|
|
The bowl of integers is the arrangement of two solid spheres of radius 1/2 inside a hollow sphere of radius 1, with
the two smaller spheres touching at the center of the
enclosing sphere and tangent to it at opposite ends of
a diameter (Soddy 1937). In Soddy's construction, the packing
of spheres generated by the following replacements has
integral bends. For five mutually tangent
spheres, replacing one by the other sphere tangent
to the remaining four replaces its bend 
by 
. Starting with the five bends 
, 2, 2, 3, and 3 therefore preserves
integrality.
In this packing, the spheres tangent to the enclosure and exactly one of the initial bend-2 spheres,
together with the initial bend 2, have distinct bends
2, 5, 6, 9, 11, 14, 15, 18, 21, 23, 27, 29, 30, 33, 38, ... (OEIS A046160).
Let 
for integers 
and 
, so that 
is a Löschian number.
Inversion at a point of contact of the enclosure and
an initial sphere transforms this subpacking to equal
spheres on a triangular
lattice between parallel planes. Transforming back
gives 
,
excluding 
,
which gives the six equatorial spheres of bend
3 touching both initial spheres.
The table gives the first few distinct bends, with the enclosing sphere prepended. Here 
are cylindrical
coordinates of the centers, with the initial centers at 
. For 
, 
and 
. The listed angles
are representatives, repeated under rotations by multiples
of 
.
Reflection in 
gives the corresponding lower-half spheres.
 |  |  |  |  |
| 1 |  | 0 | 0 | -- |
| 2 | 2 |  | 0 | -- |
| 3 | 5 |  |  |  |
| 4 | 6 |  |  | 0 |
| 5 | 9 |  |  |  |
| 6 | 11 |  |  | 0 |
| 7 | 14 |  |  |  |
| 8 | 15 |  |  |  |
| 9 | 18 |  |  | 0 |
| 10 | 21 |  |  |  |
| 11 | 23 |  |  |  |
| 12 | 27 |  |  | 0 |
| 13 | 29 |  |  |  |
| 14 | 30 |  |  |  |
| 15 | 33 |  |  |  |
| 16 | 38 |  |  | 0 |
|
|
|
|
The equatorial spheres have centers in the plane tangent to the two initial spheres of bend
2, each of radius 1/2 (Soddy 1937). For this subpacking, inversion
at the origin gives equal spheres on a triangular
lattice, and their bends are 
for positive Löschian
numbers 
.
They are 3, 11, 15, 27, 35, 47, 51, 63, 75, 83, ... (OEIS A046159).
For five mutually tangent spheres, setting 
gives the larger solution
![kappa(kappa_1,kappa_2)
=1/2(4+kappa_1+kappa_2+sqrt(3[kappa_2(8-kappa_2)+2kappa_1(kappa_2+4)-kappa_1^2])).](/images/equations/BowlofIntegers/NumberedEquation1.svg) |
For example, 
,
, 
, 
, and 
. The table gives the first few distinct equatorial
bends, with the enclosure and initial spheres
prepended. Equatorial centers have 
and 
. The listed angles
again represent rotations by multiples of 
.
 |  |  |  |
| 1 |  | 0 | -- |
| 2 | 2 | 0 | -- |
| 3 | 3 |  | 0 |
| 4 | 11 |  |  |
| 5 | 15 |  | 0 |
| 6 | 27 |  |  |
| 7 | 35 |  | 0 |
| 8 | 47 |  |  |
| 9 | 51 |  |  |
| 10 | 63 |  | 0 |
| 11 | 75 |  |  |
| 12 | 83 |  |  |
| 13 | 99 |  | 0 |
| 14 | 107 |  |  |
| 15 | 111 |  |  |
| 16 | 123 |  |  |
| 17 | 143 |  | 0 |
| 18 | 147 |  |  |
| 19 | 155 |  |  |
| 20 | 171 |  |  |
The analogous problem of placing two circles of bend 2 inside a circle of bend 
and then constructing chains of mutually
tangent circles was considered by B. L. Galebach
and A. R. Wilks (OEIS A042944). It
gives the Apollonian gasket with initial bends 
(Graham et al. 2003). The circles
have integral bends given by 
, 2, 3, 6, 11, 14, 15, 18, 23, 26, 27, 30, 35, 38, ... (OEIS
A042944). The positive bends
are congruent to 2, 3, 6, or 11 (mod 12). The first
positive integers in these classes missing from this
sequence are 78, 159, 207, 243, 246, 342, ... (OEIS A042945),
a sequence conjectured to be finite for this particular Apollonian
gasket (Haag et al. 2024).
N. Hannachi (pers. comm., Mar. 10, 2006) found a bowl of three spheres with bend 6 and one with bend 7
inside a sphere of bend 
.
