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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

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The Medusa algorithm for polynomial matings
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by Suzanne Hruska Boyd and Christian Henriksen PDF
Conform. Geom. Dyn. 16 (2012), 161-183 Request permission


The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image of its Julia set. Whether these approximations converge is answered using Thurston’s topological characterization of rational maps.

This algorithm was designed by John Hamal Hubbard, and implemented in 1998 by Christian Henriksen and REU students David Farris and Kuon Ju Liu.

In this paper we describe the algorithm and its implementation, discuss some output from the program (including many pictures) and related questions. Specifically, we include images and a discussion for some shared matings, Lattès examples, and tuning sequences of matings.

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Additional Information
  • Suzanne Hruska Boyd
  • Affiliation: Department of Mathematical Sciences, University of Wisconsin Milwaukee, PO Box 413, Milwaukee, Wisconsin 53201
  • Email:
  • Christian Henriksen
  • Affiliation: Department of Mathematics, Building 303, Technical University of Denmark, Denmark – 2800 Kgs. Lyngby, Denmark
  • Email:
  • Received by editor(s): February 24, 2011
  • Published electronically: June 26, 2012
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 16 (2012), 161-183
  • MSC (2010): Primary 37F10; Secondary 37M99
  • DOI:
  • MathSciNet review: 2943594