COXETER ON 'FIRMAMENT'
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 
n such that every 4 consecutive
members are mutually tangent, the inversive distance 
n
between 0 and n
(or between m and m+n
for any m ) is given in terms of the
Fibonacci numbers fn: 1, 1, 2, 3, 5, ..., by the formula
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 n
are
| n |
= |
1, |
2, |
3, |
4, |
5, |
6, |
7, |
| cosh n |
= |
1, |
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/x3,
1/x2, 1/ x, 1, x, x2, x3 ,
where x is the root,
between 1 and 2, of
the quintic equation
x5 - x4 - x3 - x2 - x + 1 = 0.
This equation has a root -1 and the remaining quartic is easily solvable as a quadratic in
x + 1/x to give x as
or approximately 1.8832. This gives the radii previously described.
Donald Coxeter, January 1997
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©Mathematics and Knots/Edition Limitee 1996
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