On Jul 24, 10:18 am, John Stafford <n...@droffats.ten> wrote: > In article > <f1aa9214-a7ca-4553-a7fb-db1c437b1...@c10g2000yqi.googlegroups.com>, > "Tim Golden BandTech.com" <tttppp...@yahoo.com> wrote: > > > > > On Jul 21, 11:29 am, John Stafford <n...@droffats.net> wrote: > > > > There is a strange outlier in crystal formation. Perhaps it is best to > > > call it a quasi-crystalline structure that has a tiling pattern that > > > cannot possibly be built in the traditional atom-to-atom, linear manner > > > (symmetric translation). See the work of Dany Shechtman, 1984. > > > > The point illustrated by this quasi-crystal is that in order to form its > > > five-fold symmetry, all the atoms in the solution would have to > > > simultaneously organize. It's a non-local action. Spooky stuff, as the > > > man said. > > > I found a SIAM article covering Shechtman's discovery. Pretty neat. > > I've got a copy of Kittel's solid state physics which specifically > > rules out the 5-fold symmetry. I do have a hard time with the Bravais > > breakdown because it seems so cartesian based. I do have an > > alternative lattice style in polysign: > > http://bandtechnology.com/PolySigned/Lattice/Lattice.html. > > > How much of a space can we actually have? Some work that I've done > > exposes that we can have more or less than tradition will allow: > > http://bandtechnology.com/ConicalStudy/conic.html > > Perhaps there is a way around the simultaneous organization > > requirement here. > > > I've never fully followed the crystallographic X-ray patterning, which > > is supposed to be the boon of analysis, even under the Shechtman > > discovery, but am happy to consider that there could be some > > electromagnetics in diffraction that is being overlooked too > > conveniently. We don't see any photograph of the aluminum and > > manganese alloy, which I suppose does not look very impressive. Should > > there be some attempt to grow one of these and see if there is some > > growth pattern? I couldn't find any photos of the material, or even a > > name for it. Didn't work too hard at it though. > > > - Tim > > I found a photo in Roger Penrose's _ Emperor's New Mind_, page 564 in > our library copy. It is early, and different from the later > representations. If you surf for "penrose aluminum-manganese alloy" > (sans quotes), you should come up with some good information. > > I struggle to follow Penrose, but that's my shortcoming. He's a very > good instructor and writer.
I have an old copy, but it is from 1989, and I guess he's added this in a later edition. My copy has only 466 pages so he's added quite alot. I didn't realize such a book could go stale.
Googling for your phrase in quotes I've come to http://intendo.net/penrose/info_4.html whose last figure reminds me of the signon packing pattern. Ah. Here is your reference in a google book: http://books.google.com/books?id=oI0grArWHUMC&pg=PA564 I think its a bit dubious to overlook the rise in freedom in 3D from a 2D model like the tiling. According to polysign a natural five-fold will be in 4D. We can simply project back down to 2D or 3D and have a five-based symmetry, though in terms of depth the lattice is going to look very complicated, except at carefully chosen angles where the lattice will overlap. Also the actual packing arrangement will be packing http://bandtechnology.com/PolySigned/Lattice/P5Signon.gif so that this object centered at the origin has adjacent copies centered on [| 2,0,1,1,1] within the simplex coordinate system, where the bar indicates permutations of this coordinate. So that's alot more than five adjacent neighbors, but projections of this should contain some five- fold kaleidoscopic patterning.
There is an issue embedded in these considerations of traversing dimensions that is not well addressed. The very way that we construct multiple dimensions is problematic. There may be places within existing theory where such dimensional manipulations are going on without being fully addressed.
One simple catch: when we claim to have constructed a two dimensional space (x,y) we accept that x is independent of y, yet we place a dependence on their orientation of being orthogonal to each other in order to claim independence of their values. This marriage of two one dimensional spaces to construct a two dimensional space can be investigated for consistency. I am not in doubt that the positions on the face of a piece of paper can be uniquely addressed this way, but I do find that the formal construction of the cartesian product is not necessary, and so I cast doubt on that construction. At this level of fundamental inquiry even the word 'dimension' deserves careful consideration, and its marriage to the real number is always the context that I use it in, which is important to the discussion I believe.