Isosceles Triangle Containers of Triangles - 3

This post continues a somewhat old story narrated here and here . Note: the basic question was, given a general triangle T, to find the least area(perimeter) isosceles triangle that contains T.

Here is a preprint by Kiss, Pach and Somlai answering the question of finding the smallest isosceles (smallest area; smallest perimeter is pending) triangle container of any given triangle. Even as I try to work my way through it, let me record a couple of thoughts:

- In this post are pointers to works on these 2 questions: for a given convex polygon, find the least area (least perimeter) general triangle that contains it.

Questions:
- To find the least area isosceles triangle that contains a given convex polygon P: Will this work: 1)first find the least area general triangle that contains P and then 2) find the smallest area isosceles triangle that contains this least area general triangle container?

- Given any isosceles triangle T, how does one find that triangle of smallest area(perimeter) such that T is the least area(perimeter) isosceles triangle that contains it? Is the smallest convex region for which T is the smallest isosceles container necessarily a triangle? Note: here we are basically turning the original isosceles triangle container question inside out.

Note: All these thoughts can be floated with right triangles replacing isosceles. And what happens when one goes to Hyperbolic geometry?

Update (9th March 2020): One can also ask about covering a given general triangle T with 2 isosceles triangles with least total area. Then replace 2 by 3 and so on (Replacing isosceles by right makes this question trivial). Is this question related to finding the largest isosceles triangle that can be contained by T?

Cutting the Unit Square into Rational-sided Pieces

This post continues this old thread and this recent post here. The immediate trigger was an email exchange with Dirk Frettloh (Thanks!)

The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational.

1. To cut a unit square into n, a finite number of triangles with all sides of rational length. For which values of n can it be done if at all?

2. To cut a unit square into n right triangles with all sides of rational length. For which values of n can it be done if at all?

3.To cut a unit sauare into n isosceles triangles all sides of rational length. For which values of n can it be done if at all?

Now, one can add the requirement of mutual non-congruence of all pieces to all these questions. And one can add rationality of area of pieces etc... and so on...

A 3D Packing Question

Basic Question: Is there a convex polygon that can tile the plane only such that the arrangement is not vertex to vertex? Note: In a vertex to vertex arrangement, any vertex of any unit touches any other unit only at a vertex and not at an intermediate point of an edge.

The answer appears to be yes. See this: https://en.wikipedia.org/wiki/Marjorie_Rice
3 of the 4 pentagonal tiles discovered by Marjorie Rice appear to tile only in a non-vertex-to-vertex fashion. However, the three non-regular hexagons that tile the plane (discovered by Reinhardt; see this: https://arxiv.org/pdf/1803.06610.pdf ) appear to give v-to-v tiles.
And of course, all triangles and all quadrilaterals tile but they all can tile the plane in v-to-v manner.

Question: Going on to 3D, are there polyhedral units (convex or otherwise) that can pack 3D space without gaps only such that the arrangement is NOT vertex to vertex or edge to edge?