> 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