Non-Congruent Tilings of the Plane - a bit more

. The basic question of whether one can tile the plane with mutually non-congruent triangles all of equal area and perimeter has been answered in the negative by Kupaavski, Pach and Tardos (as already noted in this earlier post here: )

Another question: If one looks for tiling the plane with non-congruent triangles of equal area and equal diameter (diameter of a triangle is its longest side) what happens? I the answer is in negative, can the plane be tiled with convex pieces of equal area and diameter with no constraint on the numbers of sides? And what about other combinations of quantities maintained constant over the mutually non-congruent tiles?

I don't know if an answer to these questions follows naturally from the above mentioned paper. A few variants of the basic question answered in the above-mentioned paper had been recorded in earlier posts here (especially this post) but if I am not mistaken, the "area & diameter constant" case wasn't mentioned.

On Aperiodic Packing - and Covering

In continuation of this this earlier post here:

a simple question:
"Are there convex 2D shapes which cannot tile the plane and for which the best packing fraction is achieved by an aperiodic arrangment and not by a lattice one?" (the question can be posed for general -non-convex shapes as well).

If such regions exist, which among them shows max difference in packing fraction between its best lattice and best aperiodic pack?

Mathworld says: "Gauss proved that the hexagonal lattice is the densest plane lattice packing with unit circles and in 1940, L. Fejes Tóth proved that the hexagonal lattice is indeed the densest of all possible plane packings."

Fejes Toth has also proved that a lattice arrangement is the best packing for any centrally symmetric shape. So one has to look beyond such shapes for answers to our question.
And just as aperiodic packing, one can ask about (convex and general) shapes (in 2D and beyond) for which aperiodic covering beats lattice covering.