[math-fun] Delian Brick, a 3D 2-rep-tile
A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile. https://math.stackexchange.com/questions/2822566/ Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}] 3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare. I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before? --Ed Pegg Jr
Funny you should mention this; a few weeks ago I was reading in a post by Joel Hamkins about the 1-by-2-by-4 bricks that apparently are used in math education, and I read the claim that you can use two of these bricks to make a scaled up brick of the same kind, and I thought to myself “No, that would be a 1-by-2^(1/3)-by-2^(2/3) brick”. Jim Propp On Wednesday, June 20, 2018, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile.
https://math.stackexchange.com/questions/2822566/
Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before?
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This should be the standard for packing and shipping boxes, instead of all the odd sizes we have that don't pack together nicely. Maybe shrunk by a tiny (and scaled!) bit so the boxes could be nested like fractal matryoshka dolls. This could be bigger than the 1:4:9 monolith was to the apes in 2001. On Wed, Jun 20, 2018 at 8:12 AM James Propp <jamespropp@gmail.com> wrote:
Funny you should mention this; a few weeks ago I was reading in a post by Joel Hamkins about the 1-by-2-by-4 bricks that apparently are used in math education, and I read the claim that you can use two of these bricks to make a scaled up brick of the same kind, and I thought to myself “No, that would be a 1-by-2^(1/3)-by-2^(2/3) brick”.
Jim Propp
On Wednesday, June 20, 2018, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile.
https://math.stackexchange.com/questions/2822566/
Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before?
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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It is, of course, the proper three-dimensional generalisation of the aspect ratio of international standard paper sizes: https://en.wikipedia.org/wiki/Paper_size#A_series The D0 box size should have a volume of 1 m^3, and therefore be 2^(-1/3) by 2^0 by 2^(1/3) approxeq 0.7937 * 1.0000 * 1.2599. Each subsequent box size has half the volume of the previous one. (This is actually more elegant in odd dimensions d, because the median of the side-lengths is equal to their geometric mean, and therefore equal to the dth root of the volume.) Best wishes, Adam P. Goucher
Sent: Wednesday, June 20, 2018 at 5:53 PM From: "Tomas Rokicki" <rokicki@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Delian Brick, a 3D 2-rep-tile
This should be the standard for packing and shipping boxes, instead of all the odd sizes we have that don't pack together nicely. Maybe shrunk by a tiny (and scaled!) bit so the boxes could be nested like fractal matryoshka dolls.
This could be bigger than the 1:4:9 monolith was to the apes in 2001.
On Wed, Jun 20, 2018 at 8:12 AM James Propp <jamespropp@gmail.com> wrote:
Funny you should mention this; a few weeks ago I was reading in a post by Joel Hamkins about the 1-by-2-by-4 bricks that apparently are used in math education, and I read the claim that you can use two of these bricks to make a scaled up brick of the same kind, and I thought to myself “No, that would be a 1-by-2^(1/3)-by-2^(2/3) brick”.
Jim Propp
On Wednesday, June 20, 2018, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile.
https://math.stackexchange.com/questions/2822566/
Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before?
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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And one of the faces is very close (within 2%) of a golden rectangle (easily close enough for the eye). Using Adam's D0, every box will have one side an exact power of two times a meter. How many sheets of A0 fit inside such a box of D0, with optimal packing and keeping the sheets perfectly flat? On Thu, Jun 21, 2018 at 9:39 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
It is, of course, the proper three-dimensional generalisation of the aspect ratio of international standard paper sizes:
https://en.wikipedia.org/wiki/Paper_size#A_series
The D0 box size should have a volume of 1 m^3, and therefore be 2^(-1/3) by 2^0 by 2^(1/3) approxeq 0.7937 * 1.0000 * 1.2599. Each subsequent box size has half the volume of the previous one.
(This is actually more elegant in odd dimensions d, because the median of the side-lengths is equal to their geometric mean, and therefore equal to the dth root of the volume.)
Best wishes,
Adam P. Goucher
Sent: Wednesday, June 20, 2018 at 5:53 PM From: "Tomas Rokicki" <rokicki@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Delian Brick, a 3D 2-rep-tile
This should be the standard for packing and shipping boxes, instead of all the odd sizes we have that don't pack together nicely. Maybe shrunk by a tiny (and scaled!) bit so the boxes could be nested like fractal matryoshka dolls.
This could be bigger than the 1:4:9 monolith was to the apes in 2001.
