[math-fun] A COMPLETELY different kind of "squared square"
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice. I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares. (If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that N^2 = A^2 + B^2 for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square. Can there be a smaller NxN squared torus than the smallest squared square? But how about the famous expression 1^2 + 2^2 + 3^2 + ... + L^2 = N^2 for L = 24 and N = 70, the unique nontrivial such equation. QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ??? Possibly in a non-parallel fashion? —Dan
Any square torus of side C can certainly be tiled with can certainly be tiled with two squares, one of side A and one of side B, as long as A^2 + B^2 = C^2. There are probably similar examples for any higher number of squares as well. On Fri, Jul 3, 2020 at 11:48 AM Dan Asimov <dasimov@earthlink.net> wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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Is love to see a picture of that torus tiling On Fri, Jul 3, 2020, 10:48 AM Dan Asimov <dasimov@earthlink.net wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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I am not good at putting together images. But when you unwrap this torus into a tiling of the plane, you get these guys: https://en.wikipedia.org/wiki/Pythagorean_tiling On Fri, Jul 3, 2020 at 12:53 PM Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Is love to see a picture of that torus tiling
On Fri, Jul 3, 2020, 10:48 AM Dan Asimov <dasimov@earthlink.net wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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Lots of cubes can be represented as the sum of three cubes (https://oeis.org/A023042); which of these extend to tilings of 3-space? On Fri, Jul 3, 2020 at 11:58 AM Allan Wechsler <acwacw@gmail.com> wrote:
I am not good at putting together images. But when you unwrap this torus into a tiling of the plane, you get these guys: https://en.wikipedia.org/wiki/Pythagorean_tiling
On Fri, Jul 3, 2020 at 12:53 PM Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Is love to see a picture of that torus tiling
On Fri, Jul 3, 2020, 10:48 AM Dan Asimov <dasimov@earthlink.net wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
If none of those work, how about four or more? On Fri, Jul 3, 2020 at 1:46 PM Mike Stay <metaweta@gmail.com> wrote:
Lots of cubes can be represented as the sum of three cubes (https://oeis.org/A023042); which of these extend to tilings of 3-space?
On Fri, Jul 3, 2020 at 11:58 AM Allan Wechsler <acwacw@gmail.com> wrote:
I am not good at putting together images. But when you unwrap this torus into a tiling of the plane, you get these guys: https://en.wikipedia.org/wiki/Pythagorean_tiling
On Fri, Jul 3, 2020 at 12:53 PM Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Is love to see a picture of that torus tiling
On Fri, Jul 3, 2020, 10:48 AM Dan Asimov <dasimov@earthlink.net wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
I take back my intemperate guess that a square toroidal dissection into _any_ number of distinct squares is possible. In fact I don't know what the next smallest number of parts is, after 1 and 2. On Fri, Jul 3, 2020 at 1:58 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am not good at putting together images. But when you unwrap this torus into a tiling of the plane, you get these guys: https://en.wikipedia.org/wiki/Pythagorean_tiling
On Fri, Jul 3, 2020 at 12:53 PM Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Is love to see a picture of that torus tiling
On Fri, Jul 3, 2020, 10:48 AM Dan Asimov <dasimov@earthlink.net wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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Here's a site dedicated to squaring the torus: http://www.squaring.net/sq/spt/st/st.html On Fri, Jul 3, 2020 at 1:48 PM Allan Wechsler <acwacw@gmail.com> wrote:
I take back my intemperate guess that a square toroidal dissection into _any_ number of distinct squares is possible. In fact I don't know what the next smallest number of parts is, after 1 and 2.
On Fri, Jul 3, 2020 at 1:58 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am not good at putting together images. But when you unwrap this torus into a tiling of the plane, you get these guys: https://en.wikipedia.org/wiki/Pythagorean_tiling
On Fri, Jul 3, 2020 at 12:53 PM Paul Palmer <paul.allan.palmer@gmail.com> wrote:
Is love to see a picture of that torus tiling
On Fri, Jul 3, 2020, 10:48 AM Dan Asimov <dasimov@earthlink.net wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
Thanks, Mike -- I had actually found that already. He does not show any other perfect squared *square* tori, though, beyond the two types we know already (squared squares, and the Pythagorean tilings). On Fri, Jul 3, 2020 at 3:51 PM Mike Stay <metaweta@gmail.com> wrote:
Here's a site dedicated to squaring the torus: http://www.squaring.net/sq/spt/st/st.html
On Fri, Jul 3, 2020 at 1:48 PM Allan Wechsler <acwacw@gmail.com> wrote:
I take back my intemperate guess that a square toroidal dissection into _any_ number of distinct squares is possible. In fact I don't know what
the
next smallest number of parts is, after 1 and 2.
On Fri, Jul 3, 2020 at 1:58 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am not good at putting together images. But when you unwrap this torus into a tiling of the plane, you get these guys: https://en.wikipedia.org/wiki/Pythagorean_tiling
On Fri, Jul 3, 2020 at 12:53 PM Paul Palmer < paul.allan.palmer@gmail.com> wrote:
Is love to see a picture of that torus tiling
On Fri, Jul 3, 2020, 10:48 AM Dan Asimov <dasimov@earthlink.net wrote:
A squared square <https://en.wikipedia.org/wiki/Squaring_the_square> is an NxN square that's tiled with smaller KxK squares, where no K occurs twice.
I've long wondered about squared *tori*, (using the NxN square torus), which may be easier to find than squared squares.
(If you don't require the K to be different, there is an amusing example of squared tori — where the tiles are all 1x1 squares — for any Pythagorean hypotenuse, i.e., any N such that
N^2 = A^2 + B^2
for integers A, B.) Amusing because the sides of the tiles are not parallel to those of the NxN square.
Can there be a smaller NxN squared torus than the smallest squared square?
But how about the famous expression
1^2 + 2^2 + 3^2 + ... + L^2 = N^2
for L = 24 and N = 70, the unique nontrivial such equation.
QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ≤ K ≤ 24 ???
Possibly in a non-parallel fashion?
—Dan
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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participants (4)
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Allan Wechsler -
Dan Asimov -
Mike Stay -
Paul Palmer