The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph
of type Anβ determines planar fractal sets obtained by infinite dissections
of a given triangle. All triangles appearing in the dissection process have
angles that are multiples of Ο/(n+1). There are usually several possible
infinite dissections compatible with a given n but a given one makes use of
n/2 triangle types if n is even. Jones algebra with index [4Β cos2n+1Οβ]β1 (values of the discrete range) act naturally on vector spaces
associated with those fractal sets. Triangles of a given type are always
congruent at each step of the dissection process. In the particular case n=4,
there are isometric and the whole structure lead, after proper inflation, to
aperiodic Penrose tilings. The ``tilings'' associated with other values of the
index are discussed and shown to be encoded by equivalence classes of infinite
sequences (with appropriate constraints) using n/2 digits (if n is even)
and generalizing the Fibonacci numbers.Comment: 14 pages. Revised version. 18 Postcript figures, a 500 kb uuencoded
file called images.uu available by mosaic or gopher from
gopher://cpt.univ-mrs.fr/11/preprints/94/fundamental-interactions/94-P.302