We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.
Deep Dive into On the number of inscribed squares of a simple closed curve in the plane.
We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.
It is an open problem if for every simple closed curve in the plane there are four points from the curve that form the vertices of a square. Such a square is called inscribed in the curve (though it is not required that it is contained in the region bounded by the curve). The problem is simply stated, old, and has only partial positive solutions. See [5] for a list of papers, and for comments.
The present note answers in the negative what we interpret as a conjecture posed by Jason Cantarella on his web page [2]. The web site comments on his joint work with Elizabeth Denne and John McCleary on this problem. Their results have been announced in [3]. The author has recently been informed by Elizabeth Denne and Jason Cantarella that the preprint presenting the results announced in [3] is not yet ready to be released. Our recent discussion with Jason Cantarella and Elizabeth Denne on some of the ideas presented in [2] and [4] has been helpful to the author, yet the following statement made at the web site [2] has not been yet clarified:
‘Our results prove that there are an odd number of squares in any simple closed curve which is differentiable or “not too rough”.’
Apparently the exact statement of the above result would appear in the forthcoming paper by Jason Cantarella, Elizabeth Denne and John McCleary.
The purpose of the present note is the proof of the following.
Theorem 1. For every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.
This seems to indicate (though we provide some “evidence” only, and no complete proof) that the following conjecture about the number of inscribed squares of an immersed in the plane curve (self intersections allowed) made at the same web site, is not valid, if only differentiability is assumed:
‘We might guess that the number of squares is equal to St+(J + -J -)+1 mod 2.’ As indicated in [2], St, J + and J -denote the invariants of the curve called strangeness, positive jump, and negative jump, introduced by Arnold. See [1].
First we sketch the construction of an infinitely differentiable simple closed curve in the plane that has exactly two inscribed squares.
Clearly the unit circle has infinitely many inscribed squares. On the other hand it is easy to modify the unit circle to obtain a (non-differentiable) simple closed curve which has exactly two inscribed squares. The construction is shown on Figure 1, left. The arc determined by central angles 5π 4 and 7π 4 is removed from the unit circle, and replaced by the semi-circle y =
The reader may verify that there are only two inscribed squares, as shown on Figure 1, left.
What looks like the equal sides (though they are not line segments) of an isosceles triangle on that picture is the set of points, that are endpoints of the base of a square such that the top side of the square has endpoints that are symmetric about the y-axis, belong to the unit circle, and have y-coordinates ≥ 1 √
) To get a differentiable example we replace that arc with the graph of
Notice that this graph for positive and not too big values of c intersects the unit circle only at the end-points of the arc that was removed. Among these values of c, for larger c the graph intersects each of the two equal sides of that isosceles triangle in two points (we do not count the endpoints of the arc that was removed). For smaller values of c the graph does not intersect the equal sides of the triangle (except at the endpoints of the removed arc). Therefore for a certain value of c (approximately 1.18264) on each side of the triangle there is a unique point that belongs to the graph (apart from the endpoint). We sketch the proof that the two inscribed squares shown on Figure 1, right, are the only ones.
The above considerations show that these two squares are the only inscribed squares that have a horizontal side. Assume S is an inscribed square with no horizontal side. If S has three vertices on the 3 4 -circle (i.e. on the union of arcs AB, BC, CD, see Fig. 2) then it follows that the fourth vertex would be on the unit circle, on the arc that was removed from our curve, a contradiction. Let DA denote the graph of ( * ). Let the vertices of S be E, F, G, H (in this order) and consider the case when E, F belong to the 3 4 -circle, and G, H belong to DA. We only consider two typical cases.
The next set of examples is also based on the idea that we may replace a certain arc of the unit circle. It will eventually lead to a differentiable convex curve with a number of inscribed squares specified in advance.
The idea is very simple, and the proofs are easy (though might be technical) so we omit some of the details.
Start with the unit square and this time remove the arc [ -π 4 , π 4 ]. For convenience we identify any real number P with the corresponding point on the unit circle, if we treat P as an angle. Pick any P ∈ ( -π 4 , π 4 ) and connect
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
