A Conversation for SashaQ's NaJoPoMo 2017: Q's the Word

SQ NaJoPoMo 2017 - 15: Q is for Quasicrystals

Post 1

SashaQ - happysad

Next in the Q list is A1078346 Quarters - a Drinking Game. That doesn't interest me, particularly, apart from noting that there are quite a few drinking games in the Edited Guide as they are linked to from that one!

After that in the list is A2026766 The Quasi-war Between the United States and France 1797 - 1800. Vintage Florida Sailor from 2004 - very informative about the fledgling US Navy smiley - ok

The main Entry for today, though is A6582675 Quasicrystals from 2005. Not only is the Entry well written, giving a good flavour of the subject in two- and three-dimensions, but also I like it because it reminds me of an article wot I wrote in 2004 to help me describe to people the motivation behind my PhD thesis smiley - biggrin


SQ NaJoPoMo 2017 - 15: Q is for Quasicrystals

Post 2

Gnomon - time to move on

That's an interesting entry on Quasicrystals but it doesn't mention the thing that puzzled Penrose himself.

When you're constructing a non-periodic tiling, as you add pieces you often reach a point in the pattern where you can't proceed. It's like a jigsaw where some of the pieces almost fit. You only realise they're in the wrong place when further along you come to a dead end in which you can't go any further. You have to back-track by a few pieces and start again.

This trial and error process seems to be the only way of creating a large-scale non-periodic pattern. But Penrose couldn't see a way in which this could happen without intelligence, so how do quasi crystals form?

He came up with a solution involving quantum-mechanical collapse of wavefunctions which I didn't understand and then went on to propose it as a way in which a human brain can be intelligent by solving problems which are not mathematically computable. I didn't understand that bit either and reckon that neither did Penrose.


SQ NaJoPoMo 2017 - 15: Q is for Quasicrystals

Post 3

SashaQ - happysad

Yes, that is an interesting point about the Penrose pattern and similar, that it makes a very hard jigsaw!

There are two ways of creating large scale quasiperiodic patterns that I know of - one is Substitution and the other is Projection. The Penrose tiling can be generated from both methods.

Substitution is like fractals, where each tile is divided up into smaller tiles in a certain way, then each small tile is expanded to the size of the original tile. Doing this again and again covers larger and larger areas.

Projection is where a periodic pattern in higher dimensional space is sliced at an irrational angle to create a non-periodic pattern in the lower dimensional space. For example, if I recall correctly, the Penrose tiling is formed on a plane sliced through a 5-dimensional cubic pattern, and the Octagonal Tiling is a plane sliced through a 4-dimensional cubic pattern. A 3-dimensional slice through a 6-dimensional cubic lattice could generate non-periodic solids. Sounds a bit like it could be related to quantum-mechanical collapse of wavefunctions, but I don't know anything about that in relation to what I did.

smiley - biggrin


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