6 ms·

Impossible Cookware and Other Triumphs of the Penrose Tile

64 points by prostoalex 8y ago

ncmncm 8y ago
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-quasicrystalline-tilings-medieval-islamic-architecture http://www.peterlu.org/content/decagonal-and-quasicrystallin...
- microtherion 8y ago
  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.
briteside 8y ago
Anybody manufacture actual Penrose tiles?
- jbeda 8y ago
  If you want a toy here is some code to generate SVGs that you can laser cut: https://github.com/jbeda/penrose-svg https://github.com/jbeda/penrose-svg
  - alanbernstein 8y ago
    This is neat, do you have any other projects on tiling, or other plane geometry designs?
- wallflower 8y ago
  I can’t tell if you are talking about puzzle tiles or floor tiles. Given that, these are not Penrose but check out these floor tiles. https://officeinsight.com/officenewswire/tarkett-officially-launches-pentagonals-rubber-flooring-collection/ https://officeinsight.com/officenewswire/tarkett-officially-...
  - burfog 8y ago
    All of us nerdy homeowners are looking for ceramic, marble, granite, quartzite, glass, or similar. We need a few sizes. Our square tiles are 18x18 inches (3/8 thick, 3/8 grout lines) in larger rooms and about 2x2 or 3x3 inches (1/8 thick, 1/8 grout lines) in showers. Tiles of generally similar area would be good. We also need wall tiles for kitchens (behind the sink) and for bathroom walls. The rhombic Penrose tiling is basic and nice. The Voronoi transform of it is better I think, being less likely to chip or injure feet because there are no acute angles.
- doodpants 8y ago
  You can buy some here [0]. Really nifty laser-cut acrylic pieces in a variety of colors, and you can even get a vinyl mat and set of rules for a board game that uses them. (Disclaimer: I personally know the designer of the game, and helped her playtest it.) [0] http://gamepuzzles.com/pentuniv.htm#KD http://gamepuzzles.com/pentuniv.htm#KD
- litigilicious 8y ago
  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 8y ago
    You could do it with webbing. The installer would have to pop a few tiles off the edge to match things up. Those tiles could then be used to fill in gaps. Depending on webbing size and installer skill, perhaps 80% of the tiles could be left attached to the webbing.
  - tropo 8y ago
    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.
jfengel 8y ago
The tl;dr on the headline: Penrose tiling -> quasicrystals -> nonstick cookware.
tobylane 8y ago
"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
charlysl 8y ago
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)
mudil 8y ago
What cookware on the market actually has quasicrystal coating? Any recommendations?
- nate_meurer 8y ago
  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.
BobMackay 8y ago
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 8y ago
  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 8y ago
    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.
  - BobMackay 8y ago
    I don't have a 3d printer - Shapeways is much more convenient and cheaper!

64 points by prostoalex 8y ago

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

microtherion 8y ago
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.

briteside 8y ago

Anybody manufacture actual Penrose tiles?

jbeda 8y ago
If you want a toy here is some code to generate SVGs that you can laser cut: https://github.com/jbeda/penrose-svg https://github.com/jbeda/penrose-svg
- alanbernstein 8y ago
  This is neat, do you have any other projects on tiling, or other plane geometry designs?
wallflower 8y ago
I can’t tell if you are talking about puzzle tiles or floor tiles. Given that, these are not Penrose but check out these floor tiles. https://officeinsight.com/officenewswire/tarkett-officially-launches-pentagonals-rubber-flooring-collection/ https://officeinsight.com/officenewswire/tarkett-officially-...
- burfog 8y ago
  All of us nerdy homeowners are looking for ceramic, marble, granite, quartzite, glass, or similar. We need a few sizes. Our square tiles are 18x18 inches (3/8 thick, 3/8 grout lines) in larger rooms and about 2x2 or 3x3 inches (1/8 thick, 1/8 grout lines) in showers. Tiles of generally similar area would be good. We also need wall tiles for kitchens (behind the sink) and for bathroom walls. The rhombic Penrose tiling is basic and nice. The Voronoi transform of it is better I think, being less likely to chip or injure feet because there are no acute angles.
doodpants 8y ago
You can buy some here [0]. Really nifty laser-cut acrylic pieces in a variety of colors, and you can even get a vinyl mat and set of rules for a board game that uses them. (Disclaimer: I personally know the designer of the game, and helped her playtest it.) [0] http://gamepuzzles.com/pentuniv.htm#KD http://gamepuzzles.com/pentuniv.htm#KD
litigilicious 8y ago
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 8y ago
  You could do it with webbing. The installer would have to pop a few tiles off the edge to match things up. Those tiles could then be used to fill in gaps. Depending on webbing size and installer skill, perhaps 80% of the tiles could be left attached to the webbing.
- tropo 8y ago
  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.

jfengel 8y ago

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

tobylane 8y ago

"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

charlysl 8y ago

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)

mudil 8y ago

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

nate_meurer 8y ago
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.

BobMackay 8y ago

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 8y ago
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 8y ago
  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.
- BobMackay 8y ago
  I don't have a 3d printer - Shapeways is much more convenient and cheaper!