What is the smallest-area convex set in the plane inside which 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!

**Presentation Suggestions:**

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*. <https://www.math.hmc.edu/funfacts>.

**References:**

E. Stein, *Harmonic Analysis. *MAA film; Falconer, p. 95.

**Fun Fact suggested by: **

Lesley Ward