

Impossible Cookware and Other Triumphs of the Penrose Tile - dnetesn
http://nautil.us/issue/13/symmetry/impossible-cookware-and-other-triumphs-of-the-penrose-tile

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jacquesm
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.

~~~
justinator
> 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.

His artwork is beautiful, by the way,

[http://www.clarkrichert.com/](http://www.clarkrichert.com/)

~~~
nkoren
Clark Richert's artwork is great. But do you have a source for this Penrose
lawsuit? Google is failing to turn up anything about it.

~~~
jacquesm
[http://science.slashdot.org/comments.pl?sid=148921&cid=12503...](http://science.slashdot.org/comments.pl?sid=148921&cid=12503476)

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taliesinb
Penrose gave a talk at the Royal Institution about his periodic filings. Well
worth a watch: [http://youtu.be/th3YMEamzmw](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!

~~~
thangalin
For those pressed for time, jump to the 26.5-minute mark:

[https://www.youtube.com/watch?v=th3YMEamzmw#t=1596](https://www.youtube.com/watch?v=th3YMEamzmw#t=1596)

Fascinating to see Moiré patterns appear.

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jessriedel
My favorite part of Shechtman's story:

> ... 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."

[https://en.wikipedia.org/wiki/Dan_Shechtman](https://en.wikipedia.org/wiki/Dan_Shechtman)

~~~
mikeash
The quote from Linus Pauling is ironic given his steep descent into kookery
which was well underway by that time.

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tripzilch
> 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](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

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pndmnm
_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.

~~~
matoffk
In general, I think that the connection between Penrose and cookware is
romanticised just for the sake of a pop-sci narrative.

~~~
happyscrappy
It's a good article, you should read it.

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moomin
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.

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jbeda
I wrote some golang code to generate Penrose P2 tiling in svg. Plan is to
laser cut these but haven't had a chance yet.

Code:[https://github.com/jbeda/penrose-svg](https://github.com/jbeda/penrose-
svg)

