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use the symbol bc to denote a point on a bdiagonal D which is the cth among the intersectionpoints of D with the other bdiagonals, counting from the midpoint of D. This is illustrated in Figure 2. A point which is simultaneously bc and de has symbol bcde. Finally, an astral (n4) configuration which has as points the m = n/2 vertices bbdd of a regular mgon and m points bcde, will be designated by the symbol m#bcde.
With this notation, we have the following result:
Theorem 1. There exist two infinite families of astral (n4) configurations. One consists of the configurations (6k)#(3kj)3k2j(2k)j where k e" 2, 1 d" j d" 2k1 but j `" k and, if k is even, j `" 3k/2 as well. The other family consists of configurations (6k)#(3k2j)j(3kj)2k where k e" 2 and 1 d" j d" k1.
Figure 2. In the notation explained in the text, the points A, B, C, E have the symbols 44, 43, 42, 41, respectively.
It may be noted that if k is even, the configurations of the two families coincide for j = k/2.
To prove the theorem, we first note that the midpoint M of a selected bdiagonal D, together with the intersection points of the other bdiagonals with D, determine angles which are multiples of b = /m. Therefore, in the notation of Figure 2 and with distance OA equal 1, the
distance from O to bc is EQ \F(cos (b/m),cos (c/m)) , where 1 d" c < b d" [ EQ \F(m1,2) ] . Putting EQ \F(,m) = EQ \F(,6k) = b, and noting that the existence of a point bcde implies EQ \F(cos (b/m),cos (c/m)) = EQ \F(cos (d/m),cos (e/m)) , the existence of a configuration of type (6k)#(3kj)3k2j (2k)j is easily seen to be equivalent to
cos jb cos (3k  j)b = cos 2kb cos (3k  2j)b.
Since cos 2kb = cos EQ \F(,3) = 1/2 and since
2 cos a cos b = cos (a + b) + cos (a  b),
this is equivalent to
cos 3kb + cos (3k  2j)b = cos (3k  2j)b .
This relation holds since cos 3kb = cos /2 = 0. A similar argument establishes the existence of the configurations in the second family.
It is easily seen that for every positive integer s, given any astral configuration m#bcde there is an astral configuration m*#b*c*d*e*, where m* = sm, b* = sb, c* = sc, d* = sd, and e* = se. We shall call each such configuraion a multiple of m#bcde. The multiple is obtained by symmetrically placing s copies of m#bcde. For the configurations of the families listed in Theorem 1 all multiples are configuration of the same family.
The configurations listed in Theorem 1 are by no means the only astral (n4) configurations. There exist also a number of "sporadic" configurations, specified in Theorem 2.
Theorem 2. The following sporadic configurations exist, together with their multiples:
30#41 76 , 30#61 74 , 30#62 86 ,
30#61 1110 , 30#72 1211 , 30#81 132 ,
30#101 116 , 30#106 1210 , 30#107 1312 ,
30#112 127 , 30#116 1413 , 30#121 138 ,
30#124 1412 , 30#127 1310 , 30#136 1411
42#61 1312 , 42#116 1817 , 42#121 136 ,
42#125 1918 , 42#176 1811 , 42 # 172 1914 ,
60#92 2221 , 60#125 2524 , 60#143 2726 ,
60#212 229 , 60#245 2512 , 60#263 2714 .
The proof of Theorem 2 proceeds along lines similar to the proof of Theorem 1, but utilizing specific information about the values involved. For m=30 we have b= EQ \F(,30) =6. The existence of 30#62 86 is equivalent to:
cos2 36 = cos 48cos12,
which is the same as
1 + cos 72 = cos 60 + cos 36,
or
1 + 2 cos 72 + 2 cos 144 = 0.
This relation is valid, as can be seen without computations, since it expresses the fact that the origin coincides with the centroid of a regular pentagon inscribed in the unit circle, with one vertex at (1,0). Completely analogous reasoning establishes the existence of configurations with symbols 30#61 1110 and 30#124 1412 .
For the other sporadic cases with m = 30, explicit values of the cosines can be used. With
cos 36 = EQ \F(1 + \R(5),4)
and
cos 12 = EQ \F(1 + \R(5) + \R(30 + 6\R(5)),8)
it is easy to verify the existence of the remaining irreducible sporadic astral (604) configurations.
Similar calculations establish the existence of the sporadic (1204) configurations.
