Charles H. Jepsen

Equidissections of Trapezoids

Suppose a plane polygon is dissected into triangles of equal areas. What numbers of triangles are possible? Given a polygon K, a dissection of K into m triangles of equal areas is called an m-equidissection. The spectrum of a polygon K, denoted S(K), is the set of integers m such that K has an m-equidissection. If S(K) consists of all multiples of a single integer m, we say that S(K) is principal and write S(K)= . Monsky [2] showed that if K is a square, then S(K)= . Kasimatis [1] showed that if K is a regular n-gon with n 2 5, then S(K) = . A discussion of this problem for other polygons is contained in Chapter 5 of [4]. What can we say about the spectrum of a trapezoid? We may confine our attention to trapezoids with vertices (0, 0), (1, 0), (0, 1), (a, 1), a > O, since any trapezoid is affinely equivalent to such a trapezoid. Denote this trapezoid by T(a). The following are shown in [4] (pp. 121-122):

Dissections into 1:2 rectangles

Abstract Suppose a square or a 1:2 rectangle is dissected into 1:2 rectangular elements of different sizes. What orders (i.e., numbers of elements) are possible? We show: There are dissections of squares of all orders ⩾ 9. There are dissections of 1: 2 rectangles of all orders ⩾ 8. There are no dissections of smaller order.

Constructing equidissections for certain classes of trapezoids

We investigate equidissections of a trapezoid T(a), where the ratio of the lengths of two parallel sides is a. (An equidissection is a dissection into triangles of equal areas.) An integer n is in the spectrumS(T(a)) if T(a) admits an equidissection into n triangles. Suppose a is algebraic of degree 2 or 3, with each conjugate over Q having positive real part. We show that if n is large enough, n is in S(T(a)) iff n/(1+a) is an algebraic integer. If, in addition, a is the larger root of a monic quadratic polynomial with integer coefficients, we give a complete description of S(T(a)).

Dissections of p : q rectangles

We determine all simple perfect dissections of p:q rectangles into at most twelve p:q rectangular elements. A computer search shows there are only eight such dissections, two of order 10, three of order 11, and three of order 12.

Billiard Balls and a Number Theory Result.

At the beginning of Mathematics, A Human Endeavor, Jacobs considers the following situation (see [1]). A ball is hit from one corner of a rectangular billiard table so that it travels at a 450 angle with the sides of the table and continues to rebound off the cushions until it hits a corner and stops. To fix notation used throughout this paper, let us assume that the table is a units wide and b units long, a and b positive integers, and that the ball starts in the lower left corner. See Figure 1 for an example. Jacobs asks the reader to experiment with tables of different shapes and answer the following three questions.

Making Squares from Pythagorean Triangles

Here a complete answer is not known, but Laczkovich [2] has shown that a square cannot be dissected into 30?-60?-90? triangles. In fact, he proved that a square cannot be dissected into a finite number of triangles all of whose angles (measured in degrees) are even. (These two types of dissection problems form the content of chapters 5 and 6 of Stein and Szabos recent book [510 Now, suppose a square is dissected into Pythagorean triangles?right triangles with integer sides:

On colorings of finite projective planes

Abstract Using counting arguments we extend previous results concerning the coloring of lines in a finite projective plane of order n whose points are n -colored.

Dissections of p:q rectangles into 13 p:q rectangular elements

We determine all simple perfect dissections of p:q rectangles into 13 p:q rectangular elements. A computer search shows there are 26 such dissections. Previous work yielded only eight such dissections into at most twelve rectangular elements.