7 research outputs found
Tilings with noncongruent triangles
We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that any tiling of a convex polygon with more than three sides with finitely many triangles contains a pair of triangles that share a full side. © 2018 Elsevier Lt
Tiling the plane with equilateral triangles
Let T be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then T is periodic and it consists of translates of only at most three different triangles. As a corollary, we prove a theorem of Scherer and answer a question of Nandakumar. The same result has been obtained independently by Richter and Wirth
Tilings of the plane with unit area triangles of bounded diameter
There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar. © 2018, Akadémiai Kiadó, Budapest, Hungary
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