On Wed, Jun 20, 2018 at 8:12 AM James Propp <jamespropp@gmail.com> wrote:
Funny you should mention this; a few weeks ago I was reading in a post by Joel Hamkins about the 1-by-2-by-4 bricks that apparently are used in math education, and I read the claim that you can use two of these bricks to make a scaled up brick of the same kind, and I thought to myself “No, that would be a 1-by-2^(1/3)-by-2^(2/3) brick”.
Jim Propp
On Wednesday, June 20, 2018, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile.
https://math.stackexchange.com/questions/2822566/
Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before?
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Great observations! That depends on the thickness. I guess the precise way of asking Tom's question is: "What is the supremum of values H such that we can fit a union of 2^(-1/4) by 2^(1/4) by h_i cuboids inside a 2^(-1/3) by 2^0 by 2^(1/3) bounding box, where H = h_1 + ... + h_n is the total depth of paper used?" The trivial bounds are 2^(-1/3) <= H <= 1, where the former is attained by a single stack of paper and the latter is a volume bound. Best wishes, Adam P. Goucher
Sent: Thursday, June 21, 2018 at 5:45 PM From: "Tomas Rokicki" <rokicki@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Delian Brick, a 3D 2-rep-tile
And one of the faces is very close (within 2%) of a golden rectangle (easily close enough for the eye).
Using Adam's D0, every box will have one side an exact power of two times a meter.
How many sheets of A0 fit inside such a box of D0, with optimal packing and keeping the sheets perfectly flat?
On Thu, Jun 21, 2018 at 9:39 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
It is, of course, the proper three-dimensional generalisation of the aspect ratio of international standard paper sizes:
https://en.wikipedia.org/wiki/Paper_size#A_series
The D0 box size should have a volume of 1 m^3, and therefore be 2^(-1/3) by 2^0 by 2^(1/3) approxeq 0.7937 * 1.0000 * 1.2599. Each subsequent box size has half the volume of the previous one.
(This is actually more elegant in odd dimensions d, because the median of the side-lengths is equal to their geometric mean, and therefore equal to the dth root of the volume.)
Best wishes,
Adam P. Goucher
Sent: Wednesday, June 20, 2018 at 5:53 PM From: "Tomas Rokicki" <rokicki@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Delian Brick, a 3D 2-rep-tile
This should be the standard for packing and shipping boxes, instead of all the odd sizes we have that don't pack together nicely. Maybe shrunk by a tiny (and scaled!) bit so the boxes could be nested like fractal matryoshka dolls.
This could be bigger than the 1:4:9 monolith was to the apes in 2001.
On Wed, Jun 20, 2018 at 8:12 AM James Propp <jamespropp@gmail.com> wrote:
Funny you should mention this; a few weeks ago I was reading in a post by Joel Hamkins about the 1-by-2-by-4 bricks that apparently are used in math education, and I read the claim that you can use two of these bricks to make a scaled up brick of the same kind, and I thought to myself “No, that would be a 1-by-2^(1/3)-by-2^(2/3) brick”.
Jim Propp
On Wednesday, June 20, 2018, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile.
https://math.stackexchange.com/questions/2822566/
Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before?
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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For rectangles into similar rectangles and paper sizes, I recently found an amazing A-size paper dissection. It's at the following like, along with many other rectangle dissections with various ratios. These are irreptiles here, where all shapes are similar but with different sizes. https://math.stackexchange.com/questions/2709153/ On Thu, Jun 21, 2018 at 12:00 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Great observations!
That depends on the thickness. I guess the precise way of asking Tom's question is:
"What is the supremum of values H such that we can fit a union of 2^(-1/4) by 2^(1/4) by h_i cuboids inside a 2^(-1/3) by 2^0 by 2^(1/3) bounding box, where H = h_1 + ... + h_n is the total depth of paper used?"
The trivial bounds are 2^(-1/3) <= H <= 1, where the former is attained by a single stack of paper and the latter is a volume bound.
Best wishes,
Adam P. Goucher
Sent: Thursday, June 21, 2018 at 5:45 PM From: "Tomas Rokicki" <rokicki@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Delian Brick, a 3D 2-rep-tile
And one of the faces is very close (within 2%) of a golden rectangle (easily close enough for the eye).
Using Adam's D0, every box will have one side an exact power of two times a meter.
How many sheets of A0 fit inside such a box of D0, with optimal packing and keeping the sheets perfectly flat?
