[math-fun] The bathroom square dance?
A well-known tiling of the plane employs offset square tiles of two sizes. See the example below using tiles of 2 and 3 units, somewhat perturbed on this occasion by proportional spacing rather than incompetent home improvement skills. [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] I can't resist remarking that such diagrams provide a particularly elegant demonstration of Pythagoras' theorem, apparent when the period along a horizontal axis is calculated. A particularly humiliating programming exercise involves generating a matrix in which unoccupied cells above are represented by 0 and asterisked cells by +1 or -1 , chosen to satisfy the local constraints on an integer number wall (frame theorems). But all that's completely irrelevant to my purpose, which is simply to enquire if these tilings have an established name --- preferably one rather more euphonious than my current nomenclature: The bathroom floor tiling! Fred Lunnon
This reminds me of PostScript's type 10 halftone dictionaries. In ancient history, when I was implementing halftones in firmware, I realized that any rational-tangent halftone could be represented by a single orthogonal rectangle of values that repeated horizontally but was offset vertically. Later, PostScript 3 appeared with type 10 halftone dictionaries, defined by two orthogonal squares tiling the plane. It took me a little while to realize that these approaches are equivalent. (Halftones are normally defined by a single square of values with some arbitrary angle to the coordinate axes. When the angle has a rational tangent, we can get either of the above techniques.) I guess I didn't answer your question. --ms On 28-May-17 11:13, Fred Lunnon wrote:
A well-known tiling of the plane employs offset square tiles of two sizes. See the example below using tiles of 2 and 3 units, somewhat perturbed on this occasion by proportional spacing rather than incompetent home improvement skills.
[ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ]
I can't resist remarking that such diagrams provide a particularly elegant demonstration of Pythagoras' theorem, apparent when the period along a horizontal axis is calculated.
A particularly humiliating programming exercise involves generating a matrix in which unoccupied cells above are represented by 0 and asterisked cells by +1 or -1 , chosen to satisfy the local constraints on an integer number wall (frame theorems).
But all that's completely irrelevant to my purpose, which is simply to enquire if these tilings have an established name --- preferably one rather more euphonious than my current nomenclature: The bathroom floor tiling!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
As on page 494 of the handy 900-page http://paulbourke.net/dataformats/postscript/psref.pdf for instance. I cannot honestly pretend that "PostScript type 10 halftone dictionary" constitutes a great improvement on "bathroom floor tiling". Where are the ancient Greeks, the Moors (Moroccan, not Yorkshire), the Alhambra when we need them? WFL On 5/28/17, Mike Speciner <ms@alum.mit.edu> wrote:
This reminds me of PostScript's type 10 halftone dictionaries. In ancient history, when I was implementing halftones in firmware, I realized that any rational-tangent halftone could be represented by a single orthogonal rectangle of values that repeated horizontally but was offset vertically. Later, PostScript 3 appeared with type 10 halftone dictionaries, defined by two orthogonal squares tiling the plane. It took me a little while to realize that these approaches are equivalent. (Halftones are normally defined by a single square of values with some arbitrary angle to the coordinate axes. When the angle has a rational tangent, we can get either of the above techniques.)
I guess I didn't answer your question.
--ms
On 28-May-17 11:13, Fred Lunnon wrote:
A well-known tiling of the plane employs offset square tiles of two sizes. See the example below using tiles of 2 and 3 units, somewhat perturbed on this occasion by proportional spacing rather than incompetent home improvement skills.
[ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ]
I can't resist remarking that such diagrams provide a particularly elegant demonstration of Pythagoras' theorem, apparent when the period along a horizontal axis is calculated.
A particularly humiliating programming exercise involves generating a matrix in which unoccupied cells above are represented by 0 and asterisked cells by +1 or -1 , chosen to satisfy the local constraints on an integer number wall (frame theorems).
But all that's completely irrelevant to my purpose, which is simply to enquire if these tilings have an established name --- preferably one rather more euphonious than my current nomenclature: The bathroom floor tiling!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Can a square-tiling of the plane, like a square-tiling of a square, be converted into an electrical network (a la Brooks, Smith, Stone, and Tutte)? That might give a picture that would be suggestive of a different name than "bathroom floor tiling". Incidentally, I've always thought that the tiling of the plane by squares and octagons was the "bathroom floor tiling": https://www.houzz.com/photos/48092404/Victorian-Octagon-With-White-Dot-Porce... Jim Propp On Sun, May 28, 2017 at 7:24 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As on page 494 of the handy 900-page http://paulbourke.net/dataformats/postscript/psref.pdf for instance.
I cannot honestly pretend that "PostScript type 10 halftone dictionary" constitutes a great improvement on "bathroom floor tiling".
Where are the ancient Greeks, the Moors (Moroccan, not Yorkshire), the Alhambra when we need them?
WFL
On 5/28/17, Mike Speciner <ms@alum.mit.edu> wrote:
This reminds me of PostScript's type 10 halftone dictionaries. In ancient history, when I was implementing halftones in firmware, I realized that any rational-tangent halftone could be represented by a single orthogonal rectangle of values that repeated horizontally but was offset vertically. Later, PostScript 3 appeared with type 10 halftone dictionaries, defined by two orthogonal squares tiling the plane. It took me a little while to realize that these approaches are equivalent. (Halftones are normally defined by a single square of values with some arbitrary angle to the coordinate axes. When the angle has a rational tangent, we can get either of the above techniques.)
