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Tangent Loeschian Spheres

Posted 9 years ago

The Loeschian numbers (OEIS A003136) are those of the form x^2+x y+y^2 where x and y are integers. These numbers can be used to make a nice set of tangent spheres.

min = 0; max = 40;
Loeschian = Drop[Union[{#[[1]]^2 + #[[1]] #[[2]] + #[[2]]^2, #[[1]], #[[2]]} & /@ (Sort /@ Tuples[Range[min, max], {2}])], 1];
LoeschianSpheres = Take[Flatten[Union[With[{a = #[[1]], b = #[[2]], c = #[[3]]},
   {{2 b + c, c Sqrt[3], 1}, {2 c + b, b Sqrt[3], 1}, {a + b - c, (a - b - c) Sqrt[3], 1}, 
   {a - b + c, (a - b - c) Sqrt[3], 1}, {2 a - 2 c - b, b Sqrt[3], 1}, {2 a - 2 b - c, c Sqrt[3], 1}}/a]
   ] & /@ Loeschian, 1], 600];
Graphics3D[{Sphere[#, Last[#]] & /@ Take[LoeschianSpheres, num]}, SphericalRegion -> True, ImageSize -> {600, 460}, 
   PlotRange -> {{0, 2}, {0, 13/8}, {0, 1/4}}, Boxed -> False, ViewPoint -> {0, 0, -3}, ViewAngle -> Pi/11]

Loeschian spheres

If the min has a negative value, an image like the following can appear:

Loeschian sphere with negatives

These can be seen as a 3D version of Ford circles, and are also related to the Circles of Descartes. Over at math.stackexchange, I posted a question about sphere tangencies.

Question about sphere tangencies

Can anyone answer that question at the end? Does anyone else have any nice sphere (or circle) tangency programs?

POSTED BY: Ed Pegg
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