Loeschian Spheres
Loeschian Spheres
The Loeschian numbers (OEIS A003136) are numbers of the form , where and are integers. For example, , with (28,2,4) as the related Loeschian triplet.
x+xy+y
2
2
x
y
28=4+2×4+2
2
2
But first, something else. A different sequence is the Farey sequence, which at order 5 is
0,,,,,,,,,,1
1
5
1
4
1
3
2
5
1
2
3
5
2
3
3
4
4
5
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 as a Loeschian triplet satisfying , the sphere centers are at one of six locations.
(a,b,c)
a=b+b×c+c
2
2
,,,
2b+c
a
c
3
a
1
a
,,,
b+2c
a
b
3
a
1
a
,,,
a+b-c
a
(a-b-c)
3
a
1
a
,,,
a-b+c
a
(a-b-c)
3
a
1
a
,,,
2a-b-2c
a
b
3
a
1
a
,,.
2a-2b-c
a
c
3
a
1
a