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On Isoclinal Sequences of Spheres proceedings of the american mathematical society Volume 88. Number 4, August 1983 ON ISOCLINAL SEQUENCES OF SPHERES ASIA IVlt WEISS Abstract. In «-dimensional inversive geometry it is possible to construct an infinite sequence of ( n — 1)-spheres with the property that every n + 2 consecutive numbers are isoclinal. For every such sequence there is a Möbius transformation which advances the spheres of the sequence by one. 1. Introduction. In the beautiful paper [1], Coxeter showed that in «-dimensional inversive space (n > 2) there is an infinite sequence of (n — l)-spheres such that every n + 2 consecutive members are mutually tangent. Given such sequence there is a Möbius transformation mapping the sequence onto itself (see [4]). Mauldon in [3] considered sets of spheres with the same mutual inclination, and Gerber in [2] extended these sets to infinite sequences. Here we give a recursion formula for computing coordinates of the spheres in such sequences. Using this formula we show that such a sequence can, as well as a tangent one, be mapped onto itself by a Möbius transformation. 2. Notation and definitions. In considering «-dimensional inversive geometry we can use vectors with « + 2 coordinates to denote points and spheres. Let 2 be a unit sphere in R"+1 centred at the origin, and II an «-flat through the origin. Stereographic projection establishes one-to-one correspondence between II and 2\{north pole}. To every point X in n we can assign an (« + 2)-tuple (x, 1) where x is the coordinate of the stereographic projection of X. Similarly, to every sphere Y in n we can assign an (« + 2)-tuple (csc 0 • y, cot 0), where y is the coordinate of the centre and 0 the angular radius of the stereographic projection of Y. The coordinates are positive homogeneous. In connection with these coordinates we introduce the Lorentz bilinear form A * B = (ax,...,a„+2)*(bi.b„+2) = axbx + ••• +a„+xb„+x - a„+2b„+2. A vector A represents a point if and only if A * A — 0 and a„ + 2 > 0, and it names a sphere when A * A = 1. A point A lies on a sphere B if A * B = 0. If A and B are spheres then A * Bis called the inclination between A and B. Furthermore: A * B — ± 1 if ^4 and B axe internally and externally tan- gent respectively. A * B = cos 0 if ^4 and B intersect at angle 8. Received by the editors June 11, 1982. 1980 Mathematics Subject Classification. Primary 51B10. ©1983 American Mathematical Society 0002-9939/82/0000-1305/$02.50 665 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 666 A. I. WEISS A * B = ±cosh S if A and B axe disjoint with nested and dis- joint interiors respectively, and their inversive distance is 8. If D is a sphere such that A — 2(D * A)D = B then D is a midsphere, i.e. inversion in D interchanges A and B. For proofs see [5]. 3. Sets of isoclinal spheres. A set of spheres is said to be isoclinal if any two spheres from the set have the same mutual inclination. If the inclination is y we say that the set is y-clinal. In particular, a set of « + 2 tangent spheres is called a cluster. Spheres of any cluster form a basis for (« + 2)-space (see [5]). Moreover, Theorem I. Any ordered set of n + 2 y-clinal (« — \)-spheres, y^ 1, forms a basis for (n + 2)-space. Proof. Let iß = {2?0, Bx,...,Bn+x] be a y-clinal set. Define a matrix B = (bi/) = (B, * Bj). Then B = (1 - y)I + yJ, where / is the (« + 2) X (« + 2) identity matrix and J is the (n + 2) X (n + 2) matrix with all entries 1. It is easy to verify that (1 — y)"'/ — (1 — y)~l(n + 1 + y"1) '7 = B'\ and hence B is nonsingular, so that the vectors in 9> are linearly independent. Clearly for any y < 1 there is a set of « + 1 y-clinal (« — l)-spheres (spheres with centres at the vertices of a simplex). If there is a set of n + 2 y-clinal spheres it can be inverted into (n + l)-spheres centred at the vertices of a simplex and a sphere centred at the centre of the simplex. To see that, let {B{), Bx,...,Bn+x) be y-clinal. Then D0, =[1/2(1 — y)]]/2(B() — Bx) is a midsphere for B() and Bx and it is perpendicular to B2,...,Bn + 2. Similarly we define DX2,... ,D„(). It is easy to see that there is a point P = a(B0 + • • • +B„) + ßB„+] (for some a and ß) such that P is on all Z),,+ ,. Inversion in