Equidissections of Polygons
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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 85 (1990) 281-294 281 North-Holland EQUIDISSECTIONS OF POLYGONS E.A. KASIMATIS Department of Mathematics, California State University, Northridge, CA 91325, USA S.K. STEIN Department of Mathematics, University of California, Davis, CA 95616, USA Received 26 September 1988 A dissection of a planar polygon K is a family of triangles whose union is K and such that the intersection of the interiors of any two distinct triangles is empty. If all the triangles have equal areas, the dissection is called an equidissection of K. An equidissection of K into m triangles is called an m-equidissection of K. It is known that in any m-equidissection of a square m is even. It has also been shown that in any m-equidissection of the regular n-gon, n P 5, m is a multiple of n and that in any m-equidissection of a centrally symmetric polygon m is even. In the present paper we explore equidissections of more general polygons. For instance, we show that for almost all polygons no equidissection exists. Furthermore, there are polygons for which the integers m such that the polygon has an m-equidissection are not the multiples of a single integer. 0. Preliminaries Let K be a polygon. Its spectrum, S(K), consists of those integers m for which K has an m-equidissection. S(K) may or may not be empty. If S(K) is not empty, K is called equidissectuble. Observe that if m E S(K) and k is a positive integer, then km E S(K). A set B c S(K) is a generating set for S(K) if S(K) = {kb: k 3 1, b E B}; we then write S(K) = (B). If S(K) h as a generating set consisting of just one element, b, we have S(K) = (b) and call both K and S(K) principal. For example, a triangle is principal with generating set {l}, a square is principal with generating set (2) ([9,10,14]), and for n 2 5, a regular n-gon is principal, with generating set {n} ([6]). The main tool for examining the spectrum of a polygon situated in the plane R2 is obtained by considering certain partitions of lR2 that are determined in terms of the ‘ord’ functions, which are defined for each positive prime integer p. They are also called ‘exponential valuations’ and are closely linked to non-Archimedean valuations. (See [2, p. 671 or [l].) Let p be such a prime. An ‘ordP’ function is any function, x*x’, with domain R and range R U {a~}, that has these properties: (a) p’ = 1, (b) (xy)’ =x’ + y’ for any real numbers x and y, (c) (X + y)’ 2 min{x’, y’}, with equality holding when x’ # y’, and (d) n’ = m if and only if x = 0. We write x’ = ord&), or, if the prime p is understood from the context, x’ = ordx. A function defined on a subfield of R’ and satisfying these four conditions will also be called an ord, function. 0012-365X/90/$03.50 @ 1990- Elsevier Science Publishers B.V. (North-Holland) 282 E. A. Kasimatis, S. K. Stein For a given prime p there is an infinite number of ord, functions, with domain R , all of which agree on the set of rational numbers: if n is an integer and a, b # 0 are integers relatively prime to p, then ord,(p”a/b) = IZ. An ord, function defined on Q extends in either one or two ways to an ord, function defined on a quadratic extension Q(a), when d > 1 is a square-free positive integer. The function ord, has two extensions to CD(a) if d = 1 (mod 8); otherwise it has only one extension. If p is an odd prime that does not divide d and x2= d (modp) is solvable in Z, then there are two extensions of ord, to O(a); otherwise there is only one. (See [5, pp. 189-1901.) If F is a subfield of R and a E R is transcendental over P, an ord, function defined on F can be extended to F(u) with the value of ord,(a) prescribed to be, say, c E R’ by defining (C:&x’)’ = min, {f: + TC}. Thus the values of ord,, can be prescribed at a set of algebraically independent numbers. An ord, function defined on IR can be used to partition U&”into three sets, PO, PI, and P2, as follows: PO= {(x1, x2): xi, xJ>O}, PI = {(x1, x2): xl ~0, x;cx;}, P2 = {(XI, x2): x; s 0, xi >x;}. For example, with p = 5, (0, 5/3) E PO, (l/25, 0) E PI and (0, 1) E P2. Note that the origin is in POfor all p. A point in I&‘*that lies in Z$ will be labelled P. A triangle in IL!’ whose vertices