[math-fun] Hey kids, bathroom stuff!
I'm pretty sure this is the pattern in Oren's Hummus bathrooms on Castro St; Can you find a short proof or disproof that the dark tiles form a square grid? —Bill (During the 70s, I stared at (slowly, in those days) evolving Life patterns so habitually that the AI Lab bathroom tiles would run a couple of steps when I looked down. But those were square tiles, randomly dark and light.)
I love this tile pattern, which is pretty common in old institutional bathrooms, like in train stations or aging university buildings. When our downstairs bathroom flooded a few years ago, I had the tile replaced with this pattern. The color in the picture is pretty blown out, but in the lower right corner you can see the pattern of black and white hex tiles. If the tiles are regular and the edges of the hexagons have length 1, the diagonal pitch of the black "squares" in one direction is 6 and in the other direction is 4*sqrt(3) -- almost 7. So not squares. On Sun, Jul 21, 2019 at 6:04 PM Bill Gosper <billgosper@gmail.com> wrote:
I'm pretty sure this is the pattern in Oren's Hummus bathrooms on Castro St;
Can you find a short proof or disproof that the dark tiles form a square grid? —Bill (During the 70s, I stared at (slowly, in those days) evolving Life patterns so habitually that the AI Lab bathroom tiles would run a couple of steps when I looked down. But those were square tiles, randomly dark and light.) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
[Math-fun: For pic, see recent mail] Spoiler: Ideal would be a proof that there is no square grid in a honeycomb tiling, analogous to no equilateral triangles in a square grid. (Despite note 3 to 5 o'clock, 6 to 8 o'clock <http://gosper.org/p24backrakes.png>) Rectangle Arithmetic (see pp 12+13 of 14) <http://gosper.org/rectarith12.pdf> And yet contractors are condemned to build rectangular bathrooms. Perhaps they bribe the general contractor to make the house very slightly rhombic. —rwg On Sun, Jul 21, 2019 at 6:03 PM Bill Gosper <billgosper@gmail.com> wrote:
I'm pretty sure this is the pattern in Oren's Hummus bathrooms on Castro St;
Can you find a short proof or disproof that the dark tiles form a square grid? —Bill (During the 70s, I stared at (slowly, in those days) evolving Life patterns so habitually that the AI Lab bathroom tiles would run a couple of steps when I looked down. But those were square tiles, randomly dark and light.)
I see, it's crucial to the puzzle that we can't actually see the boundaries between the white tiles ... otherwise we could count the hexes. However, perhaps you could use the "give" of the tile grout to squeeze the rhumbuses into squares? On Mon, Jul 22, 2019 at 1:26 PM Bill Gosper <billgosper@gmail.com> wrote:
[Math-fun: For pic, see recent mail] Spoiler: Ideal would be a proof that there is no square grid in a honeycomb tiling, analogous to no equilateral triangles in a square grid. (Despite note 3 to 5 o'clock, 6 to 8 o'clock <http://gosper.org/p24backrakes.png>) Rectangle Arithmetic (see pp 12+13 of 14) <http://gosper.org/rectarith12.pdf> And yet contractors are condemned to build rectangular bathrooms. Perhaps they bribe the general contractor to make the house very slightly rhombic. —rwg
On Sun, Jul 21, 2019 at 6:03 PM Bill Gosper <billgosper@gmail.com> wrote:
I'm pretty sure this is the pattern in Oren's Hummus bathrooms on Castro St;
Can you find a short proof or disproof that the dark tiles form a square grid? —Bill (During the 70s, I stared at (slowly, in those days) evolving Life patterns so habitually that the AI Lab bathroom tiles would run a couple of steps when I looked down. But those were square tiles, randomly dark and light.)
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?? No! You got a picture, but de-rezzed? The tile boundaries are nearly washed out in the center, but clear in the lower right, no? Try this: Graphics[{EdgeForm[LightGray], White, Table[RegularPolygon[{3 j/2, \[Sqrt]3 j/2 + k \[Sqrt]3}, 1, 6], {j, 0, 7}, {k, 0, 10}], Brown, Table[RegularPolygon[{3 3 j/3, 4 \[Sqrt]3 j/2 + 4 k \[Sqrt]3}, 1, 6], {j, 0, 3}, {k, 0, 2}]}] (Assuming I haven't erred,) the removal of perspective clearly looks non-square, which is fairly easy to confirm by computing the height of a unit hexagon. (This really was all I was asking the kids.) For math-fun, the problem is to show that no such grid can be square. On 2019-07-22 11:43, Allan Wechsler wrote:
I see, it's crucial to the puzzle that we can't actually see the boundaries between the white tiles ... otherwise we could count the hexes.
However, perhaps you could use the "give" of the tile grout to squeeze the rhumbuses into squares? On a sphere? In French? Don't drink and drive. —rwg
On Mon, Jul 22, 2019 at 1:26 PM Bill Gosper <billgosper@gmail.com> wrote:
[Math-fun: For pic, see recent mail] Spoiler: Ideal would be a proof that there is no square grid in a honeycomb tiling, analogous to no equilateral triangles in a square grid. (Despite note 3 to 5 o'clock, 6 to 8 o'clock <http://gosper.org/p24backrakes.png>) Rectangle Arithmetic (see pp 12+13 of 14) <http://gosper.org/rectarith12.pdf> And yet contractors are condemned to build rectangular bathrooms. Perhaps they bribe the general contractor to make the house very slightly rhombic. —rwg
On Sun, Jul 21, 2019 at 6:03 PM Bill Gosper <billgosper@gmail.com> wrote:
I'm pretty sure this is the pattern in Oren's Hummus bathrooms on Castro St;
Can you find a short proof or disproof that the dark tiles form a square grid? —Bill (During the 70s, I stared at (slowly, in those days) evolving Life patterns so habitually that the AI Lab bathroom tiles would run a couple of steps when I looked down. But those were square tiles, randomly dark and light.)
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participants (4)
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Allan Wechsler -
Bill Gosper -
rwg -
Tom Duff