4,719 research outputs found
An enumeration of equilateral triangle dissections
We enumerate all dissections of an equilateral triangle into smaller
equilateral triangles up to size 20, where each triangle has integer side
lengths. A perfect dissection has no two triangles of the same side, counting
up- and down-oriented triangles as different. We computationally prove W. T.
Tutte's conjecture that the smallest perfect dissection has size 15 and we find
all perfect dissections up to size 20.Comment: Final version sent to journal
Triangular dissections, aperiodic tilings and Jones algebras
The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph
of type 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 There are usually several possible
infinite dissections compatible with a given but a given one makes use of
triangle types if is even. Jones algebra with index (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 ,
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 digits (if 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
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