On Coxeter’s Loxodromic Sequences of Tangent Spheres

Numerical distances among the spheres in a loxodromic sequence

“…For the numerical distance Dn between two spheres at sequential distance n in a loxodromic sequence, the expression (8) w i t h t h e u n f a m i l i a r i n i t i a l v a l u e s u0 = -1, Ul = 2, y i e l d i n g u 2 = 3 , u 3 = 8 , u 4 = 1 9 , . .…”

On the aesthetics of inversion and osculation

“…lib). Coxeter (1968) has shown that points of contact of consecutive pairs of spheres lie on a loxodrome (Weiss 1981). If/~ is a constant angle, and 0 and ~b the longitude and latitude of a point on the loxodrome, its equation m a y be written x = sin q5 cos 0, y = sin q~ sin 0, z = cos ~b, where 0 = -tan fl log tan(~b/2).…”

On isoclinal sequences of spheres

“…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.…”

“…A * B = 0), and M is a dilative rotatory reflection; finally for « odd and -(« + 2)(«2 + 3« + l)"1 < y < -(« + 1)"' we have one fixed sphere and M is a rotary reflection. All the angles of rotation are given as solutions of the equation (4). Coefficients of dilation for the dilative rotation and the dilative rotatory reflection are solutions of (5), (6) or (7) depending on y and « as indicated above.…”

“…When y = -1, so that «+ 1 consecutive spheres are tangent, choose any point of contact A between two spheres; the set of points {AM'}iez lies on the curve (8) y(t) = e2S'{axe2^,...,a"/2e2^') (see [4]). Now if A0 is any point on B0 and A¡ = A0M', i E Z, then Aj is a point on Bi since 0 = A0 * B0 = A0M' * B0M' = A¡ * B¡, and the set of points {A¡}¡eZ lies on a curve from the family (8) for a proper choice of ax,... ,an/2.…”