Figures Inscribed in Curves
A short tour of an old problem

by
Mark J. Nielsen
Associate Professor of Mathematics
University of Idaho

These pages give a brief and informal introduction to one of my favorite unsolved mathematics problems -- the so-called "inscribed squares problem". The question can be informally stated as follows:

Notice that we don't care whether or not the square "stays inside" of the curve -- in fact, its sides may cross the curve many times. Nor do we care whether or not the corners of the square appear on the curve in any particular order.

A more exact statement of the question requires a couple of (fairly simple) definitions.

Intuitively, this is exactly what you'd expect a "closed curve" to be -- you start drawing at some point and trace a curve without lifting your pencil from the paper and without making your curve cross itself, ending at the point where you began. The word "simple" refers to the fact that the curve doesn't cross itself. The inscribed squares problem can now be stated efficiently as: To me, this is an appealing problem for several reasons: The problem was first posed by Toeplitz in 1911, and while many interesting partial and related results have been given, no one knows the answer to the question above. In fact, the "range of ignorance" with regard to this question is quite astounding! For all the work that has been done on this question, we don't yet know enough to rule out either of the following extreme possibilities: Given that we can't rule out either of these extremes, how much is known? Some of the first efforts to answer the question resulted in proofs that all polygons, differentiable curves, and curves bounding convex regions had inscribed squares. The best result to date is probably the following theorem (1989) due to Walter Stromquist. Here, "nice enough" means the following: for each point P on the curve there must be a coordinate system for the plane in which some piece of the curve containing P is the graph y = f(x) of a continuous function.

At least one prominant mathematician I have met believes that Stromquist's theorem is good enough to make the statement "All simple closed curves have inscribed squares" morally true -- presumably because any curve you could actually draw with pencil and paper is nice enough to be covered by Stromquist's theorem. In other words, this theorem more-or-less settles the informal version of our original question -- any curve you draw on a sheet of paper will have an inscribed square. But the more formal version of the question is far from settled, because "most" simple closed curves have fractal-like behavior and are not "nice" at all. And while it is certainly good to have an answer for the pencil-and-paper version, the topologist in me cares about those curves that exist only in our imaginations. The universe of mathematics is much larger than just what we can draw on paper -- if there are curves that don't admit inscribed squares, I want to know about them.

But even if we are never able to ultimately answer the big question (which, by the way, I believe we will be able to do!), investigating the inscribed squares problem has led to many interesting side results. Below you will find links to six of these. I have chosen them for the simplicity and/or appeal of their proofs, which you can read by clicking on the theorem name. I have stated the theorems and proofs in their simplest, most intuitive versions, leaving off the messy details that are often necessary to state and prove a theorem in its most general form. If you want all the details, the reference to the relevant research paper is provided.

The first theorem shows that the "nice" behavior demanded by Stromquist's Theorem can be replaced by symmetry -- a different type of "nice" behavior. You will notice in the proof that the symmetry property reduces the number of points to be located on the curve from four to two.

The next two theorems don't require any special behavior on the part of the curve, but also don't quite guarantee inscribed squares. Instead, they say we can always find inscribed four-sided figures that are related to squares. Finally, many interesting results can be proved about triangles inscribed in curves. This is, of course, a bit easier to deal with than squares, since the number of points to grapple with is only three. If you like geometry, you may enjoy reading a few of the proofs of the above theorems (just follow the links). For a more thorough introduction to the inscribed squares problem, as well as many more references to related papers, consult pp.58-65 and 137-144 in the excellent book Old and New Unsolved Problems in Plane Geometry and Number Theory (1991, Mathematical Association of America) by Victor Klee and Stan Wagon.

You may even feel inspired to try your own hand at chipping away toward a solution to the inscribed squares problem. Below you will find a few exercises to get you warmed-up, followed by some open-ended (and as-yet-unanswered) questions. Good luck -- as they say in TV's The X-files, "the truth is out there somewhere."


Exercises

  1. Looking for examples is always a good way to warm up to a mathematics problem!
      (a) Find an example of a simple closed curve that has exactly one inscribed square. (b) The circle is an example of an inscribed square in which every point is a vertex in an inscribed square. Find another such simple closed curve.
  2. From Theorems D and E we know that every simple closed curve has an inscribed equilateral triangle (lots of them, in fact!) and the whole point of the inscribed squares problem is to figure out if every simple closed curve has an inscribed square. Show (by giving a negative example) that not all simple closed curves have inscribed regular pentagons.

  3. Suppose we form a simple closed curve by adjoining the segment of the x-axis from x=0 to x=1 to the graph of a continuous function
    f: [0,1] ---> R
    such that f(0) = f(1) = 0 and f(x) > 0 for 0 < x < 1. (See the figure at right.) You don't need Stromquist's Theorem to show that this curve has an inscribed square -- all you need is the Intermediate Value Theorem and Extreme Value Theorem from calculus. The proof makes a nice application of these facts -- try it!

  4. Use the "mountain climbing" technique (as in the proof of Theorem B) to prove the following relative of Theorem A:
    THEOREM: Every simple closed curve that is symmetric across some line has an inscribed square.


Open Questions

  1. Can you extend Theorem E say "given any simple closed curve J and any triangle T there are at most two points of J that are not a vertex in some inscribed triangle similar to T"? (This is true in the case that T is an equilateral triangle -- see the note at the end of Theorem E's proof.)

  2. Can you prove (or disprove) a theorem similar to Theorem E for curves in Rn?

  3. Can you find a way to do away with the requirement in Theorem F that the curve have a tangent line at one point? (This seems like an artificial hypothesis, but is needed to make the current proof work.)


Comments or Questions about this page can be directed to me at markn@uidaho.edu.

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