What is the smallest-area convex set in the plane
a needle (unit straight line segment) can be reversed
(spun around 180 degrees)?
Answer: an equilateral triangle of unit height.
OK, now what if you allow non-convex sets? What is the
smallest area set in which you can reverse a needle?
For instance, try a smaller 3-cusped hypercycloid.
See Figure 1.
In fact, you can try a similar idea with n-cusps.
Suprisingly, there exists sets of arbitrarily small
area in which a needle can be reversed!
Draw pictures. Have people think about the second question
for a minute.
The Math Behind the Fact:
This Fun Fact is easy to present but involves
some deep mathematics. The construction of
arbitrarily "small" sets (sets of small measure)
containing a needle in all directions is
a detailed analytical construction, and
the general study of Kakeya sets is currently
an active area of research in analysis.
You can learn
more about measure theory
after taking a course in real analysis.
How to Cite this Page:
Su, Francis E., et al. "Kakeya Needle Problem."
Math Fun Facts.