External arguments in the multiple-spiral medallions of the Mandelbrot set

10 pages, 10 figures.-- PACS nr.: 05.45.Df.-- Printed version published on Jun 2006.The discs of a small copy of the Mandelbrot set \mathcal{M} have the same kinds of antennas or shrubs as the discs of \mathcal{M}, with the obvious difference that the shrubs of the small copy are “attached” to tendrils. Beautiful tendril decorations exist, known as multiple-spiral medallions, and these medallions are formed by an infinite number of baby Mandelbrot sets. Computer graphics and the external arguments theory provide two invaluable tools to study the structure of these decorations. In this work, we conjecture that the baby Mandelbrot sets of a multiple-spiral medallion have a "parent" and a "gene". This conjecture allows us to write the binary expansions of the external arguments of the baby Mandelbrot sets in an easy and structured way.This work was supported by Ministerio de Educación y Ciencia of Spain, Research Grant SEG2004-02418.Peer reviewe