There are several ways to illustrate the effect Persistence of Vision has on how we view rapid changed in images, one of the simplest being a thaumotrope, a kind of toy popular during the Victorian times, which uses two separate images, which are merged into one image when flicked rapidly between the two. I made one myself to get a better idea of how they work.
There is a separate illusion that comes into play, coupled with the ability to cancel out the blanks between images, that allows us to perceive the actual movement between stationary images. This is called the Phi Phenomenon. When two or more stationary lights flash on and off, one after the other, something appears to move between the lights. So while in reality the lights are simply just blinking, our brains tell us that they actually possess direction and movement.
For example, in the gif below, what in reality is just a series of blinking dots, we perceive to have a circular movement, even though nothing is actually moving at all. That is why in animation when stationary images are flashed before us one after the other, we see action.
Combining the two natural optical illusions, we get the illusion of life in animation.
It fascinates me that these natural psychological perceptions allow us to see movement- it means that our automatic response to sequential stimuli is to see life and direction even when there is none. It's almost as if advancements in technology and animation techniques are catching up to our innate desire to see movement.
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