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Discrete Comput Geom 34:295312 (2005) Discrete & ComputationalDOI: 10.1007/s00454-005-1167-1Geometry 2005 Springer Science+Business Media, Inc.Dissection of a Triangle into Similar TrianglesAndrzej ZakFaculty of Applied Mathematics, AGH University of Science and Technology,
al. Mickiewicza 30, 30059 Krakow, [email protected]. We prove that the number of non-similar triangles T which can be dissected
into two, three or five similar non-right triangles is equal to zero, one and nine, respectively.
We find all these triangles. Moreover, every triangle can be dissected into n similar triangles
whenever n = 4 or n 6. In the last section we allow dissections into right-triangles but
we add another restriction. We prove that in any perfect, prime and simplicial dissection
into at least three tiles, the tiles must have one of only three possible shapes.1. IntroductionLet P and P be polygons in the Euclidean plane. A dissection (tiling) of P into P is
a decomposition of P into finitely many, internally disjoint polygons P1, ..., P
n (n 2)such that all of the Pi are similar to P. A dissection is perfect if the P
i are pairwise
incongruent. The perfect decomposition of rectangles and squares into squares has been
extensively studied (see [1] and also [3] for a detailed account of the history of this
problem). The first example of a dissection of a rectangle into nine pairwise incongruent squares was given by Moron [7] in 1925. In 1939 Sprague [10] found the perfect
squaring of a square into 55 tiles (see also [2]). Less is known about dissections of polygons other than the square. Tutte [11] proved that an equilateral triangle has no perfect
dissections into smaller equilateral triangles. In 1991 Kaiser [5] observed that every
non-equilateral triangle has a perfect dissection into at least six or eight tiles similar to it.
His elementary construction is shown in Fig. 1. Laczkovich [6] showed that the number
of non-similar triangles such that a non-equilateral triangle T has a dissection into ,
is at most six (he also showed that for the equilateral triangle T there are infinitely many
such ).In this paper we consider the following question: for every n (n 2) find those
triangles T that can be dissected into n similar non-right triangles. We denote by f (n)
the number of...