Report on

Tilings with noncongruent triangles

by Andrey Kupavskii, János Pach, Gábor Tardos.

R. Nandakumar asked whether there is a tiling of the plane by pairwise
noncongruent triangles of equal perimeter and equal perimeter. This
problem and its several variants have inspired a few papers recently.
This one is dedicated to the original question above. It is proven
that the answer is negative: there is no such tiling. One point in
the proof is that in such a tiling no two tiles can share a common
edge (beacuase then they are congruent already). This is lead to a
contradiction. In particular three results in this paper prove that
under some more general conditions there are two triangles sharing
a common edge. The results are new and interesting, the paper is well
written and the topic fits perfectly into the scope of the European
Journal of Combinatorics. So I suggest publication after a minor
taking care of the points below. Most of them are typos, and the few
serious remarks are fixed easily.

One colour used in Fig 1, Fig 3, Fig 4 and Fig 5 is red. The caption of
Fig 5 refers to "red". If the printed article does not use colours it
would be better to replace "red" by "grey".

p3 line 1: "Fig.2" -> "Fig. 2"; and "The tiling on" -> "The tiling in"

p4 caption: maybe it would help some readers if you write "a stretch
of size 4 with" rather than "a stretch with".

p5 l-9: "the Theorem" -> "Theorem"

p5 l-8: "of tiling" -> "of the tiling"

p6 l6: "Since every triangle has area at least ε > 0, the tiling must
be locally finite." Literally taken, this is not true: It must be "Since
every triangle has area at least ε > 0 and equal perimeter, the tiling
must be locally finite." 

p7: "Observe that sides contained in the boundary of P (full boundary
edges) do not belong to any stretch" I guess this is not true. By
definition, two full sides on the boundary of P can be collinear, making
the union of these two sides a stretch. In order to fix this it might be
good to exclude these constellation by the definition of a stretch.
Then - I guess - nothing in the sequel will be affected.

next line: "the total (euclidean) length" I guess it must be "the total
number", since t and e_full are counting objects, not measuring lengths.

p9: "so by (9) and (9)" -> "so by (4) and (9)"

p10 caption: "Illustartion" -> "Illustration"

p10: "a third vertex w'" I guess this must be "a third vertex z"

xxx

p11: [4] has a reference now: , in: M.D.E. Conder, A. Deza, A.I. Weiss (Eds.):
Discrete Geometry and Symmetry, Veszprém, Hungary, June 2015, Springer
Proceedings in Mathematics & Statistics, Vol. 234 (2018) 

[6] and [7] are not consistent: , vs :, . vs ,, Blog vs blog,
date before/after url.