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Nandakumar R
Nandakumar R
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Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?

Remark: Every trapezoid is affine equivalent to every other trapezoid with same base ratio (ratio between lengths of mutually parallel pair of sides). So one can find keep tiling the plane in parallel strips of same width; indeed, each strip can be tiled with mutually non-congruent trapezoids that all have same height and parallel sides that only need to be some constant ratio. So, the question is whether there is any other solution - perhapsideally, a solution not involving trapezoids?

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?

Remark: Every trapezoid is affine equivalent to every other trapezoid with same base ratio (ratio between lengths of mutually parallel pair of sides). So one can find keep tiling the plane in parallel strips of same width; indeed, each strip can be tiled with mutually non-congruent trapezoids that all have same height and parallel sides that only need to be some constant ratio. So, the question is whether there is any other solution - perhaps, a solution not involving trapezoids?

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?

Remark: Every trapezoid is affine equivalent to every other trapezoid with same base ratio (ratio between lengths of mutually parallel pair of sides). So one can find keep tiling the plane in parallel strips of same width; indeed, each strip can be tiled with mutually non-congruent trapezoids that all have same height and parallel sides that only need to be some constant ratio. So, the question is whether there is any other solution - ideally, a solution not involving trapezoids?

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Nandakumar R
Nandakumar R
  • 6k
  • 3
  • 7
  • 20

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?

Remark: Every trapezoid is affine equivalent to every other trapezoid with same base ratio (ratio between lengths of mutually parallel pair of sides). So one can find keep tiling the plane in parallel strips of same width; indeed, each strip can be tiled with mutually non-congruent trapezoids that all have same height and parallel sides that only need to be some constant ratio. So, the question is whether there is any other solution - and perhaps, a solution not involving trapezoids?

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?

Remark: Every trapezoid is affine equivalent to every other trapezoid with same base ratio (ratio between lengths of mutually parallel pair of sides). So one can find keep tiling the plane in parallel strips of same width; indeed, each strip can be tiled with mutually non-congruent trapezoids that all have same height and parallel sides that only need to be some constant ratio. So, the question is whether there is any other solution - and perhaps, a solution not involving trapezoids?

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?

Remark: Every trapezoid is affine equivalent to every other trapezoid with same base ratio (ratio between lengths of mutually parallel pair of sides). So one can find keep tiling the plane in parallel strips of same width; indeed, each strip can be tiled with mutually non-congruent trapezoids that all have same height and parallel sides that only need to be some constant ratio. So, the question is whether there is any other solution - perhaps, a solution not involving trapezoids?

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Nandakumar R
Nandakumar R
  • 6k
  • 3
  • 7
  • 20

Tiling the plane with quadrilaterals that are mutually non-congruent and affine equivalent

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?

Remark: Every trapezoid is affine equivalent to every other trapezoid with same base ratio (ratio between lengths of mutually parallel pair of sides). So one can find keep tiling the plane in parallel strips of same width; indeed, each strip can be tiled with mutually non-congruent trapezoids that all have same height and parallel sides that only need to be some constant ratio. So, the question is whether there is any other solution - and perhaps, a solution not involving trapezoids?