Go to a mirror, and stand very close to it with your head up straight. Look carefully at your eyeballs, then slowly tip your head to one side. Your eyeballs rotate the opposite way to your head so they remain on the horizontal plane!

Not side-to-side, or up-and-down, but spin around the axis that goes through the pupils. I never knew eyes could do this!

Someone! Tell me what this phenomenon is called!

An addendum to this, months later... If you do this too much, your eyes may break free and begin spinning freely. Very freaky.

Just kidding.
Chameleons' eyes rotate independently of one another, so the little lizards can actually sit still and see what's behind them and what's in front of them at the same time. Nifty.
Almost as nifty, any human who can cross their eyes can also make it look like they're crossing just one eye at a time. Impress your friends!
1) Cross your eyes.
2) Then, look really, REALLY hard to your right. (With both eyes.)
3) Cross your eyes again.
4) Look really, REALLY hard to your left.
I know this is silly as shit, but it's also sort of neat, especially if you're 11 years old, or you have to babysit, or you're on a reasonably bad date.
I'd call the phenomenon 'looking at a pair of reflective, symmetrical, rotating objects in front of a mirror'. If you've been up way too long, and have lots of nice visible blood vessels, try this experiment:

- Look at the mirror, and align your finger with a blood vessel of your choice
- Tip your head to one side
- Note that your finger and the blood vessel don't line up anymore.

What might be happening here is a combination of many factors:

1 - Your brain has a mechanism whereby any voluntary movement of your head or eyes is compensated for, and so the world doesn't seem to move when your head does.

Nifty experiment:
- Close one eye, and look at something
- Through the eyelid of the open eye, push on the cornea, so the eye is moved to look at something else
- The world seems to move.

(When drunk, the alcohol in the bloodstream changes the density of the fluid in the ear, making the balance sensors in it also detect gravity (Try tipping your head the other way when the world is spinning... It will start to spin the other way...). This also breaks the visual system, which is why the world appears to spin.)

2 - Eyeballs have rotational symmetry, making it difficult to observe rotation about their axis, unless they have a convenient marking.

3 - Eyeballs are wet and shiny; pretty good mirrors. They reflect objects pretty well, and when they are rotating, the reflected image will stay still, creating the impression that the eyeballs are rotating.

4 - You're expecting it to happen, so it does.

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