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?
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