### Doubts on Covering

Consider trying to cover the largest scaled copy of a region C with n instances of a fixed tile T (iow, we were asking how big a circle or square or... (C) could be covered with n copies of an equilateral triangle or square or circle .... (T) (see Erich's Packing Center for several families of such questions: (https://www2.stetson.edu/~efriedma/packing.html)

Note 1: An example of the non-convex case mentioned above:

In above picture, the shaded non-convex region is the tile T and C is the circle. If the angle of the 'fan' portion of T divides 360, a suitable number n of unit Ts can cover the big circle. Then the 'core' of the big circle could have n T's overlapping. Note that the two circles are concentric and the inner one - core - is quite small. And the same n units don't seem to be able to cover any bigger circle.

Note 2: One could also go to higher dimensions and ask if there is an upper bound on the number of tiles (which are 3d regions now) that can overlap in some part of space when n tiles cover a convex region C of maximum volume.

'Erich's Packing Center' has lists of coverings such as 'equilateral triangles covering squares, squares covering circles and so on...'

Could we have any intuitive 'large scale' results such as (say):

a. "For any n, among all unit area triangular tiles of unit area, the equilateral triangle tile is the the one such that n units cover the largest square" . And if this claim is invalid, is it that for every n, that the triangular unit such that n units thereof covers the largest square has a different shape?

OR

b. "For any n, among all unit area rectangles, the square is the one such that n unit squares cover the largest circle"

.. indeed many such claims can be made but how many are valie?

**Question 1:**In the best covering layout for a given C with n T's, consider regions where where more than one T unit overlap. The observation is that if T is non-convex, the best layout could have regions where an arbitrarily large number of T units overlap (see below). However, but for the case when T is convex, no known best covering layout (as seen at say, the Packing Center), seems to have regions where more than 3 unit Ts overlapping. Is this a provably strict bound?Note 1: An example of the non-convex case mentioned above:

In above picture, the shaded non-convex region is the tile T and C is the circle. If the angle of the 'fan' portion of T divides 360, a suitable number n of unit Ts can cover the big circle. Then the 'core' of the big circle could have n T's overlapping. Note that the two circles are concentric and the inner one - core - is quite small. And the same n units don't seem to be able to cover any bigger circle.

Note 2: One could also go to higher dimensions and ask if there is an upper bound on the number of tiles (which are 3d regions now) that can overlap in some part of space when n tiles cover a convex region C of maximum volume.

**Question 2:**'Erich's Packing Center' has lists of coverings such as 'equilateral triangles covering squares, squares covering circles and so on...'

Could we have any intuitive 'large scale' results such as (say):

a. "For any n, among all unit area triangular tiles of unit area, the equilateral triangle tile is the the one such that n units cover the largest square" . And if this claim is invalid, is it that for every n, that the triangular unit such that n units thereof covers the largest square has a different shape?

OR

b. "For any n, among all unit area rectangles, the square is the one such that n unit squares cover the largest circle"

.. indeed many such claims can be made but how many are valie?