Quasi-Cystals like Penrose tiling

[Charles Pegge]

Regular symmetry in higher dimensions.

http://en.wikipedia.org/wiki/Quasicrystal

http://www.google.co.uk/search?hl=en&sugexp=lttmoc,n%3D60&cp=6&gs_id=m&xhr=t&q=quasicrystals&gs_sm=&gs_upl=&bav=on.2,or.r_gc.r_pw.,cf.osb&biw=983&bih=658&um=1&ie=UTF-8&tbm=isch&source=og&sa=N&tab=wi

[kryton9]

Beautiful formations and designs, thanks for the links.

[danbaron]

This subject is somewhat frustrating to me.

It is not apparent to me how a particular set of tiles can be shown to be able to tile the entire plane, and to be aperiodic.

If, for instance, the set consists of one tile, a square, then, I see how it can tile the plane, but, only periodically.

[danbaron]

I'm not saying that I don't believe it, I do.

No one has to convince me about how smart Roger Penrose is.

But, intuitively, I don't see how the first guy could have come up with a set of 20,000 tiles that can aperiodically tile the plane - 20,000?

I would like to be able to instantaneously understand everything that interests me, but, unfortunately, for me that's not how it works.

For a subject like this, I (and, maybe most people) have to weigh my interest against, my limited time and energy, and, the looming possibility that even if I tried to understand the associated mathematics, I could not.

Therefore, the result is some degree of frustration.

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Additionally, I think there is another problem which is common to humanity.

The degree that something has value, is proportional to how much it (or, at least knowledge of it) can be shared with other people.

For instance, my speculation is that Andrew Wiles would not have spent 7 years solving Fermat's Last Theorem, if, he knew beforehand that no one else would ever see his result.

In other words, humans, whether they like it or not, are a social species.

If suddenly everyone disappeared except for one person, then, effectively, that person would own the planet.

But, my guess is that he would be very unhappy, because, he was alone.

How would you like to live solitarily, in a house with a thousand rooms?

[kryton9]

I always think of this quote. It holds true in so many ways often in my life.
"Ignorance is bliss."

[danbaron]

http://en.wikipedia.org/wiki/Penrose_tiling
(http://en.wikipedia.org/wiki/Penrose_tiling)
"It is non-periodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original."

I don't want to try to understand all of the mathematical legalities concerning what constitutes non-periodicity.

But, I'll just go from the above quote.

So, we're talking about no translational symmetry.

If I look at one white square on a chessboard, and one black square, then, the pattern is non-periodic, because, the two squares are different colors.

But, as soon as I look at any three squares, the pattern becomes periodic, because, at least two of the squares are the same color, and all three are translationally symmetric (meaning that they all have the identical angular orientation).

So, concerning Penrose tiles, in which no pattern repeats, I think a question becomes, what constitutes a pattern?

If a pattern consists of a single tile (either a "kite", or a "dart"), then, I would bet everything I have that, Penrose Tiles are not aperiodic.

Because, in that case, it would mean no two of the infinity of kites tiling the plane could have the same angular orientation, and likewise for the darts.