Fibonacci-Like Sequences of Apollonian Circle Packings
Janet
McShane and Michael Ratliff
Northern Arizona University
We
consider an Apollonian Circle packing of the unit circle and illustrate a novel
way to number the circles in the packing. We then investigate several
interesting properties of this circle packing including: a 4-color tiling of
the plane; the fact that curvatures and Graham’s center*curvature coordinates
satisfy Descartes equation; and the derivation of certain integer quadruples
from the “polynomial” curvatures.
Certain subsequences of the circle packing are
studied and we investigate how those sequences depend on their so-called basis
circles. In one case, this dependence is typified by the emergence of
Fibonacci number coefficients. Finally, we generalize our results to a
much wider class of circle packings.
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Janet M. McShane, PhD
Associate Professor
Department of Mathematics and Statistics
Northern