### Non-Congruent Tiling - an Ongoing Story

News Update: Dirk Frettloh has written a preprint on some interesting fresh work of his and consolidating whatever is known on the non-congruent equipartition problem here:
http://arxiv.org/abs/1603.09132

Recording some speculations and thoughts on similar lines:

1. It appears that a plane cannot be tiled by mutually non-congruent, equal area rectangles if the following constraint is applied: The thinnest of the rectangles has length less than twice its thickness. The proof for this goes on similar lines to the proof that with near equilateral triangles all with same area and perimeter, the plane cannot be tiled (mentioned in an earlier post). One is not sure if relaxing the constraint partially (ie with some upper bound on the length to width ratio) will enable such a tiling.

2. Can the plane be tiled with non congruent right triangles all of the same area? (With equal area rectangles, one can have spiral arrangements; with right triangles, one is not sure)

3. Can the plane be tiled with non congruent right triangles all with the same hypotenuse (ie all those right triangles inscribable in the same semi circle)? Can one manage with non congruent triangles with same area and same longest side?

4. Can the plane be tiled with equilateral triangles, all of different sizes? More generally, can a plane be tiled by non congruent but mutually similar triangles? These questions let go of the equal area requirement.

Note 1: As was mentioned in an earlier post (http://nandacumar.blogspot.in/2015/01/filling-plane-with-non-congruent-pieces.html), is is not very hard to tile the plane with squares all of different sizes. Method: a way to dissect a square into 21 squares all of different sizes is known. Do this to a unit square, then along side this dissected unit square, patch another unit square, then build outwards a Fibonacci spiral of squares of increasing size. Of course, this will need arbitrarily large squares. See the Wiki article on 'squaring the square' for details.

It looks very likely that *any* such tiling of the plane with unequal squares has to have either arbitrarily large or arbitrarily small squares - even if we use squares of irrational side. Not sure of this! At least as shown in the Wiki article mentioned above, we can have a lower bound on the size of the squares used.

Note 2: With equilateral triangles, such a tiling seems doable but unlike in the square case, a lower bound on the size of the tiles may not exist. Not sure!

5. CAn the plane be filled with rectangles none of which share either dimension? This appears doable with spiral arrangements starting with a Blanche dissection. But then, what is one wants some upper bound on the length to width ratio?

Shall update this page as answers come up...

Recording some speculations and thoughts on similar lines:

1. It appears that a plane cannot be tiled by mutually non-congruent, equal area rectangles if the following constraint is applied: The thinnest of the rectangles has length less than twice its thickness. The proof for this goes on similar lines to the proof that with near equilateral triangles all with same area and perimeter, the plane cannot be tiled (mentioned in an earlier post). One is not sure if relaxing the constraint partially (ie with some upper bound on the length to width ratio) will enable such a tiling.

2. Can the plane be tiled with non congruent right triangles all of the same area? (With equal area rectangles, one can have spiral arrangements; with right triangles, one is not sure)

3. Can the plane be tiled with non congruent right triangles all with the same hypotenuse (ie all those right triangles inscribable in the same semi circle)? Can one manage with non congruent triangles with same area and same longest side?

4. Can the plane be tiled with equilateral triangles, all of different sizes? More generally, can a plane be tiled by non congruent but mutually similar triangles? These questions let go of the equal area requirement.

Note 1: As was mentioned in an earlier post (http://nandacumar.blogspot.in/2015/01/filling-plane-with-non-congruent-pieces.html), is is not very hard to tile the plane with squares all of different sizes. Method: a way to dissect a square into 21 squares all of different sizes is known. Do this to a unit square, then along side this dissected unit square, patch another unit square, then build outwards a Fibonacci spiral of squares of increasing size. Of course, this will need arbitrarily large squares. See the Wiki article on 'squaring the square' for details.

It looks very likely that *any* such tiling of the plane with unequal squares has to have either arbitrarily large or arbitrarily small squares - even if we use squares of irrational side. Not sure of this! At least as shown in the Wiki article mentioned above, we can have a lower bound on the size of the squares used.

Note 2: With equilateral triangles, such a tiling seems doable but unlike in the square case, a lower bound on the size of the tiles may not exist. Not sure!

5. CAn the plane be filled with rectangles none of which share either dimension? This appears doable with spiral arrangements starting with a Blanche dissection. But then, what is one wants some upper bound on the length to width ratio?

Shall update this page as answers come up...