Penrose tiles are fascinating in that they are a very direct proof that our intuition about geometry can be spectacularly wrong. Who (Roger Penrose excepted) would have guessed that it would be possible to make a set of times so incredibly small and deceptively simple that would have these extraordinary properties.
Every time I read about them or even look at them I'm blown away by this and I end up looking at the patterns (staring, really) for minutes on end to see 'how they work'.
Simply mindblowing, it's like watching an expert juggler make the impossible seem so easy.
> Who (Roger Penrose excepted) would have guessed that it
> would be possible to make a set of times so incredibly
> small and deceptively simple that would have these
> extraordinary properties.
Well, one of those people was my professor in school, Clark Richert, who "discovered" the tiles at just about the same time. Penrose sued, my professor won the case.
I don't find it overly surprising. There are many simple calculation specifications which exhibit chaotic behavior, such as prime numbers and rule 110.
I think those qualify as counterexamples to our intuition as well. That simple rules might lead to complex behavior was basically unknown until not too long ago.
Sure. But specifications and prime numbers as well as rule 101 are not tangible objects.
Rule 101 for instance requires a whole pile of infrastructure to even understand it, a prime number requires you to understand what prime numbers are and chaos theory is something many people claim to understand (you get all kinds of talks about butterflies) but to really grok what 'sensitivity to initial conditions' really means you have to dive pretty deep.
Contrast that with being given a pile of tiles and finding that no matter what you do as long as you lay the tiles according to the (very simple) instructions you can't help but discover that try as you want you can't make a repeating pattern.
Anyway, I'm glad to see you don't find it overly surprising, I'm a bit naive when it comes to things like this and find endless enjoyment in such simplicity.
But isn't there about as much infrastructure in 2-dimensional geometry? There are 5 Euclidean axioms and you probably need some topological axioms too for the notion of neighborhoods and space-filling tilings. As for prime numbers you need 9 axioms. Rule 110 probably requires the least amount of infrastructure of these three examples.
We just happen to live in a space in which we obtain intuition about 2D geometry from very early on.
You need formal axioms to do precise and rigorous mathematics, but you don't need any axioms to see and manipulate geometric patterns in our heads. We seem to naturally posses this ability (probably because our brains use a lot of pattern-matching).
We already have the geometry manipulation program installed in our head, and maybe this program is even based on Euclid's (or Hilbert's) axioms. But our brains just run this program, they can use this abstraction without having been thought geometric axioms beforehand.
No. Geometric intuition is widely believed to be only a very rudimentary native skill. There appear to be structures in our brain that allow us to think spatially [1], but infants are generally unable to perform even very simple geometric reasoning. Our environments are full of geometric theorems however, that our brains become acustomed to within the first few years of our lifes.
When I was looking at the sample of tiles, I got the same feeling I get when I'm in total darkness. My brain is trying to organize and compartmentalize it but none of the available templates are working, so it just tries them over and over again.
Penrose gave a talk at the Royal Institution about his periodic filings. Well worth a watch: http://youtu.be/th3YMEamzmw
It's light on math and heavy on hand-drawn transparencies being lined up on an overhead projector. He has these inscrutable registration marks that he matches by hand -- reminds one of the alien code in the movie Contact.
One of my favorite parts concerns some mysterious scribblings by Kepler that seem to presage his work!
> ... Shechtman experienced hostility from him toward the non-periodic interpretation. "For a long time it was me against the world," he said. "I was a subject of ridicule and lectures about the basics of crystallography. The leader of the opposition to my findings was the two-time Nobel Laureate Linus Pauling, the idol of the American Chemical Society and one of the most famous scientists in the world. For years, 'til his last day, he fought against quasi-periodicity in crystals... Linus Pauling is noted saying "There is no such thing as quasicrystals, only quasi-scientists."
As a result, quasicrystal coatings have found their way into nonstick cookware.
I remember reading about these pans back a decade or so (Sitram Cybernox). At the time I tried to find a place where I could actually test one out (they got mixed reviews), but wasn't able to, and eventually just got good at cooking on stainless & cast iron. It looks like their site is still up but I recall seeing something a couple years ago about the pans in question being discontinued or unavailable.
There's some real rubbish in here. De Moivre had demonstrated the link between the golden ratio and the Fibonacci sequence. Euclid knew the geometry of a pentagon involved the golden ratio. The discovery that a new geometrical structure encompassing 5-symmetry threw up the three of them isn't surprising. It would be surprising if it didn't.
