Definition: The Mondrian problem consists of dissecting a square of side length n (an integer) into mutually non-congruent rectangles with integer length sides such that the difference d(n) between the largest and the smallest areas of the rectangles into which the square is partitioned is minimum.
It is suspected that there is no square of integer side that can be partitioned into noncongruent rectangles (whatever be n and whatever be the number of rectangles) all of equal area and with integer length sides. IOW, there probably is no 'perfect' Mondrian dissection - one with d(n) = 0.
Basic Problem: To find a set of rectangles all of same perimeter (let us call such rectangles isoperimetric rectangles) but with different areas which together form a neat big rectangle.
For n = 7, 8, 9 there seem to be solutions with mutually non-congruent isoperimetric rectangles all with rational (and hence integer) sides which can be put together to form a big rectangle.
Eg: for n = 7, we have the 7 rectangles with dimensions: {10.000000, 9.500000}, {16.000000, 3.500000}, {6.000000, 13.500000}, {15.500000, 4.000000}, {5.500000, 14.000000}, {18.500000, 1.000000} and {2.500000, 17.000000} which together form a spiral layout that is a big 24 X 18 rectangle.
Are there rectangles which can be cut into less than 7 or more than 9 isoperimetric and non-congruent tiles with all dimensions integers?
Is there any dissection for a square of integer sides into isoperimetric and non-congruent rectangles (this can be seen as the perimeter analog of the 'perfect' Mondrian partition)? Guess: there probably isn't one.
What can one say if the ratios between dimensions of the isoperimetric rectangles are allowed to be irrational?
Note: These questions were recorded at: https://arxiv.org/ftp/arxiv/papers/1307/1307.3472.pdf (page 4-5).
Further Question: Has the classical Mondrian problem been studied with a rectangle (instead of a square) being partitioned?
