It is clear that dividing the perimeter of a square of side 1 by its area results in a ratio of 4. Doing the same for two adjacent unit squares that share an edge results in a smaller ratio, in this case 3. So what can be said about this ratio in the case of an arbitrary union of (possibly overlapping) unit squares in the plane? Keleti's Perimeter to Area Conjecture was that this ratio never exceeds 4, although this is now known to be false. The picture shows a counterexample to the conjecture in which the ratio is approximately 4.28.
A problem related to this one appeared as Problem 6 on the famous Hungarian Schweitzer Competition in 1998. That problem asked for a proof that the perimeter-to-area ratio of a union of unit squares in the plane has an upper bound. Impressively, several Hungarian undergraduates were able to prove this for the competition. In the same year, Tamás Keleti published his Perimeter to Area Conjecture that the bound was exactly 4. This is now known to be incorrect; the best known bound is about 5.551 and was found by Keleti's student Zoltán Gyenes in his 2005 Master's thesis.
I found out about this problem from the paper Bounded – yes, but 4? (http://arxiv.org/abs/1507.08536) by Paul D. Humke, Cameron Marcott, Bjorn Mellem and Cole Stiegler. This paper was posted recently but is dated November 2013, and it does not mention progress on the problem that has been made since then. The 2014 paper Unions of regular polygons with large perimeter-to-area ratio (http://arxiv.org/abs/1402.5452) by Viktor Kiss and Zoltán Vidnyánszky proves that Keleti's conjecture is false. The picture here comes from Kiss and Vidnyánszky's paper.
The counterexample in the picture uses 25 unit squares, but Kiss and Vidnyánszky use a systematic method to construct (in Theorem 2.4) a counterexample using only five unit squares, and use an ad hoc method to construct (in Section 3) a counterexample using only four squares. They also (in Theorem 2.8) show that the analogue of Keleti's conjecture for equilateral triangles is false, by exhibiting a counterexample involving four triangles. The authors conjecture that similar constructions can be made for other regular polygons; more specifically, that their systematic method can be used to produce an analogous configuration of n+1 regular n-gons that has a greater perimeter-to-area ratio than a single regular n-gon.
Kiss and Vidnyánszky pose a number of other interesting questions in their closing section. For example, is n the minimal possible number of n-gons in a counterexample? And does something analogous happen in three dimensions, with regular polyhedra?
Although Keleti's conjecture is false, it is known to be true under certain additional hypotheses. In his Master's thesis and in a 2011 paper, Gyenes proved that the conjecture holds (a) if the squares have a common centre, or (b) if all the squares have sides parallel to the x or y axes, or (c) if only two squares are involved.
Gyenes' Masters thesis: http://www.cs.elte.hu/~dom/z.pdf
A mathoverflow discussion of Keleti's Perimeter to Area Conjecture from 2010 (http://mathoverflow.net/questions/15188/) includes some interesting commentary on the problem from various people, including .
The picture contains an oblique reference to Betteridge's law of headlines (https://en.wikipedia.org/wiki/Betteridge's_law_of_headlines) which states that Any headline that ends in a question mark can be answered by the word “no”. This principle also applies to the title of the paper by Humke et al.
#mathematics #sciencesunday #spnetwork arXiv:1507.08536
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