Mathematics Awareness Month 2014: Mathematics, Magic, and Mystery

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Thinky the Dragon and Hollow Face Illusions

Thinky the Dragon is a magical beast who seems to be alive, following us around the room in a seemingly impossible way as we move. All the while our brain tries to figure out his secret.

Thinky belongs to the class of hollow face illusions, which are a subcategory of ambiguous images. They all share the same ability to challenge the brain by presenting it with more than one plausible model for what it sees. Hollow face illusions usually work best when you close one eye, or stand far enough away so the brain can’t use depth perception to analyze the illusion. (The Thinky illusion works for some people with both eyes open.) If we turn Thinky too far and see behind his head, a singularity occurs and the illusion breaks down: the brain sees that it has been fooled.

Taking it Further

This magical illusion effect has to be seen to be believed. Click on any of the images below to download the template for Thinky, now available in six colors for Math Awareness Month! Then follow the video directions below to assemble Thinky.

We’ve also developed three brand-new illusions specifically for Math Awareness Month. Click on any of the images below to download the templete for the illusion. Links to assembly videos are below each image.

How to assemble the duck. How to assemble the rocket. How to assemble the block.

The Underlying Mathematics

In the following video, ThinkFun president and original Thinky handler Bill Ritchie explains the math behind how the illusion works.

The math behind Thinky’s surprising movement involves the combination of a subtle geometric effect, called reverse motion parallax (a variation of motion parallax), with the brain’s attempt to “map” Thinky’s unorthodox motion to its stored models of how faces look and act.

Like most Hollow Face illusions, Thinky is related to the Necker Cube illusion, perhaps the simplest, and oldest, example of an ambiguous object. In its 3D form, it is a wire-frame cube that seems to flip-flop as we watch it. Our brains have two plausible mappings for the image, but they’re mutually exclusive, and the brain must choose. Ambiguous image illusions usually have a “preferred” interpretation: we’re more likely to think we’re looking down on the Necker cube, for example, because we normally see objects from above rather than from below.

Here’s a simple thought experiment that might help explain why Thinky is such an effective, magical illusion: If you were to place a “normal” (i.e., non-inverted) Thinky on a table, point his face at you, then walk in a circle counterclockwise around the midpoint of his base, he would rotate clockwise away from you at the same speed you’re rotating, until his face was away from you. This is as we would expect.

Now let’s replace the “normal” Thinky with our “inverted” Thinky. (A good way to do this is to place Thinky on a plate, with his base centered on the plate and his face toward you, and either walk in a circle around the plate, or sit in place and rotate the plate about its axis.) In this case, because Thinky is an inverted figure, as we rotate counterclockwise about him, our brain misinterprets what it sees, and adds the absolute value of Thinky’s angular rotation to ours, instead of subtracting it, creating an apparent net speed twice ours. Throughout our rotation, Thinky’s head will continuously rotate in the same direction we’re moving. Because he’s moving at twice our speed, his face will appear to be rotating ahead of us, until we reach the edge of the model, and the illusion breaks down. (A wire-frame cube will exhibit these same behaviors as long as your brain “switches” its interpretation from a “positive” cube to a "negative cube".)

Adding to the richness of this effect is the “reverse motion parallax” effect, which is the opposite of motion parallax. Motion parallax is best illustrated by imagining you’re looking out the side window of a moving car: objects that are closer will seem to be moving faster than those that are farther away. In the case of reverse motion parallax, the opposite is true: objects that are actually farther away will be perceived as being closer, and will therefore be seen to be moving faster.

Reverse motion parallax helps the brain to believe that the normal rules of angular motion still apply, despite the inverted structure of the model. (This is partly due to viewing Thinky with one eye closed: the brain loses its three-dimensional clues.) For example, Thinky’s snout is behind his ears in the physical model, but we see it as being in front of the ears. The snout is farther away from Thinky’s axis of rotation than his ears, so although both are rotating at the same angular speed, the snout appears to be moving faster than his ears. (It’s worth noting that Thinky’s eyes may appear to be tracking us during this experiment, but they’re not. It’s the same optical illusion that causes the eyes of portraits on a wall to seem to follow us as we move.)

Reverse motion parallax helps illusions like Thinky to exhibit independent, life-like motion that adds to the magical effect. His features subtly shift or morph at different rates during the rotation, and in unexpected ways, adding to the illusion. The illusion also works when you tip the model forward and backward, as well as from side to side. Thinky not only appears to wobble in your hand: parts of the illusion may unexpectedly shrink or grow as you move. Reverse motion parallax works better for some hollow face illusions than for others: it’s up to the designer to make the most of this powerful tool!

The hollow-face illusion: Object-specific knowledge: general assumptions or properties of the stimulus?,” by Harold Hill and Alan Johnston, (Perception, 2007, Vol. 36) delves more deeply into the psychology of the hollow face illusion.

In “Knowledge in Perception and Illusion” (Philosophical Transaction of the Royal Society B: Biological Sciences, 1997, 352, pp. 1121–1128), Richard Gregory discusses the Angular Rotation Effect and many other illusions.

Amanda Turner designed the Rocket, Block, and Duck illusions, and the Thinky “Montage” and “How to Assemble” videos. Chris Morgan created the “How to Assemble” videos for the Rocket, Block, and Duck.

The design for Thinky was inspired by Jerry Andrus. In this video Jerry demonstrates his illusions.