Loeschian Spheres
Loeschian Spheres
The Loeschian numbers (OEIS A003136) are numbers of the form +xy+, where and are integers. For example, , with (28,2,4) as the related Loeschian triplet.
2
x
2
y
x
y
28=+2×4+
2
4
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×c+
2
b
2
c
2b+c
a
3
ca
1
a
b+2c
a
3
ba
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
ba
1
a
2a-2b-c
a
3
ca
1
a