and wants to take a few seconds to work it out.)

Huh. I looked at each frame, and found that when she is facing the observer, she is clearly on her left foot. Weird.

Every frame that can be interpreted as her facing the observer and standing on her left foot can also be interpreted as her facing away from the observer and standing on her right foot.


I'm in the same boat as you.


I'm not having near so much trouble today. Yay neuroplasticity!


What works for me is to look at my finger while I spin it around, then the dancer will seem to spin in the same direction.

An interesting application of priming effects!

Hey, that works for me!

I was shocked that this didn't work for me.

I had my two older kids look at it, and that unscientific sample suggested that people may vary widely in a) their initial perceptions (one saw it CW, the other CCW), b) the fixity of their initial perceptions (one got two flips within a minute or two, I had to instruct the other).

Dunno if there has been any systematic study of this, but I'd sure be interested in how these characteristics are distributed, whether they are fixed or variable in a given individual, and so on.

Focus on her hips and hands, and when she's facing directly to the left or the right, mentally pop the far hip and hand to the near side.

Interestingly enough, illusions like this don't faze my visual artist friends. Realistically rendering a scene requires the ability to see a 3 dimensional scene as the two-dimensional projection that actually hits your eyes, and applying that skill to an illusory 3-d image is easy for them.

I usually can't get her to shift, but she shifted once for me today, and now she's stuck again. (Yes, I know it's really my mind, not her.)

I had to block out all but a tiny, tiny piece of the picture (the foot, as suggested) with my hands and concentrate in order to make the flip happen.

It seems like an unfortunate choice of example.

How do you mean?

Err, whats that mean? Do you mean that the dancer is a bad example?

Some problems turn out to be easier to solve in a domain which is the dual of that where they were originally formulated... I don't recall offhand any examples that I feel familiar enough with to discuss here

One of the most famous dualities is the Fourier transform (and the family of similar transforms). Inter alia, this allows one to easily convolve two functions, which is often valuable in the applied sciences and engineering (or just doing a Gaussian blur in Photoshop).

Some problems turn out to be easier to solve in a domain which is the dual of that where they were originally formulated. Unfortunately, my math having gone rusty for quite a while, I don't recall offhand any examples that I feel familiar enough with to discuss here

It's a little advanced, but I like Fourier's solution to the heat equation: any function satisfying certain properties can be thought of as a sum of waves (frequency being dual to position), and the heat equation is easy to solve if your initial data is such a wave. Constructing the solution from this perspective leads us to the heat kernel, which can be calculated without even breaking the function into waves in the first place!


I'll finish the article when I have a little more time, but I wanted to note that, when I read the intro part on the new feed, she was spinning on her right foot/going counter clockwise, up until I read that she could spin both ways. Then my mind flipped her to the left foot/going clockwise and I can't flip it back. I'll see what happens when I return again and read the full article tonight or tomorrow night.

Update: My brain seems to set to whatever direction I was perceiving last, and I can switch it once by looking at only the shadows, but never back to what I originally saw [not without leaving for a while and doing the shadow trick again]. Trying the shadow trick two times in a row oddly doesn't work.

Update 2: My mind still switches at will every now and again, but looking at only the shadows for a while works too, and now works a few times.

At one point, I had the dancer going back and forth instead of rotating, so that her moving foot was always behind her pivot foot.

I can get that too. Whenever I focus on her pivot foot, it seems like the toes on that foot are pointed away from me. If I want to switch the direction, I just focus on that foot and she seems to bounce back in the other direction, and if I keep focusing on that foot she seems to bounce back and forth with her stuck-out leg farther away than her pivot leg.

Yeah, I can get that. It was a lot harder but I finally got her moving foot in front of her pivot foot instead. Mental exercise for the day. I wonder if there would be external value in training your brain on adjusting these ambiguous images (the Necker Cube being a non-animated one).

Try getting her leg and her body spinning in different directions... it's pretty trippy :)

Where do people get this "no depth cues" claim? The way her lifted leg moves suggests so obviously a clockwise motion that even Alicorn's link can't make my brain see the counterclockwise motion for longer than a round or two at most.

I mean, the only way the perspective makes any sense is if the lifted leg is furthest away when it's the highest up in the 2d image. Yes, the shadow/reflection is all wrong but for some reason my brain just refuses to give that priority.

What cues do others use? I'd love to see variations of this image with different cues present/absent/reversed.

ETA: I'm no longer sure that the reflection is wrong. But something about the image is off.

Edit: typos

The Wikipedia page has more info, including a breakdown into individual frames. It looks as if it shouldn't be too hard to use an animated GIF editor to modify the original, adding clues as desired.

If there is interest, in a follow-up I might explore exploiting ambiguity in problem-solving a little more formally, touching on "schemas" for formal decision processes (as discussed e.g. in Drescher) and some problems you run into when you consider what I'd call "reified schemas" which abound around us, objects which embody particular assumptions. Some problem-solving entails introducing new schemas which break such assumptions, which I view as a form of "exploiting ambiguity".


If you can't switch easily, this will probably help: http://i.imgur.com/jEwD9.gif