In the classic video we’re sharing here, you see two concentric circles drawn on a rolling wheel, all sharing the same center. As the wheel rolls along the ground, it appears that the two inner circles and the edge of the wheel cover the same linear distance in one full rotation. Strange, right? This seems counterintuitive—The inner circles have a smaller circumference, so how can they travel the same distance?
Here’s what’s really happening:
🔹 The outer wheel touches the ground and rolls without slipping. It covers a distance exactly equal to its circumference.
🔹 The inner circles don’t touch the ground. They rotate along with the wheel but don’t roll independently. Instead, they’re passively dragged along—combining rotation with slipping, not true rolling.
To help illustrate this, the diagram below replaces circles with concentric hexagons. As the outer blue hexagon rotates, it carries the smaller ones by making them slip—this slipping is shown by the dashed lines.
A Mathematical Perspective
Mathematically, the “paradox” shows that a one-to-one correspondence between points on three distinct rotating paths doesn’t imply equal arc lengths. While each point on the smaller circles aligns with a point on the larger one, their trajectories differ due to the nature of their motion.
Conclusion
Aristotle’s Wheel Paradox isn’t a true paradox, but a reminder that intuition can mislead when dealing with motion and geometry. The apparent equal travel of the concentric circles and the wheel results not from identical rolling behavior, but from the interplay between rotation, slipping, and perception.