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 fourdimensional 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> n3q 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,). (XZ,)}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 YjZk = 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 .