# 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