This paradox, usually ascribed to Aristotle but possibly even older, is from an Ancient Greek text, Mechanica, from the 4th century BC.


Aristotle's Wheels
All circles have the same circumference!
Drag the coloured circles and watch the outsides unwind. How can it be that both red lines are the same length?

Use the ratio buttons to change the relative sizes of the circles. The illusion becomes less convincing as the inner circle becomes smaller.

Choose a ratio: Animation:

Click here for an explanation.

It is of course impossible for both wheels to mark out the same length on a single rotation. There are two ways to resolve this paradox:
  1. The inner wheel must slip as it rotates, i.e. it is not a a pure rotation, but is also being dragged along by the rotation of the outer wheel
  2. The inner and outer wheels can be decoupled and thus able to rotate at different speeds. Use the button below to see this happening.

       

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About the activity

According to Aristotle, a metaphor involves taking something appropriate to one class of things and applying it to something alien.

This view of a metaphor as a challenging juxtaposition of ideas is somewhat akin to paradox, (meaning in Greek "contrary to expectation", or more colourfully, "beyond belief") and relates well to the experience of enjoyment in tackling the obvious mistakenness of a paradox, and the satisfaction of a resolution. Paradoxes motivate us to think intensely, and critically examine our existing knowledge, in an effort to incorporate something new and unexpected.

In "Essay on what I think about most", the poet and classicist Anne Carson reflects that metaphor teaches us:

Not only that things are other than they seem,
and so we mistake them,
but that such mistakenness is valuable.
Hold onto it, Aristotle says,
there is much to be seen and felt here.
Metaphors teach the mind
to enjoy error
and to learn