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Loeschian Spheres

​
number of spheres
100
show curvatures
The Loeschian numbers (OEIS A003136) are numbers of the form
2
x
+xy+
2
y
, where
x
and
y
are integers. For example,
28=
2
4
+2×4+
2
2
, with (28,2,4) as the related Loeschian triplet.
But first, something else. A different sequence is the Farey sequence, which at order 5 is
0,
1
5
,
1
4
,
1
3
,
2
5
,
1
2
,
3
5
,
2
3
,
3
4
,
4
5
,1
.
In two dimensions, the Farey sequence can make Ford circles, each with the number as a radius, centered above their number, and tangent to the zero line and each other.
Loeschian spheres can be made, each with the number as a curvature (the reciprocal of the radius). With
(a,b,c)
as a Loeschian triplet satisfying
a=
2
b
+b×c+
2
c
, the sphere centers are at one of six locations.
2b+c
a
,
3
c
a
,
1
a
,​​
b+2c
a
,
3
b
a
,
1
a
,​​
a+b-c
a
,
3
(a-b-c)
a
,
1
a
,​​
a-b+c
a
,
3
(a-b-c)
a
,
1
a
,​​
2a-b-2c
a
,
3
b
a
,
1
a
,​​
2a-2b-c
a
,
3
c
a
,
1
a
.
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