Spinning Your Wheels

Sometimes the world is all backwards.  And sometimes it just looks that way.  Exhibit A: car commercials.

Did you catch it?  If not, watch again, and this time focus on the wheels.  At certain points they seem to be spinning in the wrong direction even though the professional driver isn't spitting out pieces of his transmission all over the closed course, which suggests that the wheels are in fact rotating forward.  So what gives?

You might assume your brain is just getting overwhelmed by all the crazy spinning.  It happens.  And that seems pretty reasonable until you think back and realize that you've never actually witnessed this phenomenon in person.  (Right?)  Spinning wheels can certainly look blurry in real life, but always in a forwards-blurry kind of way.  Then again, we're not talking about real life.1  Something's up with cameras.

Your TV screen, as you know, is one big optical illusion.  Nothing on there is actually moving; it's just a series of still images shown in quick-enough succession to force your brain into interpolation mode.  In other words, you have to infer the motion of an object based on its jump in position from one frame to the next.  In order for the illusion to be convincing, the time between frames has to be relatively short, so cameras typically record at around 30 frames per second.  That means the time between successive frames is about 1/30th of a second.

To recap: the video camera captures a single frame, the car's wheels continues to rotate (unrecorded) for 1/30th of a second, and then the camera captures another frame. Rinse and repeat.  It's important to note that the wheels are moving while the camera isn't looking, so how far they move in between frames depends on how fast the wheels are spinning.

Like so.  Imagine this wheel is attached to a car that's driving to the right.  (We'll color one of the spokes green just to make it easier to keep track of the wheel's rotation.)  If the wheel moves a just a little bit forward in between frames, it'll look something like this:


If the wheel moves a lot forward between frames, though, it might look something like this:


So now we see where the illusion comes from!  If the wheel is moving fast enough, if it moves far enough in that 1/30th of a second, it actually appears to tick backwards a tiny bit in each subsequent frame, tricking your brain into thinking it's rotating in reverse.

Another interesting thing happens if the wheel is moving just fast enough to get through exactly 1/5th of a rotation between frames.  Because of the wheel's five-fold radial symmetry, rotating the figure by 1/5th of a full turn doesn't change anything about its apparent orientation.  A wheel spinning at this rate, shot at 30 fps, wouldn't appear to be moving at all:


Here's an animated diagram so you can better see what's going on.  Play around with the number of rotations the wheel makes each 1/30th of a second and see how you can get its apparent motion to change directions...or even stop entirely.

So the mystery reveals itself.  At any rate that's a leads to multiple of 1/5 of a rotation, the wheel will appear stationary; slightly more and it will appear to creep forward; slightly less and it will appear to creep backward. That explains why film of an accelerating car shows its wheels appearing to cycle through those three states. See for yourself in the interactive widget below, as it tracks the wheel's apparent motion from speeds of 0 through 100 mph.

Cool, right?  The moral of the story: if things seem to be moving in the wrong direction for you, try mashing your foot on the throttle.  Or the brake, I guess.  But where's the fun in that?

Teachers, want to have this conversation in your class?  Check out our new lesson, Spinning Your Wheels.

1.  It's  possible to observe this phenomenon under certain conditions in the wild, but it's pretty rare.  You can replicate the experience in everyday (non-recorded) life using strobe lights, which effectively let you manipulate your vision's "frame rate."  People have made some really cool demonstrations this way.  On a more somber note, since fluorescent bulbs are basically high-frequency strobes, they can lead to car-commercial-like illusions in potentially dangerous real-life situations.  Like, for example, tricking you into thinking a piece of machinery in your loud factory is standing still when it's actually rotating at an arm-shredding speed.

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