# Geometry & Arithmetic

Standard

A couple of weeks ago I discussed a geometric solution to an arithmetic problem. In this post, I will discuss an arithmetical solution to a geometry problem. Consider the following question:

Given a square whose sides are reflecting mirrors. A ray of light leaves a point inside the square and is reflected repeatedly in the mirrors. What is the nature of its paths?

It may happen that the ray passes through a corner of the square. In that case, we assume that it returns along its former path.

In figure, the parallels to the axis are the lines, $x = m + \frac{1}{2}$ and $y = n + \frac{1}{2}$, where $m$ and $n$ are integers. The thick square, of side 1, around the origin is the square of the problem and $E\equiv(a,b)$ is the starting point. We construct all images of $E$ in the mirrors, for direct or repeated reflection. One can observe that they are of four types, the coordinates of the images of the different types being

1. $(a+2n, b+2m)$
2. $(a+2n, -b+2m+1)$
3. $(-a+2n+1, b+2m)$
4. $(-a+2n+1, -b+2m+1)$

where $m$ and $n$ are arbitrary integers. Further, if the velocity at $E$ has direction cosines, $\lambda, \mu$, then the corresponding images of the velocity have direction cosines

1. $(\lambda, \mu)$
2. $(\lambda, -\mu)$
3. $(-\lambda, \mu)$
4. $(-\lambda, -\mu)$

where we suppose (on the grounds of symmetry) that $\mu$ is positive. If we think of the plane as divided into squares of unit side, then interior of a typical square being

$\displaystyle{n -\frac{1}{2} < x < n+\frac{1}{2}, \qquad m-\frac{1}{2}

then each squares contains just one image of every point in the original sqaure, given by $n=m=0$ (shown by the bold points in the figure). And if the image in any of the above squares of any point in the original sqaure is of type (1.), (2.), (3.) or (4.), then the image in any of the above squares of any other point in the original square is of the same type.

We can now imagine $E$ moving with the ray (shown by dotted lines in the figure). When the point $E$ meets the mirror, it coincides with an image and the image of $E$ which momentarily coincides with $E$ continues the motion of $E$, in its original direction, in one of the squares adjacent to the fundamental square (the thick square). We follow the motion of the image, in this square, until it in its turn it meets a side of the square. Clearly, the original path of $E$ will be continued indefinitely in the same line $L$ (dotted line in the figure), by a series of different images.

The segment of $L$ in any square (for a given $n$ and $m$) is the image of a straight portion of the path of $E$ in the original square. There is one-to-one correspondence between the segments of $L$, in different squares, and the portions of the path of $E$ between successive reflections, each segment of $L$ being an image of the corresponding portion of the path of $E$.

The path of $E$ in the original square will be periodic if $E$ returns to its original position moving in the same direction; and this will happen if and only if $L$ passes through an image of type (1.) of the original $E$. The coordinates of an arbitrary point of $L$ are $x=a+\lambda t, \quad y = b+\mu tf$.

Hence the path will be periodic if and only if $\lambda t = 2n, \quad \mu t = 2m$, for some $t$ and integral $n,m$, i.e. if  $\frac{\lambda}{\mu}$ is rational.

When $\frac{\lambda}{\mu}$ is irrational, then the path of $E$ approaches arbitrarily near to every point $(c,d)$ of the sqaure. This follows directly from Kronecker’s Theorem in one dimension (see § 23.3 of G H. Hardy and E. M. Wright’s An Introduction to the Theory of Numbers.):

[Kronecker’s Theorem in one dimension] If $\theta$ is irrational, $\alpha$ is arbitrary, and $N$ and $\epsilon$ are positive, then there are integers $p$ and $q$ such that $p>N$ and $|p\theta - q-\alpha|<\epsilon$.

Here, we have $\theta = \frac{\lambda}{\mu}$ and $\alpha = (b-d)\frac{\lambda}{2\mu} - \frac{1}{2}(a-c)$, with large enough integers $p=m$ and $q=n$. Hence we can conclude that

[König-Szücs Theorem]Given a square whose sides are reflecting mirrors. A ray of light leaves a point inside the square and is reflected repeatedly in the mirrors. Either the path is closed and periodic or it is dense in the square, passing arbitrarily near to every point of the square. A necessary and sufficient condition for the periodicity is that the angle between a side of the square and the initial direction of the ray should have a rational tangent.

Another way of stating the above Kronecker’s theorem is

[Kronecker’s Theorem in one dimension] If $\theta$ is irrational, then the set of points $n\theta - \lfloor n\theta\rfloor$ is dense in the interval $(0,1)$.

Then with some knowledge of Fourier series, we can try to answer a more general question

Given an irrational number $\theta$, what can be said about the distribution of the fractional parts of the sequence of numbers $n\theta$, for $n=1,2,3,\ldots$?

The answer to this question is called Weyl’s Equidistribution Theorem (see §4.2 of  Elias M. Stein & Rami Shakarchi’s Fourier Analysis: An Introduction)

[Weyl’s Equidistribution Theorem] If $\theta$ is irrational, then the sequence of fractional parts $\{n\theta - \lfloor n\theta\rfloor\}_{n=1}^{\infty}$ is equidistributed in $[0,1)$.

I really enjoyed reading about this unexpected link between geometry and arithmetic (and Fourier analysis). Most of the material has been taken/copied from Hardy’s book. The solution to the geometry problem reminds me of the solution to the Cross Diagonal Cover  Problem.