> Given that Fibonacci seems to appear everywhere in nature—from pineapples to rabbit populations—it was all the more odd that the ratio was fundamental to a tiling system that appeared to have nothing to do with the physical world. Penrose had created a mathematical novelty, something intriguing precisely because it didn’t seem to work the way nature does. It was as if he wrote a work of fiction about a new animal species, only to have a zoologist discover that very species living on Earth. In fact, Penrose tiles bridged the golden ratio, the math we invent, and the math in the world around us.
This is nonsense. The Golden Ratio is inherent in the geometry of the regular pentagon, which is the basis for Penrose's tiling.
An easy way to see this is by extending the lines of a regular pentagon to a pentagram; the ratio at which the five lines of a pentagram intersect eachother is exactly the Golden Ratio. In this case, however, the Golden Ratio appears as an exact number, because it's the solution to a geometrical problem. Therefore it has little to do with the Fibonacci sequence, which approaches the Golden Ratio in the limit, which is similar to mechanisms that you see pop up here and there in Nature.
In Nature this limit is only reached in a hypothetical plant growing perfectly undisturbed. Just because such a mechanism will, in the limit, produce ratios very close to (1 + sqrt(5)) / 2, doesn't mean it necessarily has a lot to do with this same number appearing as the exact solution in geometrical shapes with pentagonal angles.
Given that he started out with a five-fold symmetry, and tiles with angles based on the pentagon, it's no surprise that the Golden Ratio will pop up everywhere.
I love the Penrose tiling system for all its weird and quirky properties, but trying to draw a connection with the appearance of Fibonacci sequences in Nature because it happens to approach the same number as found in the geometry of five-fold symmetries, is going to need a few more arguments than just "it's the same number".
Now, in all fairness, I must add, I don't know everything about Penrose tilings there is to know. And there might be certain properties of this tiling that give rise to the Golden Ratio in the limit in a manner that is grounded in the Fibonacci sequence rather than the ratios in five-fold angles. Then still, in the spirit of scientific/mathematical honesty, it's good to draw a clear distinction between these two. For instance:
"the ratio of the area of the kite to that of the dart is the golden ratio. The ratio of the longer side of the kite to its shorter side is also the golden ratio" -- this is clearly a result of the geometrical properties of the shapes, their five-fold angles, and relation to the pentagram. As is also demonstrated by the fact that these ratios are exactly (1 + sqrt(5)) / 2, no limits to infinity, it's just the straight answer to a geometrical question.
Another property, however:
"[In an infinite Penrose tiling] the ratio of darts to kites is identical to the ratio of kites to the total number of tiles." -- this happens in a limit to infinity, and is a number approaching the Golden Ratio, similar to the Fibonacci sequence, which may (or not) be grounded in the same mechanisms. You still need to make a good argument for that case, but unlike the above geometrical properties, the possibility is there. It could, however also have to do with something yet even different, such as the Golden Ratio base numbering system ( https://en.wikipedia.org/wiki/Golden_ratio_base ).
Finally, there could be a deeper reason why these things are all connected, from the exact number appearing in geometry, to the appearance of the Fibonacci sequence in Nature, and with the Penrose tiling as an important link in between these two. Maybe. Who knows. But you're going to need to come with a better argument than "the exact solution to one is approached in the limit by another and it's the same number". Just because it's an irrational number? If it was a rational or an integer, nobody would be arguing that all places where the same small number crops up, in Nature, mathematics and geometry would somehow be intimately and mystically interconnected. Well, unless you're a Discordian, of course:
The Law of Fives states simply that: ALL THINGS HAPPEN IN FIVES, OR ARE
DIVISIBLE BY OR ARE MULTIPLES OF FIVE, OR ARE SOMEHOW DIRECTLY OR
INDIRECTLY APPROPRIATE TO 5.
The Law of Fives is never wrong.
In the Erisian Archives is an old memo from Omar to Mal-2: "I find the Law
of Fives to be more and more manifest the harder I look."
http://www.principiadiscordia.com/book/23.php
Every time I read about them or even look at them I'm blown away by this and I end up looking at the patterns (staring, really) for minutes on end to see 'how they work'.
Simply mindblowing, it's like watching an expert juggler make the impossible seem so easy.