Geodesic
Growing Apollonian packings
Journal of integer sequences, Tome 12 (2009) no. 2.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: In two dimensions, start with three mutually tangent circles, with disjoint interiors (a circle with negative radius has the point at infinity in its interior). We can draw two new circles that touch these three, and then six more in the gaps that are formed, and so on. This procedure generates an (infinite) Apollonian packing of circles. We show that the sum of the bends (curvatures) of the circles that appear in successive generations is an integer multiple of the sum of the bends of the original three circles. The same is true if we start with four mutually tangent circles (a Descartes configuration) instead of three. Also the integrality holds in three (resp., five) dimensions, if we start with four or five (resp. six or seven) mutually tangent spheres. (In four and higher dimensions the spheres in successive generations are not disjoint.) The analysis in the three-dimensional case is difficult. There is an ambiguity in how the successive generations are defined. We are unable to give general results for this case.
Keywords: circle-packing 
@article{JIS_2009__12_2_a5,
     author = {Mallows, Colin},
     title = {Growing {Apollonian} packings},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a5/}
}
TY  - JOUR
AU  - Mallows, Colin
TI  - Growing Apollonian packings
JO  - Journal of integer sequences
PY  - 2009
VL  - 12
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a5/
LA  - en
ID  - JIS_2009__12_2_a5
ER  - 
%0 Journal Article
%A Mallows, Colin
%T Growing Apollonian packings
%J Journal of integer sequences
%D 2009
%V 12
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a5/
%G en
%F JIS_2009__12_2_a5
Mallows, Colin. Growing Apollonian packings. Journal of integer sequences, Tome 12 (2009) no. 2. http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a5/