[math-fun] Motley torus tilings
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to: Puzzle: Find a tiling of the square torus by two squares of unequal sizes. Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes? Jim Propp
Oops; I just realized that a “squared square” is an answer to the question as asked! I meant to specify (in both the puzzle and the question) that the tiling must not have points where the corners of four squares meet. Jim Propp On Saturday, July 7, 2018, James Propp <jamespropp@gmail.com> wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
And did you mean "squares of two unequal sizes" ? WFL On 7/7/18, James Propp <jamespropp@gmail.com> wrote:
Oops; I just realized that a “squared square” is an answer to the question as asked!
I meant to specify (in both the puzzle and the question) that the tiling must not have points where the corners of four squares meet.
Jim Propp
On Saturday, July 7, 2018, James Propp <jamespropp@gmail.com> wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
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No, I meant two squares of unequal sizes. Jim On Saturday, July 7, 2018, Fred Lunnon <fred.lunnon@gmail.com> wrote:
And did you mean "squares of two unequal sizes" ? WFL
On 7/7/18, James Propp <jamespropp@gmail.com> wrote:
Oops; I just realized that a “squared square” is an answer to the question as asked!
I meant to specify (in both the puzzle and the question) that the tiling must not have points where the corners of four squares meet.
Jim Propp
On Saturday, July 7, 2018, James Propp <jamespropp@gmail.com> wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
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OK then, how about one 1x1 and one 2x2 tile in a rhombus: ^ / + \ / + + + \ \ + + + / \ + / v I think the answer to the second question may well be no: the vague reason being that the space between the two larger tiles cannot be filled by the smallest without repetition? WFL On 7/7/18, James Propp <jamespropp@gmail.com> wrote:
No, I meant two squares of unequal sizes.
Jim
On Saturday, July 7, 2018, Fred Lunnon <fred.lunnon@gmail.com> wrote:
And did you mean "squares of two unequal sizes" ? WFL
On 7/7/18, James Propp <jamespropp@gmail.com> wrote:
Oops; I just realized that a “squared square” is an answer to the question as asked!
I meant to specify (in both the puzzle and the question) that the tiling must not have points where the corners of four squares meet.
Jim Propp
On Saturday, July 7, 2018, James Propp <jamespropp@gmail.com> wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Yes, this is the solution I had in mind. The rhombus is actually a square. Jim Propp On Saturday, July 7, 2018, Fred Lunnon <fred.lunnon@gmail.com> wrote:
OK then, how about one 1x1 and one 2x2 tile in a rhombus: ^ / + \ / + + + \ \ + + + / \ + / v
I think the answer to the second question may well be no: the vague reason being that the space between the two larger tiles cannot be filled by the smallest without repetition?
WFL
On 7/7/18, James Propp <jamespropp@gmail.com> wrote:
No, I meant two squares of unequal sizes.
Jim
On Saturday, July 7, 2018, Fred Lunnon <fred.lunnon@gmail.com> wrote:
And did you mean "squares of two unequal sizes" ? WFL
On 7/7/18, James Propp <jamespropp@gmail.com> wrote:
Oops; I just realized that a “squared square” is an answer to the question as asked!
I meant to specify (in both the puzzle and the question) that the tiling must not have points where the corners of four squares meet.
Jim Propp
On Saturday, July 7, 2018, James Propp <jamespropp@gmail.com> wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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In regard to the second question: Start with the two-square solution. Replace the smaller square with an appropriately-scaled "squared-square" solution. As long as the squared-square has no internal 4-corners points, then the tiling as a whole will not either. This will tile a square torus with many different sized squares. The process may be repeated by doing the same thing to the new smallest square. Tom James Propp writes:
Oops; I just realized that a “squared square” is an answer to the question as asked!
I meant to specify (in both the puzzle and the question) that the tiling must not have points where the corners of four squares meet.
Jim Propp
On Saturday, July 7, 2018, James Propp <jamespropp@gmail.com> wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
Can't you just start with a tiling using identical squares, then replace one or more (but not all) of the squares with four (or nine, or sixteen, etc.) smaller identical squares? Continuing this process, any number of unequal squares can be obtained. You could constrain the sizes so that no square can be tiled by any of the other squares. The same approach would work, but would require that all squares be decomposed, using different component sizes, e.g. a 6x6 square can be decomposed into four 3x3 squares or nine 2x2 squares. Tom James Propp writes:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
Sorry I wasn’t clearer. ALL the (finitely many) squares that tile the torus should be of unequal sizes. Tom’s repeated subdivision method won’t achieve this. Jim On Saturday, July 7, 2018, Tom Karzes <karzes@sonic.net> wrote:
Can't you just start with a tiling using identical squares, then replace one or more (but not all) of the squares with four (or nine, or sixteen, etc.) smaller identical squares? Continuing this process, any number of unequal squares can be obtained.
You could constrain the sizes so that no square can be tiled by any of the other squares. The same approach would work, but would require that all squares be decomposed, using different component sizes, e.g. a 6x6 square can be decomposed into four 3x3 squares or nine 2x2 squares.
Tom
James Propp writes:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
For the puzzle, isn't that the way Adobe PostScript specifies halftones? I preferred using a single rectangle and an offset. Which probably means that if you can decompose a rectangle into squares of unequal sizes, that would solve the problem. On 07-Jul-18 18:08, James Propp wrote:
Inspired by Scott Kim’s work on motley dissections ( http://www.gathering4gardner.org/g4g13-videos/), I offer a puzzle that I know the answer to and ask a question I don’t know the answer to:
Puzzle: Find a tiling of the square torus by two squares of unequal sizes.
Question: Is there a tiling of some (not necessarily square) torus by more than two squares, all of unequal sizes?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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Fred Lunnon -
James Propp -
Mike Speciner -
Tom Karzes