 
Summary: SIMILAR DISSECTION OF SETS
SHIGEKI AKIYAMA, JUN LUO, RYOTARO OKAZAKI, WOLFGANG STEINER,
AND JšORG THUSWALDNER
Abstract. In 1994, Martin Gardner stated a set of questions concerning the dissection of
a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions
have been generalized and some of them are already solved. In the present paper, we solve
more of his questions and treat them in a much more general context.
Let D Rd
be a given set and let f1, . . . , fk be injective continuous mappings. Does
there exist a set X such that D = Xf1(X). . .fk(X) is satisfied with a nonoverlapping
union? We will prove that such a set X exists for certain choices of D and {f1, . . . , fk}.
The solutions X will often turn out to be attractors of iterated function systems with
condensation in the sense of Barnsley.
Coming back to Gardner's setting, we use our theory to prove that an equilateral
triangle can be dissected in three similar copies whose areas have ratio 1 : 1 : a for
a (3 +
5)/2.
1. Introduction
In the present paper, we deal with the dissection of a given set D into finitely many
