  The paper is an attempt to apply the theory of dessins d'enfants to the
theory of fullerenes. The classical results concerning the calculation of the
dodecahedron Belyi function are presented and then applied to the calculation
of the Belyi function of the barrel, and the euclidean geometry of the latter
is investigated. The non-existence of the fullerene with the only hexagonal
face is established by the methods of dessins d'enfants.
