{"id":429,"date":"2019-06-27T17:43:13","date_gmt":"2019-06-27T17:43:13","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=429"},"modified":"2019-11-22T22:08:42","modified_gmt":"2019-11-22T22:08:42","slug":"kakeya-needle-problem","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/kakeya-needle-problem\/","title":{"rendered":"Kakeya Needle Problem"},"content":{"rendered":"\n
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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)?<\/p>\n\n\n\n

Answer: an equilateral triangle of unit height.<\/p>\n\n\n\n

OK, now what if you allow non-convex sets? What is the smallest area set in which you can reverse a needle?<\/p>\n\n\n\n

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<\/em> in which a needle can be reversed!<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Draw pictures. Have people think about the second question for a minute.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
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<\/em> after taking a course in real analysis.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Kakeya Needle Problem.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
E. Stein, Harmonic Analysis. <\/em>MAA film; Falconer, p. 95.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Lesley Ward<\/p>\n","protected":false},"excerpt":{"rendered":"

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