# Understanding Geometry – 4

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Aleksej Ivanovič Markuševič’s book, “Remarkable Curves” discusses the properties of ellipses, parabolas, hyperbolas, lemniscates, cycloids, brachistochrone, spirals and catenaries.  Among these “lemniscates” are the ones that I encountered only once before starting undergraduate education (all other curves appeared frequently in physics textbooks) and that too just to calculate the area enclosed by this curve. So I will discuss the properties of lemniscates in this post.

Let’s begin with the well-known curve, ellipse. An ellipse is the locus of points whose sum of distances from two fixed points (called foci) is constant. My favourite fact about ellipses is that we can’t find a general formula for the perimeter of an ellipse, and this little fact leads to the magical world of elliptic integrals. This, in turn, leads to the mysterious elliptic functions, which were discovered as inverse functions of elliptic integrals. Further, these functions are needed in the parameterization of certain curves, now called elliptic curves. For more details about this story, read the paper by Adrian Rice and Ezra Brown, “Why Ellipses are not Elliptic curves“.

Lemniscate is defined as the locus of points whose product of distances from two fixed points $F_1$ and $F_2$ (called foci) is constant. Lemniscate means, “with hanging ribbons” in Latin.  If the length of the segment $\overline{F_1F_2}$ is $c$ then for the midpoint of this line segment will lie on the curve if the product constant is $c^2/4$. In this case we get a figure-eight lying on its side.

Lemniscate of Bernoulli; By Kmhkmh (Own work) [CC BY 4.0], via Wikimedia Commons

The attempt to calculate the perimeter of the above curve leads to elliptic integral, hence can’t derive a general formula for its perimeter. Just like an ellipse!

If we equate the value of the constant product not to $c^2/4$ but to another value, the lemniscate will change its shape. When the constant is less than $c^2/4$, the lemniscate consists of two ovals, one of which contains inside it the point $F_1$, and the other the point $F_2$.

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700b.gif

When the product constant is greater than $c^2/4$ but less than $c^2/2$, the lemniscate has the form of a biscuit. If the constant is close to $c^2/4$, the “waist” of the biscuit is very narrow and the shape of the curve is very close to the figure-eight shape.

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700b.gif

If the constant differs little from $c^2/2$, the waist is hardly noticeable, and if the constant is equal or greater than $c^2/2$ the waist disappears completely, and the lemniscate takes the form of an oval.

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700a.gif

We can further generalize this whole argument to get lemniscate with an arbitrary number of foci, called polynomial lemniscate.

# Midpoint Polygon Conjecture is false

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Contrary to my expectations, my previous post turned out to be like  Popular-­Lonely primes and Decimal Problem, i.e. I discovered nothing new.

My conjecture is false. Following counterexample is given on pp. 234 of this paper:

Counterexample of the conjecture, taken from: Berlekamp, E. R., E. N. Gilbert, and F. W. Sinden. “A Polygon Problem.” The American Mathematical Monthly 72, no. 3 (1965): 233-41. doi:10.2307/2313689.

As pointed out by uncombed_coconut, the correct theorem is:

Theorem (Berlekamp-Gilbert-Sinden, 1965). For almost all simple polygons there exist a smallest natural number $k$ such that after $k$ iterations of midpoint polygon procedure, we obtain a convex polygon.

The proof of this theorem is very interesting. Till now I thought that proving euclidean geometry theorems using complex numbers was an overkill. But using an N-tuple of complex numbers to represent the vertices of a closed polygon (in given order),  $\mathbf{z} = (z_1,\ldots , z_N)$,  we can restate the problem in terms of eigenvectors (referred to as eigenpolygons) and eigenvalues.  Following are the crucial facts used is the proof:

• An arbitrary N-gon (need not be simple) can be written as a sum of regular N-gons i.e. the eigenvalues are distinct.
• The coefficient of $k^{th}$ eigenvector (when N-gon is written as linear combination of eigenpolygons) is the centroid of the polygon obtained by “winding” $\mathbf{z}$ $k$ times.
• All vertices of the midpoint polygons (obtained by repeating the midpoint polygon procedure infinitely many times) converge to the centroid.
• The sum of two convex components of $\mathbf{z}$ is a polygon. This polygon is the affine image of a regular convex N-gon whose all vertices lie on an ellipse. (as pointed out by Nikhil)
• A necessary and sufficient condition for $\mathbf{z}$ to have a convex midpoint polygon (after some finite iterations of the midpoint polygon procedure) is that the ellipse circumscribing the sum of two convex components of $\mathbf{z}$ is nondegenerate. (The degenerate form of an ellipse is a point. )

For a nice outline of the proof, please refer to the comment by uncombed_coconut on previous post.

Since I didn’t know that this is a well studied problem (and that too by a well known mathematician!) I was trying to prove it on my own. Though I didn’t make much progress, but I discovered some interesting theorems which I will share in my future posts.