this point transforms B0, Bx,... ,B„ into the spheres centred at the vertices of a simplex, and since B„ + 2 is perpendicular to all £>,, t, its centre must be the centre of the simplex. The described set of « + 2 y-clinal (« — l)-spheres in R" may be regarded as the "equatorial" section of the set [BQ, Bx,.. -,B„+l} of n + 2 y-clinal «-spheres in R"+l. It is easy to see that there are exactly two nonintersecting spheres tangent to all Br ; = 0,...,«+ 1. Inverting in one of the points common to all the spheres perpendic- ular to the two disjoint spheres, we obtain two concentric spheres. B0.Bn+X are inverted into congruent spheres sandwiched between them. Hence centres of B0,...,B„+X are at the vertices of an (« + l)-simplex. We conclude now that given two ordered sets of « + 2 y-clinal spheres there is a Möbius transformation mapping one set onto the other, i.e. any two such sets are inversively equivalent. Here we give another proof of the result due to Mauldon [3] providing necessary and sufficient conditions for the existence of « + 2 y-clinal spheres. Theorem 2. In n-dimensional inversive space there exists a set of n + 2 y-clinal (n — l)-spheres if and only if y < -(« + l)"1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ISOCLINAL SEQUENCES OF SPHERES 667 Proof. If there is a set of « + 2 y-clinal (« — l)-spheres {B0, Bx.B„+\) we can think of it as a set of congruent «-caps on an «-sphere whose centres are vertices c„, c,,... ,c„+1, respectively, of an (« + l)-simplex inscribed in 2. Let C0, Cx.C„+x be congruent «-caps forming a cluster with their centres at c0, c,,...,cn+,, respectively. Then (i+i Bj = aC¡ + b 2 q, i E {0.1.« + 1}. /=o Since y ifitj< B, * B. 1 if/=/. we have a2 + 2ab(n - 1) + b2(n2 + 1) = y. a2 - 2ab(n + 1) - b2(n2 - 1) = 1. It is easy to see that a and b are real if and only if y =£ -(« + 1)"'. Theorem 3. Given a set [Bi}. /?,.B„+x) of y-clinal (« — \)-spheres where y ¥= —«"', -(« + Ir', /«ere « exactly one sphere B„ + 2 ¥= BQ such that {Bx, B-,,_B„ + -,} is y-clinal and (1) B*+i = -B0 + ——-(Bx + ---+Bn+X). « + y When y =-«"',-(«+ 1)"' f/iere is no sphere with this property. Proof. Any sphere B„+ 2 such that B„+2 * 5,= y for all / E {1.« + 1} must be of the form aB() + b( Bx + ■■ ■ + B„+, ) where a + b(n + y"') = 1, a2 + 2aèy(« + 1) + ¿>2y(« + 1)(« + y"') = 1. These equations have exactly two solutions, a = 1, b = 0 yielding B0, and a = -1, b = 2(n + y-')"1 if y ^ -«"', -(« + l)"1. Theorem 4. G/ue« a ^ei {/30, 5,,.. .,B„+X] of y-clinal (« — l)-spheres there is exactly one sphere D0 perpendicular to BX,...,B„+X if and only if y < -«"'. There is none otherwise. Proof. It is easy to see that D0 = -a(n + y'l)B0 + a(Bx + • • • +Bn + X) with a2 = (1 - y)"'(« + y-')-'(« + 1 + y"1)"1. Here a2 > 0 if and only if y < -«"'. Corollary 1. If D0 is a sphere from Theorem 4 and B„+ 2 is defined by (I), then the inversion in D0 interchanges B0 and B„ + 2. Proof. The inversion in DQmaps B0 to B0 — 2(D0 * B0)D0. It is easy to verify that this is B„+2. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 668 A. I. WEISS Corollary 2. Given a set {B0, Bx,...,Bn+x} of y-clinal (« — l)-spheres with y < -«"', there is a set of (-(« + y~x)~x)-clinal spheres {Da, Dx.A,+ i} sucn mat each D¡ is perpendicular to Bj for all j E {0,1,...,« + 1}\{/}. Proof. Since D¡ = -a(n + y-')£,. + al^j^Bj, for / ¥=k, D,* Dk = D,* aB, = aB,* D, = a[-a(n + y"1) + ay(« + 1)] = - (« + y"1)"'. This is an extension of Beecroft's result on tangent circles in the plane. Another proof due to Gerber can be found in [2], Corollary 3. B0 * Bn+2 = D0 * D„+2 = 2y(« + 1)(« + y"')"1 - 1. 4. Sequences of isoclinal spheres. In the last section we saw that any ordered set of « + 1 y-clinal spheres [Bx,... ,B„+X] with y ¥= -«"', -(« + 1)"' can be completed to a set of « + 2 y-clinal spheres in exactly two ways. Hence the spheres belong to just one sequence (2) ...,B_l,B0,Bx,B2,...,Bn+x,B„+2,... with the property that every « + 2 consecutive members are y-clinal. Hence we see that the relation (1) remains valid when all the indices are increased or decreased for any integer, so that it provides a recursion formula for computing coordinates of any sphere in the sequence (2), providing we know coordinates of B0,..
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