Non- congruent tiling - one more

Seeking a little clarification on https://arxiv.org/pdf/2004.01034

The paper establishes that there are tilings of the plane by convex quadrilaterals all mutually non congruent and with same area and perimeter.

QUESTIONS
1. given any convex quadrilateral Q can we form tilings with Q and choosing other quads of same perimeter and area as Q and all mutually non congruent?

2. If the answer is “not in general”, how will we characterise Qs which can function as ‘seed tiles’?

And can such tilings be achieved selecting from convex quads having any possible {area, perimeter} pair of values?

More soon on this hopefully...

Oriented containers - kites

This continues an old thread on oriented containers with latest episode here

How does one find the least area convex kite that contains a given convex polygon P? What about the least perimeter kite container?Isosceles triangles may be treated as degenerate kites

Which is the convex shape that maximises the difference in orientation (measured by angle between the lines of symmetry of the two containers) between the smallest area and smallest perimeter kite containers?

Note 1: Unlike in the case of the smallest area containing rectangle for a convex P, the smallest enclosing kite might not share an edge with the polygon - if P is a thin rectangle, a containing kite that shares an edge with it appears suboptimal. well not sure!

Note 2: one can also ask about the largest area/perimeter kite contained inside a given convex polygon.



if the smallest area kite container too shares an edge with P, one can ask if there is any type of container where the optimal container need not share an edge with P.