Today was the first perfect spring day, as bright as a day can get, and Stockholmers walked about their city as if they had just been released from a dark room, squinting, and a little dazed. Tourists were easy to spot — they were the ones dressed rationally, whereas the locals were shedding layers like eskimos in a sauna. And not a leaf on the trees yet, though that should change within hours.
Waiting for a train at Stockholm Södra Friday night, I started staring at the platform pavement. Penrose tiles! These were invented (or discovered, depending on your take on mathematics) by Roger Penrose: He figured out how to cover a surface with only two different kinds of tiles in a way that never repeats. It sounds impossible, but it’s not. Penrose used two kinds of rhombi, and then added some specific instructions as to how they are supposed to be placed next to each other. The result is a complex mathematical structure made from simple rules, putting it in the same seductive league as the Mandelbrot set and Wolfram’s Rule 30.
In Södra, the rhombi are disguised, but cleverly so: By curving the edges in specific ways, Penrose’s rules are embodied in the shapes: It was impossible for the workmen building the platform to accidentally create a repeating pattern (unless they were to use only one shape exclusively, I think, but I am not sure about that, and need to go have another look).
I started to wonder whether for each separate segment of pavement covered in tiles, the ratio of the frequency of the two shapes was similar. Luckily, the train arrived. Just now, I found out that for large enough areas, this ratio converges on the golden mean. That is deep in ways I have not yet begun to fathom.
I love public architecture that strives for obscure details that hide underfoot, literally. Next time, I will take my camera, take a picture, and post it here.
If the ratio converges to the golden ratio, which is irrational, you’ll know the pattern doesn’t repeat itself. You’d get a rational number if it did. I can’t remember if there are other two-tile tilings with different ratios, although I think there should be.
Roger Penrose sued Kimberly-Clark for using the Penrose Pattern, or something mathmatically indistinguishable, on its toilet paper. Apparently non-repeating soft and quilty is fluffier than the regular sort and also doesn’t bunch on the roll, which is what would happen if the pattern repeated.
Maths, its just gone down the toilet since Bertrand Russell.
Just to save everyone time, here’s where you heard all of this before:
[Fri, Nov 29 2002 – 16:27] Matthew (email) one of roger penrose’s signature achievements was the development of a non-repeating helix pattern which he claimed kimberly-clark stole for its extra-soft, three-ply toilet paper. the non-repeating pattern means the paper doesn’t bunch on the roll, apparently. he sued after spotting his pattern as it was about to be used. true story. don’t know if he won.
[Mon, Dec 02 2002 – 08:44] eurof (email) so let me get this straight. there he was standing with his trousers round his ankles and no doubt a big poo floating in the toilet, and he takes a strip of bogpaper and before he wipes his bum takes a good hard look at it and goes “but wait! that’s my special non-repeating helix pattern!! the thieving swine!!”
no way. i’d reckon you have to get really close before you see the ply. and no-one looks at bogpaper before it’s used. what WAS he doing? it could be really sick.
[Mon, Dec 02 2002 – 09:01] Charles Kenny (www) (email) Maybe Penrose has perfected the art of reading braille with his bum, and so immediately detected the unique pattern as he was wiping his buttocks with the paper? On another note, what has Penrose got to do with (1) Aristotle, (2) everything? And, Stefan, why would we be better off with a theory of everything? What has superstring theory done for me lately?
[Mon, Dec 02 2002 – 09:11] Charles Kenny (www) (email) oh, and having just looked at the article, although I haven’t got a clue what she’s going on about, I do now vaguely understand the Penrose link, and I’d also bet it falls squarely into the category of “it’s probably bollox, but she’s a woman and she’s cute” scientific reportage. How sexist, I know.
[Mon, Dec 02 2002 – 10:09] Matthew (www) (email) ye of little faith:
Mathematician Sues Kimberly-Clark Unit Over Its Toilet Paper — At Heart of the Messy Issue Is Tissue’s Quilted Design; Is It the `Penrose Pattern’?
By Matthew Rose
Staff Reporter of The Wall Street Journal
798 words
14 April 1997
The Wall Street Journal
B8A
English
(Copyright (c) 1997, Dow Jones & Company, Inc.)
LONDON — Sir Roger Penrose has seen his work on quantum physics and relativity theory celebrated in countless papers. But it was toilet paper that really got the renowned mathematician’s attention.
When Sir Roger examined the “Kleenex quilted toilet tissue,” made by the British unit of Kimberly-Clark Corp., what he saw was no ordinary piece of toilet paper. Embossed on the surface he discovered a series of interlocking diamonds. They bore an uncanny resemblance to “the Penrose Pattern,” a highly complex geometric formula he devised in the 1970s to prove that a nonrepeating pattern could exist, solving one of the great conundrums of the natural world.
“He wasn’t pleased,” says Sir Roger’s lawyer, Richard Kempner a partner at Addleshaw Booth & Co in Leeds, England. So, Sir Roger and Pentaplex Ltd., the Yorkshire, England, company that owns the licensing rights to his work, are going after the toilet paper with court papers, having sued Kimberly-Clark Ltd. for breach of copyright in the High Court in London.
“When it comes to the population of Great Britain being invited by a multinational to wipe their bottoms on what appears to be the work of a Knight of the Realm without his permission, then a last stand must be taken,” said David Bradley, a director of Pentaplex Ltd., in a statement.
