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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