I guess I didn't answer your question.
--ms
On 28-May-17 11:13, Fred Lunnon wrote:
A well-known tiling of the plane employs offset square tiles of two sizes. See the example below using tiles of 2 and 3 units, somewhat perturbed on this occasion by proportional spacing rather than incompetent home improvement skills.
[ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ]
I can't resist remarking that such diagrams provide a particularly elegant demonstration of Pythagoras' theorem, apparent when the period along a horizontal axis is calculated.
A particularly humiliating programming exercise involves generating a matrix in which unoccupied cells above are represented by 0 and asterisked cells by +1 or -1 , chosen to satisfy the local constraints on an integer number wall (frame theorems).
But all that's completely irrelevant to my purpose, which is simply to enquire if these tilings have an established name --- preferably one rather more euphonious than my current nomenclature: The bathroom floor tiling!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Time to exhume that glorious problem about the infinite chessboard grid of resistors, perhaps? http://www.mathpages.com/home/kmath668/kmath668.htm And get a new bathroom ... WFL On 5/29/17, James Propp <jamespropp@gmail.com> wrote:
Can a square-tiling of the plane, like a square-tiling of a square, be converted into an electrical network (a la Brooks, Smith, Stone, and Tutte)? That might give a picture that would be suggestive of a different name than "bathroom floor tiling".
Incidentally, I've always thought that the tiling of the plane by squares and octagons was the "bathroom floor tiling":
https://www.houzz.com/photos/48092404/Victorian-Octagon-With-White-Dot-Porce...
Jim Propp
On Sun, May 28, 2017 at 7:24 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As on page 494 of the handy 900-page http://paulbourke.net/dataformats/postscript/psref.pdf for instance.
I cannot honestly pretend that "PostScript type 10 halftone dictionary" constitutes a great improvement on "bathroom floor tiling".
Where are the ancient Greeks, the Moors (Moroccan, not Yorkshire), the Alhambra when we need them?
WFL
On 5/28/17, Mike Speciner <ms@alum.mit.edu> wrote:
This reminds me of PostScript's type 10 halftone dictionaries. In ancient history, when I was implementing halftones in firmware, I realized that any rational-tangent halftone could be represented by a single orthogonal rectangle of values that repeated horizontally but was offset vertically. Later, PostScript 3 appeared with type 10 halftone dictionaries, defined by two orthogonal squares tiling the plane. It took me a little while to realize that these approaches are equivalent. (Halftones are normally defined by a single square of values with some arbitrary angle to the coordinate axes. When the angle has a rational tangent, we can get either of the above techniques.)
I guess I didn't answer your question.
--ms
On 28-May-17 11:13, Fred Lunnon wrote:
A well-known tiling of the plane employs offset square tiles of two sizes. See the example below using tiles of 2 and 3 units, somewhat perturbed on this occasion by proportional spacing rather than incompetent home improvement skills.
[ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ]
I can't resist remarking that such diagrams provide a particularly elegant demonstration of Pythagoras' theorem, apparent when the period along a horizontal axis is calculated.
A particularly humiliating programming exercise involves generating a matrix in which unoccupied cells above are represented by 0 and asterisked cells by +1 or -1 , chosen to satisfy the local constraints on an integer number wall (frame theorems).
But all that's completely irrelevant to my purpose, which is simply to enquire if these tilings have an established name --- preferably one rather more euphonious than my current nomenclature: The bathroom floor tiling!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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That is a great problem. Is it possibly related to percolation, where you have an infinite grid of "cells" each of which is "open" or "closed", independently with prob. = p. Instead of open or closed, color the cells black and white. Then they try to answer the question: What is the inf of the p for which there exists, with probability one, an infinite connected component that is all the same color ??? (A component being an equivalence class under the relation: cell A ~ cell B when there is a monochromatic path P starting with cell A and ending with cell B, such that for each cell of P, the next cell is one sharing an edge with it.) E.g., for cells being the tiles of the hexagonal tessellation of the plane, the critical probability is p = 1/2. The field of percolation originated with models for liquid percolating through various types of rock, so maybe there is a connection with networks of electrical resistors. —Dan
On May 29, 2017, at 8:18 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Time to exhume that glorious problem about the infinite chessboard grid of resistors, perhaps?
http://www.mathpages.com/home/kmath668/kmath668.htm <http://www.mathpages.com/home/kmath668/kmath668.htm>
And get a new bathroom ... WFL
On 5/29/17, James Propp <jamespropp@gmail.com <mailto:jamespropp@gmail.com>> wrote:
Can a square-tiling of the plane, like a square-tiling of a square, be converted into an electrical network (a la Brooks, Smith, Stone, and Tutte)? That might give a picture that would be suggestive of a different name than "bathroom floor tiling".
Incidentally, I've always thought that the tiling of the plane by squares and octagons was the "bathroom floor tiling":
Me, I grew up in a house with a bathroom floor tiled by regular hexagons. Many of the bathrooms of that day (1950s) had this pattern (legend has it that this is one reason for the name of the game Hex).
participants (4)
-
Dan Asimov -
Fred Lunnon -
James Propp -
Mike Speciner