On Tiling with Isosceles Triangles
A few years back, I wrote this post .
Therein I had wondered in passing: what if the tiles are to be *isosceles and non-congruent* (with various possibilities for area and perimeter)?
Quite recently, I found this very interesting discussion :
A particularly nice observation there, from Noam Elkies: Any acute non-isosceles triangle can be tiled by three pairwise incongruent isosceles triangles, by connecting each vertex to the circumcenter.
An elegant construction follows where a tiling of the plane into pairwise non-congruent isosceles triangles is built up from the above observation.
Remark: One wonders if the construction given there leads to arbitrarily large isosceles triangles being used. Looks like a way out (tiling with non-congruent, bounded size isosceles triangles) can be found based on the construction given at the top of that very page:
Choose any acute and non-isosceles triangle T and tile the plane with copies of it. Now, put a random point in the interior of a unit T to divide it into 3 small acute triangles. Since there are infinitely many ways by which such a point can be put inside T, we can put a point inside each T unit in the tiling to create a tiling of plane with pair-wise non-congruent acute non-isosceles triangles. Now, using Elkies's observation, we could connect the vertices of each triangle in our latest tiling to its circumcenter to create a fresh tiling of the plane with isosceles triangles which appear pair-wise non-congruent.
Note: To my knowledge, no discussion has happened on what happens if in addition to pair-wise non-congruence, equal area requirement is also put on the isosceles tiles. And same appears to be the scene if we replace isosceles triangles with right triangles (It appears that a square can be cut into 4 mutually non-congruent right triangles in infinitely many ways; so to achieve only pair-wise non-congruence with right triangle tiles, we could tile the plane with squares and cut each square in a different way into 4 mutually non-congruent triangles).
Therein I had wondered in passing: what if the tiles are to be *isosceles and non-congruent* (with various possibilities for area and perimeter)?
Quite recently, I found this very interesting discussion :
A particularly nice observation there, from Noam Elkies: Any acute non-isosceles triangle can be tiled by three pairwise incongruent isosceles triangles, by connecting each vertex to the circumcenter.
An elegant construction follows where a tiling of the plane into pairwise non-congruent isosceles triangles is built up from the above observation.
Remark: One wonders if the construction given there leads to arbitrarily large isosceles triangles being used. Looks like a way out (tiling with non-congruent, bounded size isosceles triangles) can be found based on the construction given at the top of that very page:
Choose any acute and non-isosceles triangle T and tile the plane with copies of it. Now, put a random point in the interior of a unit T to divide it into 3 small acute triangles. Since there are infinitely many ways by which such a point can be put inside T, we can put a point inside each T unit in the tiling to create a tiling of plane with pair-wise non-congruent acute non-isosceles triangles. Now, using Elkies's observation, we could connect the vertices of each triangle in our latest tiling to its circumcenter to create a fresh tiling of the plane with isosceles triangles which appear pair-wise non-congruent.
Note: To my knowledge, no discussion has happened on what happens if in addition to pair-wise non-congruence, equal area requirement is also put on the isosceles tiles. And same appears to be the scene if we replace isosceles triangles with right triangles (It appears that a square can be cut into 4 mutually non-congruent right triangles in infinitely many ways; so to achieve only pair-wise non-congruence with right triangle tiles, we could tile the plane with squares and cut each square in a different way into 4 mutually non-congruent triangles).
posted by R.Nandakumar @ 10:06 PM
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