A spokeswoman for Kimberly-Clark declined to comment, saying the company hadn’t seen the entire lawsuit, which was filed on April 4. (The right to manufacture the Kleenex toilet tissue brand was sold last year to Swedish pulp and paper company Svenska Cellulosa AB. The plaintiffs say they may seek redress from SCA after the current case is resolved. SCA couldn’t be reached for comment.)
From a scientific standpoint, the Penrose Pattern was a critical breakthrough. Scientists had argued it was impossible to create a predictable pattern using adjacent geometric shapes that would never repeat itself. The best previous effort was a pattern that used several hundred thousand different shapes.
Sir Roger, a mathematics professor at Oxford University, using a notebook and pencil managed to create a nonrepeating pattern using only two different shapes: a fat diamond and a thin diamond. Some of the patterns created by his formula appear the same — a five-petaled flower, for example — but if traced far enough, always turn out to be distinguishable.
This may seem like a trivial way to pass time, but in 1974 the discovery was revolutionary. “It created a whole new phase of matter,” says Paul Steinhart, a physics professor at the University of Pennsylvania, who has researched the practical implications of the Penrose Pattern. It created “materials with properties you never thought were possible.”
Simply put, a material such as aluminum is defined by the arrangement of its crystals, which were always thought to exist in repeating patterns. Creating a substance made from “quasi-crystals,” which like the Penrose Pattern are random — or aperiodic — changes its nature, usually making it harder. For example, products containing quasi-crystals make excellent nonscratch coatings for frying pans.
The same logic also makes for better toilet paper. A premium brand launched in the United Kingdom in 1993, Kimberly-Clark’s Kleenex quilted toilet tissue is embossed with a pattern to fluff up the tissue, making it “thicker and softer,” according to company literature.
Sir Roger’s writ argues that making the tissue fluffier allows manufacturers to reduce the amount of paper used on each roll. But if the pattern repeats itself, the tissue would likely bunch up, looking unattractive. That can be corrected using a Penrose-type pattern which lets the paper sit evenly on the roll, the suit contends.
If the plaintiffs win, they can claim damages under British law equal to the Kleenex brand’s U.K. profits. They can also demand that all remaining examples of the toilet paper be destroyed.
Some lawyers, however, argue that this area of law is imprecise. Distinguishing between an idea, and the expression of that idea is difficult, says Morag Macdonald a partner at Bird & Bird who specializes in intellectual-property law. Similar dilemmas occur in legal disputes over medical breakthroughs and some types of computer-programming language.
Mr. Kempner, Sir Roger’s lawyer, says the patterns are so similar that the burden of proof lies with Kimberly-Clark to prove either that it didn’t copy the pattern, or that it came up with the idea on its own.
But this case has its own special wrinkle, Ms. Macdonald says: “If [the pattern] is nonrepeating, how can they be copying it?”
[Mon, Dec 02 2002 – 11:15] eurof (email) Some journalist you are. You missed the whole point of the story. This would have been my angle. Headline:
Does Eminent Scientist Sniff his Poo from the Paper?
Embarrassed knight: just checking the pattern. A likely story, sneers Kimberly Clark. Lawsuit ongoing.
[Tue, Dec 03 2002 – 06:12] charles No, I believe *you*, Matty (tho’ google search could not find outcome –lexis-nexis, anyone?), but I don’t believe the cute greek physicist has found the answer to everything.
[Tue, Dec 03 2002 – 09:45] Matthew (www) (email) no record of any end to the case in our database or in sec filings. assume it’s still going on or was settled.
A blog that writes itself by reusing simple content, in new, quasi-random patterns? Brilliant, Charles. Just hope Penrose doesn’t sue.
I only have one or two things to say at any point in time. This is one of them. Have forgotten the other.
The other point will be some obscure rambling on word derivation, would be my guess.
ha ha ha! i’m hilarious! that was back when blogging was funny.
This great! I missed all of Eurof’s poo-sniffing jokes the first time round.
Ref: Kepler/Penrose tiling problem.
About 20 yrs ago i painted a picture depicting a solution to the problem of creating a periodic pattern based on the Kepler/Penrose tiles derived from the dissection of a pentagon…this image can be perused on my web-site at:
http://www.peterhugomcclure.com
The image is entitled: ‘Penrose’s Conundrum’
Best regards pete mcclure.
Hi this is pete mcclure again…i do like the dialogue regarding Penrose Tiling Problem…ther seems to be 2 probems concerning “periodic patterns” and “non-periodic patterns”…so there is an infinite array of non-periodic patterns but i have not seen any periodic patterns with the exception of the one i created nearly 20 yrs ago…the question i ask is: are there an infinite number of periodic patterns and if not how many are there ?
So if my image was the first…how is it that i have not got credit…are all the math buffs blinded by the “Emperors New Clothes”.
Have a nice day pete mcclure.
Further to the Penrose Tile Conundrum…
I recently painted a picture which depicts a very simple periodic pattern using the Kepler/Penrose Tiles which is quite amazing as Roger Penrose claims it is IMPOSSIBLE.
You can peruse the image at:
http://fineartamerica.com/featured/periodic-tiling-pattern2014-peter-hugo-mcclure.html
So it is very strange such a simple solution to This Conundrum has remained Hidden from those who refuse to see!
Here’s looking at Eu-Clid… Peter Hugo Mcclure.