Imagine that the two objects in Figure 1 are
solid (with thickness) and
made of very flexible and stretchy rubber.
Question: is it possible to deform one object
into the other in a continuous motion
(without tearing or cutting)?
Surprise answer: Yes!!
Hint: it is important that the object is
solid and has thickness; this transformation
cannot be done with a one-dimensional piece of string.
It is also not possible to do this with a piece of rope
because even though the rope has thickness, it is not
flexible or "stretchy" enough.
See below for an explanation and animated gif.
Or, don't scroll down if you want to think about it a while!
Students (as well as you) may find this very hard
to believe! If you like this one, see also
The Math Behind the Fact:
One way to do this is the following.
Widen one of the loops and move one of its handles
along the stem between the two loops to the other loop
and push it through the hole so that the two loops
The reference contains a sequence of pictures of this
Graeme McRae has generously contributed the
animated gif in Figure 2, showing another solution to
this problem! (Thank you, Graeme!)
You can take a course in topology to learn
more about properties of objects that do not change
under continuous deformations.
How to Cite this Page:
Su, Francis E., et al. "Unbelievable Unlinking."
Math Fun Facts.