# 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