Non-congruent Tilings - 8
This post brings to the present (2020) - and would try to build on - the most recent update to this post begun in June 2016 (basically the post has had too many updates over the years so it would be better to bring the latest update to the present and take it forward from here):
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Here is a nice discussion, initiated by Prof. O'Rourke and dating hack to 2015, on the question of tiling the plane with non-congruent isosceles triangles (a question also raised in the above linked old post of mine in a November 29th 2017 update):
https://mathoverflow.net/questions/221431/tiling-the-plane-with-incongruent-isosceles-triangles
This mathoverflow page describes 2 constructions, both showing what appear to be tilings of the plane with pairwise non-congruent isosceles triangles with no upper bound on the size of the tiles. So, one naturally wonders if such a tiling with such an upper bound can be worked out. And of course, area and perimeter requirements on the tiles can be added!
Indeed, we can tile plane with non-congruent isosceles triangles with bounds only on edge lengths: perturb each vertex of an equilateral triangle tiling infinitesimally in a unique direction, to get a tiling with non-congruent acute triangles with every side unique and with both upper and lower bounds on max edge length.. Then, divide each triangle into isosceles triangles by joining its circumcenter with the vertices. That should be it.
However, it looks like further constraints such as equality of area/perimeter would make the problem harder
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A further variant: to tile the plane with mutually non-congruent acute isosceles triangles (without or with area/perimeter constraints).
Indeed, one can relax 'isoscelesness' and go back to the basic non-congruent general triangle tiling question - starting with "area equal among tiles and perimeter unconstrained" (see here):
http://stanwagon.com/potw/2015/p1199.html - and add acuteness of tiles as an additional requirement to generate a new sequence of questions.
Observations: Tiling with mutually non-congruent triangles of equal area and unbounded perimeter that are also all constrained to be strictly obtuse is pretty easy. In the construction shown in the page linked just above, instead of dividing the plane into 4 quadrants, begin with 3 wedges of 120 degrees each and do pretty much the same construction. However, with the tiles required to be strictly acute, things appear much harder.
------------
Here is a nice discussion, initiated by Prof. O'Rourke and dating hack to 2015, on the question of tiling the plane with non-congruent isosceles triangles (a question also raised in the above linked old post of mine in a November 29th 2017 update):
https://mathoverflow.net/questions/221431/tiling-the-plane-with-incongruent-isosceles-triangles
This mathoverflow page describes 2 constructions, both showing what appear to be tilings of the plane with pairwise non-congruent isosceles triangles with no upper bound on the size of the tiles. So, one naturally wonders if such a tiling with such an upper bound can be worked out. And of course, area and perimeter requirements on the tiles can be added!
Indeed, we can tile plane with non-congruent isosceles triangles with bounds only on edge lengths: perturb each vertex of an equilateral triangle tiling infinitesimally in a unique direction, to get a tiling with non-congruent acute triangles with every side unique and with both upper and lower bounds on max edge length.. Then, divide each triangle into isosceles triangles by joining its circumcenter with the vertices. That should be it.
However, it looks like further constraints such as equality of area/perimeter would make the problem harder
----------------
A further variant: to tile the plane with mutually non-congruent acute isosceles triangles (without or with area/perimeter constraints).
Indeed, one can relax 'isoscelesness' and go back to the basic non-congruent general triangle tiling question - starting with "area equal among tiles and perimeter unconstrained" (see here):
http://stanwagon.com/potw/2015/p1199.html - and add acuteness of tiles as an additional requirement to generate a new sequence of questions.
Observations: Tiling with mutually non-congruent triangles of equal area and unbounded perimeter that are also all constrained to be strictly obtuse is pretty easy. In the construction shown in the page linked just above, instead of dividing the plane into 4 quadrants, begin with 3 wedges of 120 degrees each and do pretty much the same construction. However, with the tiles required to be strictly acute, things appear much harder.
posted by R.Nandakumar @ 9:31 AM
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