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

, 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

.