Mathematical Research in High School: The PRIMES Experience
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Etingof_Mathematical research in high school.pdf
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Author(s)
Etingof, Pavel I
Gerovitch, Vyacheslav
Khovanova, Tanya
Date Issued
September 2015
Journal
Notices of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Citation
Etingof, Pavel, Slava Gerovitch, and Tanya Khovanova. “Mathematical Research in High School: The PRIMES Experience.” Notices of the American Mathematical Society 62, no. 08 (September 1, 2015): 910–918. © American Mathematical Society (AMS)
Version
Final published version
Abstract
Consider a finite set of lines in 3-space. A joint is a point where three of these lines (not lying in the same plane) intersect. If there are L lines, what is the largest possible number of joints? Well, let’s try our luck and randomly choose k planes. Any pair of planes produces a line, and any triple of planes, a joint. Thus, they produce L := k(k − 1)/2 lines and and J := k(k − 1)(k − 2)/6 joints. If k is large, J is about [[√2]/3]L[superscript 3/2]. For many years it was conjectured that one cannot do much better than that, in the sense that if L is large, then J ≤ CL[superscript 3/2], where C is a constant (clearly, C ≥ [√2]/3]). This was proved by Larry Guth and Nets Katz in 2007 and was a breakthrough in incidence geometry. Guth also showed that one can take C = 10. Can you do better? Yes! The best known result is that any number C > 4/3 will do. This was proved in 2014 by Joseph Zurer, an eleventh-grader from Rhode Island [Z].
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1090/noti1270
