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The square peg problem is deceptively simple: Given a continuous simple closed curve in the plane, prove that there are four points on the curve that form a perfect square. It was first posed in 1909, and remains open despite a long history of papers on the subject. The history of work on the subject reveals that it is relatively simple to prove that a ''differentiable'' closed curve has an inscribed square, but it is very difficult to extend the required arguments to the case where the curve is merely continuous. (For instance, does a closed fractal such as the Koch snowflake contain an inscribed square? In general, this is unknown.)

== Results on Inscribed Squares ==
With Elizabeth Denne of Smith College and John McCleary of Vassar College, I have worked on the problem from a new angle. Our results prove that there are an odd number of squares in any simple closed curve which is differentiable or ``not too rough``. Here is an example of such a curve, with 5 inscribed squares.

<center>
<gallery Caption="Squares inside an irregular curve" perrow=5 widths="110px">
Image:squiggle-1.png
Image:squiggle-2.png
Image:squiggle-3.png
Image:squiggle-4.png
Image:squiggle-5.png
</gallery>
</center>

Arnold categorized immersed plane curves in terms of three invariants called Strangeness <math>St</math>, positive Jump <math>J^+</math> and negative Jump <math>J^-</math>. In these terms, embedded loops like the curve above have all three invariants equal to zero. It is interesting to look at the squares inscribed in some example curves with different values of these invariants.

For instance, this figure 8 curve with <math>St = 0, J^+ = 0, J^- = -1</math> has two inscribed squares.

<center>
<gallery Caption="Squares in a nonembedded curve">
Image:Figure8Square1.png
Image:Figure8Square2.png
</gallery>
</center>

This curve, created by performing a Reidemeister II move on a loop, has <math> St = 0, J^+ = 0, J^- = -2</math> and seems to have 9 inscribed squares. I conjecture that only 3 of these are topologically nontrivial, while the other 6 cancel each other out pairwise. (I admit that the square in the upper right and the one in the lower left are pretty close to one another. But according to my computer program, they really ''are'' different.)

<center>
<gallery Caption="Squares inside a Reidemeister II curve" perrow=3>
Image:R2square0.png
Image:R2square1.png
Image:R2square2.png
Image:R2square3.png
Image:R2square4.png
Image:R2square5.png
Image:R2square6.png
Image:R2square7.png
Image:R2square8.png
</gallery>
</center>

This curve has <math>St = 1, J^+ = -2, J^- = -4</math> and 14 squares
<center>
[[Image:Strange1squares.png]]
</center>

This curve has <math>St = 2, J^+ = -4, J^- = -6</math> and 15 squares
<center>
[[Image:Strange2squares.png]]
</center>

This curve has <math>St = 3, J^+ = -6, J^- = -8</math> and 12 squares
<center>
[[Image:Strange3squares.png]]
</center>

Based on these examples, jump and strangeness seem to play a role. We might guess that the number of squares is equal to <math> St + (J^+ - J^-) + 1</math> mod 2.  Unfortunately, our theory is only a mod 2 theory, so no further results can be drawn from it as yet. However, it is clear that there is more to see: the figure 8 curve above must have a square inside each lobe of the 8, so though the theory only predicts that the number should be even we know that the number must be at least 2.

== Results on Inscribed Triangles ==

===The Ellipse===
[[Image:EllipseMovieFrame.png|thumb|175px|left|an equilateral triangle inscribed in an ellipse]] We also hope to show that there are continuous loops of certain shapes inscribed in any given curve. For example, it has been known since the 1950's that at any point <math>a</math> on a plane curve, you can find <math>b</math> and <math>c</math> so that <math>\Delta abc</math> is an equilateral triangle. This is true for any shape of triangle (but you cannot guarantee a particular size for the triangle). We conjecture that these triangles can be chosen so that they vary continuously with <math>a</math>. That is, that there is a continuous loop of equilateral triangles inscribed in any plane curve. [http://www.jasoncantarella.com/downloads/EllipseMovie.zip This animation] (requires Quicktime Player) shows the loop of equilateral triangles in the ellipse. A [http://www.jasoncantarella.com/downloads/BigEllipseMovie.zip higher resolution version] of the movie is available as well.

We can visualize the curve of equilateral triangles in the ellipse as a curve in the three dimensional torus <math>S^1 \times S^1 \times S^1 = T^3</math>, where the three coordinates give the positions of the vertices along the ellipse. If we parametrize the torus as a cube with opposite faces identified, we can see the curve as the space curve show below:
<center>
<gallery widths=175px>
Image:Ellipsesnap1.png
Image:Ellipsesnap2.png
Image:Ellipsesnap3.png
</gallery>
</center>
A three-dimensional model of this curve is provided [http://www.jasoncantarella.com/downloads/ellipsemodel.zip here]. You need the [http://www.geomview.org Geomview] viewer to spin and explore the model in 3d.

===A nonconvex curve===
In the jellybean-shaped curve shown below, our computational experiments reveal that there are seven loops of equilateral triangles, one of which is topologically nontrivial (the basepoint travels all the way around the curve) and six of which are topologically trivial loops (the basepoint does not travel all the way around the curve). 
<center>
<gallery>
Image:JellybeanFrame.png|a triangle from the nontrivial loop of triangles in a nonconvex curve
Image:JellybeanLoop2Frame.png|a triangle from a trivial loop of triangles of class A in a nonconvex curve
Image:JellybeanLoop3Frame.png|a triangle from a trivial loop of triangles of class B in a nonconvex curve
</gallery>
</center>
The topologically trivial loops can be divided into two classes: in class A, the triangles remain near the bottom lobe of the curve. There are three loops in this class, all containing the same triangles, but with different basepoints. In class B, the triangles remain near the upper lobe of the curve. Again, there are three copies of the loop, each putting the basepoint in a different location.

