On 2017-05-29 05:48, James Propp 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":
The tiles of the MIT AI Lab bathroom were just squares, randomly dark and light. I remember because, as I looked at them, they'd spontaneously run a couple of Life generations, so long had I spent watching the CRT. Strangely, the dark ones were 1 and the light ones 0. Maybe the dark ones were sparser? --rwg
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
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.
[ * * * * * * * *
] [ * * * * * * * *
] [ * * * * * * * *
] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * *
] [ * * * * * * * * ] [ * * * * * * * *
] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * *
] [ * * * * * * * *
] [ * * * * * * * *
] [ * * * * * * * *
was * * * * * * * * *
]
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