Wagon-wheel effect is a well-known optical illusion due to the persistence of vision. It happens when the spoke of a wheel rotates to a certain position after the duration of persistent vision. Depending on such position, the apparent wheel motion can have many appearances. However, is it possible to design a certain pattern of wheel entirely free from such an effect? Except for the trivial solution (no pattern or spokes at all).
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Usually the wagon wheel effect is associated with some kind of cinema. This is because a relatively sharp image is taken frequently enough that persistence of vision causes there not to be the perception of multiple images. But, when a wagon wheel is turning, successive frames might show an earlier spoke having almost caught up to where one spoke was at the previous frame, so the direction or speed of rotation can seem slower or reversed.
This phenomenon is known as aliasing in signal processing. When you sample a periodic signal at too low a frequency the period you measure might be wrong. (It's an "alias" for the true period.) In the case of a wagon wheel, 24 frames a second is the measuring frequency, and the actual period is how long it takes a wagon wheel to move to where the previous one was.
The wagon wheel effect is not generally associated with direct observation. Most people observe blurring, rather than spokes appearing to move at the wrong rate or in the wrong direction.
In general, to avoid the wagon wheel effect on film, you want the pattern on the wheel to repeat itself no more often than once every other frame. Maybe a wheel with no spokes, but a mark on or near the rim, would work well.