The existence of the sporadic (844) configurations can be established by reducing all the coincidence conditions to the relation
1 + 2cos 2g + 2cos 4g + 2cos 6g = 0,
where g = EQ \F(,7) . The validity of this equation can again be seen without calculations by noting that it expresses the coincidence of the center of a circle with the centroid of a regular heptagon inscribed in the circle. Alternatively, for direct proofs one can use the fact that cos EQ \F(,7) is the largest positive root of the equation 8y3  4y2  4y + 1 = 0, which is explicitly given by cos EQ \F(,7) = EQ \F(1,6) + z + EQ \F(7,36z) , where
z = EQ \R(3,\F(7,432) + \F(7i,48\R(3))) ;
numerically, cos EQ \F(,7) = 0.90096886790... .
This completes the proof of Theorem 2.
The following statement should be considered as having a status between wellsupported conjecture and fully proved theorem. The reasons will be explained below.
The only astral configurations (n4) are the ones given by Theorems 1 and 2.
The history of my involvement with the astral (n4) configurations started more than fifteen years ago, when I empirically found several, with small n. However, beyond the smallest ones I was not be sure of their existence, either by drawing them by hand or using MacDraw software. In particular, I found no pattern for the parameters of the ones I believed existed. Somewhat later I was initiated (by Stan Wagon, to whom I am greatly indebted for this) to Mathematica software. High precision calculations easily established that, for n d" 60, astral (n4) configurations exist only if n = 2m = 12k, for some integer k with k e" 2. However, the pattern of possible configurations remained puzzling. This is possibly best understood by considering Figure 3. In it a configuration m#bcde, where b > d, is represented by solid a solid square, centered at (b, d). A square with diagonals drawn indicates the existence of two distinct configurations with the same values of b and d.
The solution of this puzzle was suggested by additional numerical calculations, which led to results such as the ones shown in Figure 4. The pattern revealed is that one should distinguish between the two kinds of configurations, the ones that exist for every m = 6k with k e" 2, and those that exist only for certain multiples of 6. This is reflected in the above Theorems 1 and 2. Once the pattern was recognized, it was easy to establish that it holds for values beyond the ones for which the evidence was obtained through numerical calculations.
In order to turn the italicized statement into a theorem we need to show that there are no other astral configurations m#bcde. In principle, this should be possible with the information available in the literature. Bol
Figure 3. The values of b and d < b for which there exist configurations m#bcde.
[1] was the first to determine all multiple intersection points of the diagonals of regular polygons. Other determinations of these points were given by Rigby [4] and by Poonen and Rubinstein [3]; the latter gives references to the many related papers. The problem is surprisingly complicated, and the results cannot be expressed in simple terms. The astral (n4) configurations obviously arise from the intersection points of four diagonals, two of each of two spans. But the practical difficulty is to extract the descriptions of such intesection points from the general results given in the papers listed above. I have no doubt in the validity of the italicized statement, but have so far not derived a formal proof.
References
1. G. Bol, Beantwoording van prijsvraag no. 17. Nieuw Archief voor Wiskunde 18(1936), 14  66.
2. B. Grnbaum, Astral (nk) configurations. Geombinatorics 3(1993), 32  37.
3. B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon. SIAM J. Discrete Math. 11(1998), 135  156.
4. J. F. Rigby, Multiple intersections of diagonals of regular polygons. Geometriae Dedicata 9(1980), 207  238.
Figure 4. An illustration of the existing configurations m#bcde for additional values of m. The configurations correspond to the ones specified by Theorem 1.
Page PAGE 9
Geombinatorics 9(2000), 127  134
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ASTRAL (n4) configurations.
by Branko Grnbaum
University of Washington, GN50
Seattle, WA 98195
email: grunbaum@math.washington.edu
A family of n points and n (straight) lines in the Euclidean plane is said to be an (n4) configuration provided each point is on four of the lines and each line contains four of the points. A configuration may have various symmetries, that is, there may exist isometric mappings of the plane onto itself that map the configuration onto itself; all the symmetries of a configuration form its group of symmetries. It is obvious that no more than two points of a line can be in the same transitivity class with respect to the group of symmetries, and no more than two lines passing through one point can be in the same transitivity class unless all lines pass through that point. Hence, under its symmetry group each (n4) configuration must have at least two transitivity classes of points, and at least two equivalence classes of lines. An (n4) configuration is called astral if its points, as well as its lines, form precisely two transitivity classes. While it is not completely trivial that any astral (n4) configurations exist, there is, in fact, a large number of possibilities which will be precisely described below. A few examples of astral (n4) configurations were given in an earlier paper [2], and additional ones are shown in Figure 1. The aim of the present paper is to give this description, and to explain how it was found and established.
In order to proceed, we need some notation. It is easily verified that the points of any astral (n4) configuration must lie at the vertices of two concentric regular mgons, where m = n/2, and that the lines of the configuration must be determined by common diagonals of these mgons. The size b of a diagonal D of a regular mgon is the number of sides of P bridged by D; hence the angle subtended by D at the center of P is 2b/n = b/m, and we need consider only the range 2 d" b d" m1. We
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