Apollonian Circle Packings via Möbius Transformations of Hyperbolic Polygons

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Hyperbolic polygons are an interesting aspect of non-Euclidean geometry. They can be created as parametric plots using two pairs of sine and cosine functions, corresponding to the horizontal and vertical axes. Each function in the pair has an amplitude equal to the inverse of the argument. Because Möbius transformations preserve straight lines and circles, the transformations associated with a single hyperbolic polygon yield an arrangement of circular cutouts and straight line segments. When no straight lines are present, the transformations trace portions of Apollonian circle packings.

Contributed by: Alexandra L. Brosius (August 2022)
Open content licensed under CC BY-NC-SA



This Demonstration uses real and imaginary parts of


The constant is one less than the number of sides of the polygon. The colored lines show Möbius transformations of the form

in the complex plane with the corresponding matrix


The coefficients have a unique value for each set of transformations, while the coefficients have a range of values centered about zero.

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