On Thu, Jun 21, 2018 at 9:39 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
It is, of course, the proper three-dimensional generalisation of the aspect ratio of international standard paper sizes:
https://en.wikipedia.org/wiki/Paper_size#A_series
The D0 box size should have a volume of 1 m^3, and therefore be 2^(-1/3) by 2^0 by 2^(1/3) approxeq 0.7937 * 1.0000 * 1.2599. Each subsequent box size has half the volume of the previous one.
(This is actually more elegant in odd dimensions d, because the median of the side-lengths is equal to their geometric mean, and therefore equal to the dth root of the volume.)
Best wishes,
Adam P. Goucher
Sent: Wednesday, June 20, 2018 at 5:53 PM From: "Tomas Rokicki" <rokicki@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Delian Brick, a 3D 2-rep-tile
This should be the standard for packing and shipping boxes, instead of all the odd sizes we have that don't pack together nicely. Maybe shrunk by a tiny (and scaled!) bit so the boxes could be nested like fractal matryoshka dolls.
This could be bigger than the 1:4:9 monolith was to the apes in 2001.
On Wed, Jun 20, 2018 at 8:12 AM James Propp <jamespropp@gmail.com> wrote:
Funny you should mention this; a few weeks ago I was reading in a post by Joel Hamkins about the 1-by-2-by-4 bricks that apparently are used in math education, and I read the claim that you can use two of these bricks to make a scaled up brick of the same kind, and I thought to myself “No, that would be a 1-by-2^(1/3)-by-2^(2/3) brick”.
Jim Propp
On Wednesday, June 20, 2018, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile.
https://math.stackexchange.com/questions/2822566/
Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before?
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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* Ed Pegg Jr <ed@mathpuzzle.com> [Jun 21. 2018 10:53]:
[...]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
Is this really a rep-tile? I thought for those you take a set of affine maps map_i(v) := t_i + M * v where M is a rotation (the same for all maps!) and only the translations t_i are different. I created a few 3D fractals using 8 such maps a while ago, using either cubes, rhombic dodecahedra, or truncated octahedra. These things a lattice tilings. Not all of them are genus 0 in the limit. Best regards, jj
[...]
I had not heard the requirement that M be the same for all the maps. I assume the rest of the definition is that map_i(R) and map_j(R) have disjoint interiors for all i,j < n, and that union (i < n) (map_i(R)) = R. If that's the case, then we also need a scaling factor k, so map(v) = t + M(kv). If the dimension of the containing space in D, then k = (1/n) ^ (1/D) (or something like that). On Thu, Jun 21, 2018 at 5:30 AM, Joerg Arndt <arndt@jjj.de> wrote:
* Ed Pegg Jr <ed@mathpuzzle.com> [Jun 21. 2018 10:53]:
[...]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
Is this really a rep-tile? I thought for those you take a set of affine maps map_i(v) := t_i + M * v where M is a rotation (the same for all maps!) and only the translations t_i are different.
I created a few 3D fractals using 8 such maps a while ago, using either cubes, rhombic dodecahedra, or truncated octahedra. These things a lattice tilings.
Not all of them are genus 0 in the limit.
Best regards, jj
[...]
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Ah, OK: my restriction (same M) is for lattice tilings. This reduced the search space from "impossible" to "still hard work" for my computer assisted search. Best regards, jj * Allan Wechsler <acwacw@gmail.com> [Jun 22. 2018 07:32]:
I had not heard the requirement that M be the same for all the maps. I assume the rest of the definition is that map_i(R) and map_j(R) have disjoint interiors for all i,j < n, and that union (i < n) (map_i(R)) = R. If that's the case, then we also need a scaling factor k, so map(v) = t + M(kv). If the dimension of the containing space in D, then k = (1/n) ^ (1/D) (or something like that).
On Thu, Jun 21, 2018 at 5:30 AM, Joerg Arndt <arndt@jjj.de> wrote:
* Ed Pegg Jr <ed@mathpuzzle.com> [Jun 21. 2018 10:53]:
[...]
3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare.
Is this really a rep-tile? I thought for those you take a set of affine maps map_i(v) := t_i + M * v where M is a rotation (the same for all maps!) and only the translations t_i are different.
I created a few 3D fractals using 8 such maps a while ago, using either cubes, rhombic dodecahedra, or truncated octahedra. These things a lattice tilings.
Not all of them are genus 0 in the limit.
Best regards, jj
[...]
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participants (6)
-
Adam P. Goucher -
Allan Wechsler -
Ed Pegg Jr -
James Propp -
Joerg Arndt -
Tomas Rokicki