6 research outputs found
Similar dissection of sets
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 be a given set and let
be injective continuous mappings. Does there exist a set such
that is satisfied with a
non-overlapping union? We prove that such a set exists for certain choices
of and . The solutions 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
for
On monohedral tilings of a regular polygon
A tiling of a topological disc by topological discs is called monohedral if
all tiles are congruent. Maltby (J. Combin. Theory Ser. A 66: 40-52, 1994)
characterized the monohedral tilings of a square by three topological discs.
Kurusa, L\'angi and V\'\i gh (Mediterr. J. Math. 17: article number 156, 2020)
characterized the monohedral tilings of a circular disc by three topological
discs. The aim of this note is to connect these two results by characterizing
the monohedral tilings of any regular -gon with at most three tiles for any
.Comment: 17 pages, 9 figure
Trisecting a rectangle
AbstractIn this paper it is shown that it is impossible to dissect a rectangle into three congruent pieces unless those pieces are also rectangles