Talons Philosophy

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Rationalism, the Paradigm Shift, and the Penrose Tile

KITES & DARTS: British mathematician Roger Penrose created a plane of beautiful, endless variation with just two shapes, kites and darts, seen here in blue lines. Image by Dominique Fung

I came across this article in Nautilus that seems to tread some familiar territory in terms of a few of our discussions during metaphysics in the last while.

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.

However, an added piece of this idea relates directly to a concept at the heart of our epistemology unit, as relates to the idea of a paradigm shift:

It was as though Penrose’s fanciful mathematics had forced itself into the natural world. “For 80 years, a crystal was defined as ‘ordered and periodic,’ because all crystals studied from 1912 on were periodic,” Shechtman says. “It wasn’t until 1992 that the International Union of Crystallography established a committee to redefine ‘crystal.’ That new definition is a paradigm shift for crystallography.”

It was more than mere mental inertia that made it so hard to understand and absorb Shechtman’s discovery. Aperiodic crystalline structures weren’t just unfamiliar; they were supposed to be unnatural. Remember that the placement of one Penrose tile can affect things thousands of tiles away—local constraints create global constraints. But if a crystal forms atom by atom, there should be no natural law that would allow for the kind of restrictions inherent to Penrose tiles.

As we continue to argue the merits of empiricism versus rationalism, doesn’t the example of the Penrose tile present a case of rationalism leading the way?


2 Responses to Rationalism, the Paradigm Shift, and the Penrose Tile

  1. Vincent says:

    I feel that the golden number does appear in several ratios, but at the same time I feel that we are searching for that ratio in nature, rather than trying to disprove it. It feels like were saying look at all these examples that follow the golden ration (I totally support the golden ratio ideology, its just that it seems too convenient) and we aren’t pointing out any that disprove it. In science, empiricism, logic, rationalism, and basically everything that creates theories; the goal is to disprove your theory and therefore prove it.

    I don’t know of any examples of this theory appearing false of course, but I don’t know if we have searched for any. Therefore, I wonder, is there anything that disproves this golden number theory, the magical sequence of 0,1,1,2,3,5,8,13,21 . . .?

  2. David Wees says:


    I agree. It is easy to find patterns in nature when we go looking for them, but then to discount the many times when we do not see those patterns. If one does find a pattern, one should ask, is there some reason this pattern exists? And if not, set aside the phenomena and ignore it for now since it is likely just us reading into the world order that does not exist.



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