Why does the Goscube fly apart the way it does? Is there some potential energy that got stored in the Goscube in the assembly process (did the prongs have to bend a bit)? And why is it called the "GITD" Goscube? Jim Propp On Fri, Aug 23, 2013 at 7:13 PM, Bill Gosper <billgosper@gmail.com> wrote:
APG> http://cp4space.wordpress.com/2013/08/23/new-aperiodic-monotile/
Warren Smith wrote:
Brilliantly simple! Lovable!
apg>Thank you!
Recently a new phase of matter was discovered. It supposedly appears amorphous to Xray crystallography, but nevertheless exact chemical-composition ratios and "nucleation+growth" behavior are seen. I suppose your sort of lego is one possible explanation of that.
apg>X-ray crystallography would probably detect the `layered' aspect of the tiling from certain angles, but appear amorphous from other angles. Doesn't glass have an exact chemical composition (SiO2, empirically) coupled with an amorphous structure (the crystalline phase being quartz)?
SiO₂ is quartz, quite crystalline. "The most familiar type of glass, used for centuries in windows<http://en.wikipedia.org/wiki/Window>and drinking vessels <http://en.wikipedia.org/wiki/List_of_glassware>, is soda-lime glass<http://en.wikipedia.org/wiki/Soda-lime_glass>, composed of about 75% silica <http://en.wikipedia.org/wiki/Silicon_dioxide>(SiO 2) plus sodium oxide <http://en.wikipedia.org/wiki/Sodium_oxide> (Na2O) from soda ash, lime <http://en.wikipedia.org/wiki/Calcium_oxide> (CaO), and several minor additives." Oddity: Industrial lasers can cut quartz, but not glass.
apg>Aperiodic tilings are observed in quasicrystals. I seem to recall that certain alloys have local icosahedral symmetry, and generalise the Penrose tiling to three dimensions.
wds>(PS. How'd you make the cool graphics?)
apg>Wolfram Mathematica 8. I actually cheated by not including the indentations on the bottom (since they wouldn't be visible from that aspect):
Module[{block = {Cuboid[{-5, -5, 0}, {5, 5, 3}], GeometricTransformation[ Table[If[Abs[i] + Abs[j] <= 3, Cylinder[{{2 i, 2 j, 3}, {2 i, 2 j, 3.5}}, 0.6], {}], {i, -3, 3}, {j, -3, 3}], {{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}]}}, Graphics3D[{Red, block, Green, Translate[block, {10, 0, 0}], Purple, Translate[block, {10, 10, 0}], Yellow, GeometricTransformation[{block, Orange, GeometricTransformation[ block, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {0, 0, 3}}]}, {{{4/5, 3/5, 0}, {-3/5, 4/5, 0}, {0, 0, 1}}, {10, 0, 3}}]}, Boxed -> False]]
I think that Mathematica might be able to export 3D models, although it would be far preferable to rebuild the block in a dedicated CAD package.
Sincerely,
Adam P. Goucher
True and true. I exported the object in http://www.youtube.com/watch?v=pvT5ySMnO8U
as a (hugely "illegal") .STL, which took tedious wizardry on NeilB's part to actually print.
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