It is known [H.S.M. Coxeter, 'Loxodromic Sequences of Tangent
Spheres', **Æquationes Mathematicæ**, 1 (1968), pp. 112-117] that, for a sequence of circles s** _{n}** such that every 4 consecutive members are mutually tangent, the
inversive distance

For the analogous sequence of spheres, such that every 5 consecutive
members are mutually tangent, a prize is offered to the first
person who provides the analogous formula for the inversive distances
between pairs of the spheres. Meanwhile, by taking one pair of
adjacent 'spheres' to be a pair of parallel planes, one easily
finds that the values of **cosh ****d _{n} **are

**n****=****1,****2,****3,****4,****5,****6,****7,**
**cosh** **d**_{n}**=****1,****1,****1,****5,****7,****13.**

John Robinson's sculpture **FIRMAMENT** is based on seven such spheres whose radii are in geometric progression;
that is, the seven radii are proportional to

**1/x ^{3}, 1/x^{2}, 1/ x, 1, x, x^{2}, x^{3 }, **

where **x** is the root, between 1 and 2, of the quintic equation **x ^{5} - x^{4} - x^{3} - x^{2} - x + 1 = 0**. This equation has a root

or approximately 1.8832. This gives the radii previously described.

Donald Coxeter, January 1997

**© Mathematics and Knots/Edition Limitee 1996 - 2002**

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