Swiping Needle
The 2D Kakeya's Needle Problem
What is the minimum swiping area needed for a needle of length 1 to perform a 360-degree rotation?
For a period of time, mathematicians believed the best is to have one end of the needle follows a hypocycloid.
1. How does changing the number of cusps (N) affect the shape of the hypocycloid?
2. To minimize the swiping area, should N be odd or even? Why?
3. Which value of N yields the smallest area?
4. The below shows the general formula for the exact area of an N-cusped hypocycloid, with outer-radius of 1 unit. Evaluate the exact area of the needle swiped for your chosen N.
Further Information:
Hong Wang is a mathematician at NYU's Courant Institute who, alongside Joshua Zahl, made a monumental breakthrough on the Kakeya conjecture—the very problem we were just discussing! They successfully proved the 3D version of the Kakeya conjecture in Feb 2025, a famous problem that had remained unsolved for decades.