Smooth Jordan Curves Inscribe Every Rectangular Shape
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Smooth Jordan Curves Inscribe Every Rectangular Shape∗ B Sury The forced quarantine during these troubled times has brought forth a collaboration with a wonderful discovery. In May, Joshua Evan Greene and Andrew Lobb combined forces to resolve a problem that is more than a hundred years old. In simple terms, they prove that any planar closed curve which is smooth— that is, without sharp edges—and which are simple—that is, without crossings—contains four points which form the vertices of a rectangle with diagonals of any given slope. For this reason, it is known as the ‘Rectangular Peg Problem’. Before going on to describe the notions, results and methods more precisely, we digress a little to draw attention to a recurring theme in low-dimensional topology ascribing to the fact that dimension 4 often turns out differently. Although unrelated to the rectangular peg problem, we recall the spectacular fact that any space homeomorphic to Rn is also diffeomorphic to it precisely for n 4. In contrast, there are uncountably many spaces homeomorphic to R4 , any two of which are mutually differentiably inequivalent. The corresponding question for the 4-sphere S 4 is still open. Also, as a curiosity, we mention two interesting problems which involve four special points even though they are of
totally different flavours. In 1909, Syama Prasad Mukherjee proved (a special case of) the ‘Four-Vertex Theorem’ which asserts that each simple, closed planar curve other than a circle contains at least four ‘vertices’ (points where the curvature has an extremum). The other interesting problem is still open; it is a conjecture of Ron Graham asserting that if A is a subset of the integer lattice Z2 such that the sum of the series m21+n2 as (m, n) varies over A diverges, the set A must contain the vertices of some square. We mention in passing that Graham passed away on July 6th. Let us return to the rectangular peg problem, In 1911, Otto Toeplitz conjectured that any Jordan curve (simple, closed, continuous, planar curve) contains four points which form a square. This conjecture is still open in this generality. Why is this surprising? This reveals a connection between the geometry of the plane and the topology (Jordan curves are topologically but not geometrically invariant). Now, a word about the special role of 4. It is a simple exercise to prove that any Jordan curve must contain three points that form a triangle similar to any arbitrary triangle. On the other hand, dissimilar ellipses inscribe dissimilar pentagons. Also, note that two distinct ellipses meet at the most at four points. Hence, pentagons are not inscribable in general. By 1929, the conjecture of Toeplitz was solved for smooth Jordan curves. In 1977, Vaughan showed that some rectangle is always inscribed. Greene and Lobb have proved now that every rectangle has a similar rectangle inscribable in any smooth Jordan curve. A so-
∗ Vol.25, No.8, DOI: https://doi.org/10.1007/s12045-020-1030-y
RESONANCE | August 2020
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