Animations are available [http://www.jasoncantarella.com/downloads/jellybean_loops_A.zip here] for the trivial loops in class A, [http://www.jasoncantarella.com/downloads/jellybean_loops_B.zip here] for the trivial loops in class B, and [http://www.jasoncantarella.com/downloads/jellybean_movie.zip here] for the nontrivial loop.  These animations are also available at higher resolution. [http://www.jasoncantarella.com/downloads/big_jellybean_movie.zip The nontrivial loop is here,] [http://www.jasoncantarella.com/downloads/big_jellybean_movie_A.zip click here for class A], and [http://www.jasoncantarella.com/downloads/big_jellybean_movie_B.zip here for class B].

It is also interesting to visualize the curve of equilateral triangles inscribed in the jellybean curve as a curve in the configuration space of triples of points on the jellybean. This space is a version of the 3-dimensional torus, which can be seen as a cube with top and bottom, left and right, and front and back faces identified. The pictures below show the seven loops of equilateral triangles in this cube.

<gallery widths="175px">
Image:Loopsnap1.png
Image:Loopsnap2.png
Image:Loopsnap3.png
</gallery>

In these images, the main loop is colored green, the trivial loops in class A are red, and the trivial loops in class B are blue. The second picture shows the threefold symmetry of the curve (seen by relabeling an equilateral triangle). We can see in the pictures that the main (green) loop represents the homology class <math>(1,1,1)</math> in the homology of the three-torus, which the other loops are homologically trivial in the three-torus. 

We note that in these pictures, some of the trivial loops protrude from the cube. Because of the identifications which make the cube into the three-torus, these curves should appear as a collection of fragments which are glued together by the identifications on the sides of the cube. However, this makes the loops hard to see and also obscures the threefold symmetry of the picture. So we have lifted these loops into a covering space of the three-torus in order to draw them as connected space curves.

You can view a 3-dimensional model of these curves, with or without tubes around them, with the files [http://www.jasoncantarella.com/downloads/jellybeanloops.zip here], which are in the OOGL VECT and OFF formats of the [http://www.geomview.org Geomview] 3d viewer.

===A figure-8 curve===

[[Image:Figure8Triangle.png|thumb|175px|left|An equilateral triangle inscribed in a figure-8 curve]]The situation for curves with self-intersection is even more interesting. Analyzing the figure-8 shown at left shows a single topologically nontrivial loop of triangles, shown in this [http://www.jasoncantarella.com/downloads/figure8triangles.zip animation] ([http://www.jasoncantarella.com/downloads/figure8trianglesbig.zip or look at this larger version]). Somewhat surprisingly, this loop of triangles includes 'infinitesimal triangles', which happen when all three vertices approach the self-intersection of the curve. However, as we can see from the pictures below, this does not affect the homology class of the loop of triangles. The second picture, showing the threefold symmetry of the loop of triangles, is particularly interesting. It shows that while the curve may cross a subdiagonal of the cube (where two vertices come together on the curve at the intersection point), it never crosses the main diagonal (that is, we never have a triangle where all three vertices come together at the same parameter value.

<center>
<gallery widths="175px">
Image:Figure8snap1.png
Image:Figure8snap2.png
Image:Figure8snap3.png
</gallery>
</center>

As before, you can view these curves in 3d with these [http://www.jasoncantarella.com/downloads/figure8loop.tgz Geomview models].

===A Reidemeister II curve===

[[Image:Reidemeister2Triangle.png|thumb|115px|left]]There are two standard operations which one can perform on a simple closed curve to get a nonsimple closed curve-- the Reidemeister I (twist a loop of the curve to make a figure-8) and Reidemeister II (push a loop of the curve across a section of the curve) moves. (There is also a Reidemeister III move, but you need some crossings already to perform an RIII move.) 

This curve is the result of a single Reidemeister II, or a pair of Reidemeister I moves, applied to a simple closed curve. We note that again, the loop represents the same homology class in the 3-torus, again the curve of equilateral triangles is 3-fold symmetric, and (as in the case of the jellybean curve) there are two classes of trivial loops on the curve. 

The main loop of triangles is shown in this [http://www.jasoncantarella.com/downloads/r2small.zip animation] (a [http://www.jasoncantarella.com/downloads/r2big.zip larger version] is available, but it's actually so large that my computer can't play it smoothly. Perhaps yours is faster.)

As before, 3d models of these curves [http://www.jasoncantarella.com/downloads/r2models.tgz are available, too.]

<center>
<gallery widths="175px">
Image:R2snap1.png
Image:R2snap2.png
Image:R2snap3.png
</gallery>
</center>

==Results on Inscribed Quadrilaterals and N-gons==

[[Image:Ellipsequad.png|thumb|115px|left]]The movies above show loops of inscribed triangles in plane curves. It's a natural question to ask whether there are loops of 4-gons or 5-gons or n-gons in curves as well. Our data provides some evidence that this is the case. For instance, here's an [http://www.jasoncantarella.com/downloads/ellipse_quad_movie_small.zip animation] of a loop of equilateral 4-gons in an ellipse. Note that four times during the loop the equilateral 4-gons become squares. The animation is also available [http://www.jasoncantarella.com/downloads/ellipse_quad_movie.zip bigger].

[[Image:JellybeanQuad.png|thumb|115px|left]] The jellybean curve above also has a loop of equilateral 4-gons. You can see the entire loop in this [http://www.jasoncantarella.com/downloads/jellybean_quad.zip animation] of the 4-gons traversing the jellybean. Notice that at four points on the loop, the equilateral quadrilaterals form a perfect square!
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