Repeated Angles inE 4

Discrete & Computational Geometry, Mar 1988

Let there be givenn points in four-dimensional euclidean spaceE4. We show that the number of occurrences of the angleα iso(n3) ifα is not a right angle and Θ(n3) otherwise.

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Repeated Angles inE 4

Discrete Comput Geom Repeated Angles in E4 0 Yj = , 1, 0 , j , 0 1 Mathematics Department, Texas A & M University, College Station , TX 77843 , USA Let there be given n points in four-dimensional euclidean space E4. We show that the number of occurrences of the angle a is o(n 3) if a is not a right angle and ~(n 3) otherwise. F o r a c o n f i g u r a t i o n ~ o f n p o i n t s in d - d i m e n s i o n a l e u c l i d e a n s p a c e Ed, let fd(n, a, (8) d e n o t e the n u m b e r o f angles A B C that are equal to a, where 0 < a < nr a d i a n s a n d A, B, a n d C are points o f ~. Let fd(n, ~) be the s u p r e m u m o f fd(n, a, ~) t a k e n over all configurations %~o f n points. The function fd(n, a) is integer v a l u e d a n d O<--fd(n, a) <- N = [ n ( n - 1 ) ( n - 2 ) ] / 2 so that the s u p r e m u m is in fact a m a x i m u m . C o n w a y et al. [1] e s t a b l i s h e d the g r o w t h estimate f~(n, a ) = o(n 3) as n-~oo which they n e e d e d to establish the p r o p e r t i e s o f certain a n g l e - c o u n t i n g functions. P. E r d 6 s a s k e d me for which k and a fk(n, a ) = o(n3). The m o d i f i e d " L e n z " c o n s t r u c t i o n : where ~ 2 + ° U 2 = 1 , l<-i<_m, and A, /x, v > O shows that f6(n, ct)>-[n/3] 3 for 0 < a < ¢ r / 2 . In this note we show the following: Zk = (0; 0; 0; 0; v~llk; v°llk) - Theorem. l f a ~ 7r/2, then f4(n, c~) = o(n3), but f4(n, ¢r/2)>-[n/3] 3. Proofifc~ = ¢r/2. Let m = [ n / 3 ] a n d let o//,, ~V, be m solutions o f ~ 2 + ~V2 = 1. Let X, = (°//,, °U,,0, 0), and G. Purdy Zk = ( - 1 , 0 , 0 , k), l < - k < - m . T h e n ( Y j - X i ) . ( Z k - X ~ ) = angles. °R2i - 1+ °F~2 = 0 and the m 3 angles Yj.~,Zk are all right [] Proof for a # 7r/2. We shall assume a # rr/2 from now on, and we shall prove the stronger result f4(n, a ) = o ( n 3 - ~ ) , where e =~5. Suppose not. Then by the following combinatorial lemma of Erd6s [ 2 ] there are 15 points X,, Y~, Zk such that the 125 angles Y~f(,Zk all equal a, 1 ~ i, j, k -< 5. [] Combinatorial Lemma. Let H ~ A x A x A, where IAI = n and [H I>- n3-q Then there are subsets A, c A, 1 <- i <-3, such that lAd >- k and A~ x A2 x A3 c_ H, provided k 2<- 1/e. We use this lemma with k = 5 and e = ~ . We need two additional lemmas. Lemma 1. The points X~, X2, . . . , X5 are not collinear. Lemma 2. N o three Y~ are collinear and no three Zk are collinear. Proof o f L e m m a 1. The Xi are solutions to the vector equation { ( X - Y,). (X-Z,)}2=cos 2a { ( X - Y,). ( X - Y , ) } { ( X - Z , ) . ( X - Z , ) } . Suppose that the X, lie on the line X = C + t~//. Substituting into the above equation, we obtain an equation in the scalar t of the fourth degree, since cos2a # 1. Such an equation cannot have five solutions. [] Proof o f L e m m a 2. We shall show that no three Zk are collinear, and the p r o o f for the Yj is similar. Suppose, without loss o f generality, that Z~, Z2, and Z3 lie on the line I. For a fixed X~, the points X2, Y~ and the line l fit into an E3, in which the locus of points Z such that Y~X~Z = a is a cone with apex X,. A line such as l that intersects the cone in three points must pass through the apex X~. Hence the five points Xi all lie on l, contrary to the previous lemma. [] Proof o f the Theorem. Let X~ = 0 be the origin of coordiAnates and let Yjd e n o t e a unit vector in the direction of Yj and similarly for Zk. The p o i n t s Yj span an affine hull B which does not necessarily pass through X~ = 0. The Y~ are not necessarily distinct, but no three can be the same, by L e m m a 2, so there are at least three different ones. This forces B to have dimension two or more, since the ~ lie on a unit sphere, and no three o f them can be collinear. Let 1-<j,k, r ~ 5 . Then Yj-Zk = cos a = l~'r"Zk, SO that ( ~ - I?r)" Z'k = 0 , and therefore ( Y j - Y,) • Zk = 0. Thus B is orthogonal to the subspace H spanned by X, = 0 and the five points Zk. If the dimension of H is three or more, then we have the absurdity of an orthogonal E2 and E3 in E4. We also know by L e m m a 2 that H is not a line. Hence H is a plane which together with Y~ fits into three space. The cone, with apex X~ = 0, which is the locus o f points Z such that Y ~ f ~ Z = a, cuts the plane H in a pair of (possibly coincident) lines, one of which must consequently contain three of the points Zk, contrary to L e m m a 2. 73 1. J. H. Conway , H. T. Croft , P. Erd6s and M. J. T. Guy , On the distribution of values of angles determined by coplanar points , J. London Math. Soc. (2) 19 ( 1979 ), 137 - 143 . 2. P. Erd6s, On extremal problems of graphs and generalized graphs , Israel J. Math. 2 (t964) , 183 - 190 .


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George Purdy. Repeated Angles inE 4, Discrete & Computational Geometry, 1988, 73-75, DOI: 10.1007/BF02187897