So how does this illusion actually work?
The entire principle is based on cycloids; half circle shapes that are created by a fixed point on the rim of a circle. Imagine that you attached a marker to a wheel and rolled it along next to a wall. Though the wheel is spinning around in a full circle, the marker would draw something like this:
Based off of this principle, the curved line can be represented linearly. This video took things one step further and staggered eight small circles around one large circle to create the optical illusion. The motion inspires your brain to see the wheel that doesn’t exist.
You see, the curve is dependent on the sine and cosine of the horizontal and vertical position of the marker and the rate of motion is not held constant. Take another look at the above gif: notice how it seems to speed up as it approaches the top of the arc? Though each white circle is moving linearly and independently, the timing distance and changing positions relative to one another makes it look like a wheel in motion.
As Phil Plait points out, this geometry is pretty much how a Spirograph functions, except the toy works on variations of cyloids known as epitrochoids and hypotrochoids. By positioning the pen in a different spot in a wheel other than the edge, it can make various roulette curves. (Your childhood may have been lacking if it didn’t include a Spirograph and a giant pen that had 10 different colored inks inside.)
Math is really damn cool sometimes.
Read more at
http://www.iflscience.com/brain/cycl...S4t00cMIwMF.99