
Impossible Cookware and Other Triumphs of the Penrose Tile - prostoalex
http://nautil.us/issue/69/patterns/impossible-cookware-and-other-triumphs-of-the-penrose-tile-rp
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ncmncm
Amazingly (anyway I was amazed), Islamic mathematicians thoroughly explored
non-periodic tilings a thousand years ago. Our only records of this are the
tilings on the walls of mosques, and the design books for the tiles. (There
might be records in archives, written in Arabic.)

These were studied in depth by Peter J. Lu in 2007.

[http://www.peterlu.org/content/decagonal-and-
quasicrystallin...](http://www.peterlu.org/content/decagonal-and-
quasicrystalline-tilings-medieval-islamic-architecture)

~~~
microtherion
The explanation I've heard is that because Islam is strongly opposed to
figurative depiction, there was more investment into abstract ornamental art
as a creative outlet.

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BobMackay
My father (Professor Alan Mackay) and I visited Professor Penrose in Oxford in
1975, to discuss his new tiling pattern, and I was able to write a computer
program (in Algol 60) to draw it on a graph plotter. See
[http://bobmackay.com/Penrose/PenroseColour300.jpg](http://bobmackay.com/Penrose/PenroseColour300.jpg)

My father generated a diffraction pattern from this, using light rather than
X-rays, and so was able to predict the pattern eventually discovered by Dan
Shechtman some years later.

You might also be amused by a weave based on the pattern, with threads going
in five different directions:
[http://bobmackay.com/PentaTable/index.html](http://bobmackay.com/PentaTable/index.html)

I would be in the market for some of this cookware, but I suspect that it is
apocryphal.

~~~
alanbernstein
I enjoyed your CNC project gallery. What model is that wooden-frame mill from
the early photos? Do you have your own 3D printer as well?

~~~
BobMackay
It is a BlueChick, from BuildYourCNC.

I recommend it if all you wish to cut is wood. It is not really stiff enough
for aluminum. A great starter machine. Mine is starting to wear out the rails,
so I am in the course of building a new one based on 8020 tubing, which is
going to have a much larger carvable envelope.

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charlysl
Last week I read about this in "Fermat's Last Theorem":

 _While the fascinating thing about Penrose’s tiled surfaces is their
restricted symmetry, the interesting property of modular forms is that they
exhibit infinite symmetry. The modular forms studied by Taniyama and Shimura
can be shifted, switched, swapped, reflected and rotated in an infinite number
of ways and still they remain unchanged, making them the most symmetrical of
mathematical objects. When the French polymath Henri Poincaré studied modular
forms in the nineteenth century, he had great difficulty coming to terms with
their immense symmetry._

(Fermat's Last Theorem was ultimately proven by proving the Taniyama-Shimura
Conjecture)

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tobylane
"The Art of the Impossible: MC Escher and Me - Secret Knowledge" a two video
BBC documentary from Penrose on his work and communications with Escher.

[https://www.youtube.com/watch?v=f7kW8xd8p4s](https://www.youtube.com/watch?v=f7kW8xd8p4s)

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mudil
What cookware on the market actually has quasicrystal coating? Any
recommendations?

~~~
nate_meurer
Only one: Sitram Cybernox, which was discontinued about 5 years ago. They have
a metallic quasicrystal alloy coating applied over a stainless body. They have
a heavy aluminum sandwich base, similar to the Sitram Profiserie line,
possibly thicker. They're very well built, and were positioned as Sitram's top
line.

I have a couple. In terms of non-stickiness, it's somewhere between stainless
and well-seasoned cast iron. Someone expecting it to compete with PTFE
nonstick will be disappointed.

EDIT:

I remembered something. The Cybernox coating is very vulnerable to pitting
corrosion, much more so than stainless steel. Both of my Cybernox pans are
pitted from light use.

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briteside
Anybody manufacture actual Penrose tiles?

~~~
litigilicious
No, if you mean ceramic or stone tiles for floors or walls. I spent a bunch of
time digging into this when I bought my current house because I initially
could not believe that these were unobtanium and that custom cutting would be
the only way to install such a pattern in my bath.

In short, there are 2 reasons for this. The first is that at the physical
scale required for the nifty aperiodicity of the tiling to be apparent in a
typical home (<= ~100cm^2 or 16 in^2, aka the areal size of a "standard
bathroom tile" in North America,) tiles are typically sold and installed not
individually but in mats of many tiles adhered to a backing webbing. This is
not possible with an aperiodic tile pattern where the pattern does not, by
definition, repeat predictably.

So that's the first reason: practicality.

The second reason is exactly what you might expect if you have been around the
sun more than 2 dozen times: Roger Penrose is notoriously litigious. He
patented the aperiodic tilings he "discovered" in the late 70s, but famously
sued Kimberly-Clark for making toilet tissue with one of these tilings in the
90s claiming _copyright_ violation - and won. Even though the patent is long
expired, copyright lives longer.

Ironically, given that the infringing bog rolls were almost certainly roller-
embossed, Kimberly-Clark's Kompetent Counsel seems to have missed a trick -
their expression was NOT strictly a Penrose tiling as they are, by definition,
aperiodic. You can't emboss a continuous Penrose tiling from a roller.

~~~
tropo
Oh, there is another way to do this with webbing.

The tiling can be recursively generated. (called "inflation" and "deflation")
That is, starting from a Penrose tiling, another one with smaller tiles can be
generated. This means that the patches of webbing can be tiled as Penrose
tiles, yet be subdivided such that the result is a valid Penrose tiling with
smaller tiles.

In case you are trying to imagine this... good luck. The recursive generation
does not retain the large tile divisions as small tile divisions. The large
tile divisions become jagged edges. This is fine; the tile industry already
tolerates this issue with hex tiles on webbing.

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jfengel
The tl;dr on the headline: Penrose tiling -> quasicrystals -> nonstick
cookware.