have the labels PO, PI, and P2 is complete. A line segment in R* whose ends have the labels POand PI will also be called complete. The main algebraic tool for examining the spectrum of a polygon is given in the following theorem ([6,8,9]). Theorem 0.1. Let K be a polygon in R* of urea A(K) with an m-equidissection. Let p be a prime. Orient the boundary of K. If the number of complete edges on the boundary of K whose orientation by the induced labeling agrees with a given orientation does not equal the number of such edges of opposite orientation, then ord,m 2 ord,,(2A(K)). Since this theorem provides the foundation of our work and shows how the ordp functions are related to equidissection, we sketch its proof. For simplicity we assume that the equidissection is simplicial, that is, the intersection of two distinct triangles is either empty or a vertex or an edge of both of them. By Sperner’s lemma ([ll, p, 1511, [12, Chapter 21) the equidissection contains a triangle T with vertices qo, ql, and q2 labelled PO, PI and P2 respectively. Translation of T by -q. produces a triangle with vertices (0, 0), (x1, yr) E PI, and (x2, y2) E P2. The area of T is equal to $1~~y2 - x2 y,l and is also equal to A(K)/m. Since ord&) s ord,(.vr) and ord,W < ord,(x2), ord,(xry2 - x2y1) = ord,(xl y2), which equals ord,(xJ + ord,(y,), hence is less than or equal to 0. Applying the ord, function to both sides of the equation A(K)/m = 41x2y, - x1 y21 establishes the theorem. Equidissections of polygons 283 1. Two extreme cases We consider two contrasting types of polygons, equidissectable and non- equidissectable. A polygon K all of whose vertices have only rational coordinates is equidissectable: make a simplicial decomposition of K using these and, perhaps, additional vertices that have only rational coordinates. If the areas of the resulting triangles are a,/t, a,/& . , a,/& where the ai’s and t are positive integers, then K has an equidissection into a, + a2 + . - + a, triangles. The following theorem concerns the opposite extreme. Its proof makes use of the notion of ‘algebraically independent’ real numbers. The real numbers al, a2, . , a, are algebraically independent if there is no polynomial f(x1, x2, . 9 x,) with integer coefficients such that f(a,, a2, . , a,) = 0, other than the polynomial all of whose coefficients are 0. In particular, each ai, 1 c i c n, is transcendental. Theorem 1.1. Let Kc R2 be a polygon with at least four vertices. Zf the coordinates of the vertices are algebraically independent, then S(K) = 0. Proof. Let the vertices be (xi, yi), 1 6 i sn, numbered consecutively around K and let A be the area of K. Then A is a polynomial in these 2n coordinates with rational coefficients. Assume that m E S(K) and select a prime p > m. Extend ord, from the rationals to Q(x,, y,, . , x,, y,,) in such a way that xi = 0, yi = 1, xi = 1, y; = 1, xi = 1, y; =O. Thus (x1, yi), (x2, y2), (x3, y3) have the labels PI, PO, P2 respectively. Set xl = 0, 4 < i C n. Since A involves y4 it is algebraically independent of x1, x2, . , x,, yi, y,, y3. Hence we may set A’ = 1. Extend this assignment to an ord, function defined on II2 (or at least on the field generated by all the coordinates of all the vertices of the equidissection). Since K has only one complete edge, and so is complete, Theorem 0.1 shows that m’s(2A)‘=2’+A’ 3 1. Therefore, p divides m. Since p > m, this is a contradiction. 0 The theorem implies that the probability that a randomly chosen polygon in the plane is equidissectable is 0. A similar argument works for n-dimensional polytopes. 2. Construction of equidissections Theorem 0.1 provides only a necessary condition that a given integer m is in a spectrum. We now describe a method for constructing equidissections of polygons and will use it in later sections. 284 EA. Kasimatb, S.K. Stein Let K be bounded by a simple polygonal curve with n edges (adjacent edges not parallel) in the plane, furnished with a combinatorial scheme of a simplicial decomposition. The coordinates of the vertices are viewed as variables whose values are to be determined. The vertices are of three types: n vertices on the bounding curve of K, i vertices in the interior of K, and e vertices that lie on the interiors of edges on the boundary of K.