Impossible Cookware and Other Triumphs of the Penrose Tile
nautil.us
nautil.us
These were studied in depth by Peter J. Lu in 2007.
http://www.peterlu.org/content/decagonal-and-quasicrystallin...
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
I would be in the market for some of this cookware, but I suspect that it is apocryphal.
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.
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)
Given that, these are not Penrose but check out these floor tiles.
https://officeinsight.com/officenewswire/tarkett-officially-...
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.
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.
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.
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.