Mathematical Research in High School: The PRIMES Experience
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 tr...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
American Mathematical Society (AMS)
2017
|
| Online Access: | http://hdl.handle.net/1721.1/110267 https://orcid.org/0000-0002-0710-1416 https://orcid.org/0000-0002-1639-4548 https://orcid.org/0000-0003-0868-8981 |
| Summary: | 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]. |
|---|