External arguments of Douady cauliflowers in the Mandelbrot set M. Romera * , G. Pastor, G. Álvarez, and F. Montoya Instituto de Física Aplicada, Consejo Superior de Investigaciones Científicas Serrano 144, 28006 Madrid, Spain Near to the cusp of a cardioid of the Mandelbrot set, except for the main cardioid, there is a sequence of baby Mandelbrot sets. Each baby Mandelbrot set is in the center of a Douady cauliflower, a decoration constituted by an infinity of minute Mandelbrot sets and Misiurewicz points linked by filaments. A Douady cauliflower appears to have a complicated structure, and how the external rays land without intersecting in its Misiurewicz points and minute Mandelbrot sets is not really well known. In this work we study the Douady cauliflowers, giving a binary tree model to calculate the binary expansions of the external arguments of both the main cardioid of each minute Mandelbrot set and Misiurewicz points, and creating a horsetail model that explains the arrangement of its corresponding external rays. Keywords: Mandelbrot set, external arguments, external rays, cauliflowers. 1. Introduction In Fig. 1 we can see six details of the Mandelbrot set that we name Mandelbrot medallions, all of them with rotational symmetry. The unquestionable beauty of these Mandelbrot medallions have interested to many researchers as Branner [1] (a), Philip [2] (b), Misiurewicz [3] (c), Munafo [4] (d), Romera and Negrillo [5] (e), and Douady et al. [6] (f). A Mandelbrot medallion presents an infinity of decoration levels around its center. Each decoration level is formed by an infinity of minute Mandelbrot sets and Misiurewicz points with different rotational symmetry. So, the first, second, third... levels have rotational symmetries of orders two, four, eight... In the center of a Mandelbrot medallion there is a baby Mandelbrot set, not a Misiurewicz point, and for this cause the rotational symmetry of a Mandelbrot medallion in not perfect and it must be considered as a quasi-rotational symmetry. * Corresponding author. Tel: +34 915631284; fax: +34 914117651. E-mail address: miguel@iec.csic.es (M. Romera)