Image by EdPeggJr via Wikimedia Commons

# Breakthrough Pentagon Can Tile the Plane

Image by EdPeggJr via Wikimedia Commons

Ask most elementary school children what the difference between a triangle, a square, and a pentagon is, and they’ll be able to tell you with ease. Shapes are one of the easiest mathematical concepts to grasp, and among the infinite number of possible polygons, shapes with three, four, or five sides are the most basic. However, beyond the simplest, most child-friendly definition of a pentagon—“a shape that has five sides”—lurks a problem complex enough to have stumped mathematicians for nearly a century.

One of the special properties ascribed to triangles and quadrangles (all four-sided shapes, including squares, rectangles, rhombuses, and parallelograms) is their ability to “tile the plane,” i.e. perfectly cover a flat surface, leaving no gaps and creating no overlaps between each identical shape. Finding a real-world example can be as simple as glancing down at the kitchen or bathroom floor, where regular ceramic or linoleum shapes form a smooth, unbroken pattern, sometimes called a tessellation.

Although a regular pentagon (one in which all five sides and all five angles are of equal measure) can’t tile the plane, German mathematician Karl Reinhardt broke new ground in 1918 when he discovered equations for five non-regular pentagons that could, in fact, cover a flat surface sans gaps or overlaps. This introduced the possibility that there might be even more irregular pentagons out there capable of tiling the plane, if only someone could discover them. From 1968 to 1985, various contributors added to the list of tiling pentagons until there were fourteen known varieties. Those fourteen stood alone until a recent breakthrough at the University of Washington Bothell that added a fifteenth.

Married research team Jennifer McLoud-Mann and Casey Mann of the university’s School of Science, Technology, Engineering and Mathematics had been working on pentagon tiling for two years prior to their recent discovery, but it took the special expertise of a third team member to bring the fifteenth pentagon to light.

David Von Derau arrived at the University of Washington Bothell seeking an undergraduate degree, but brought with him years of experience as a professional software developer. McLoud-Mann and Mann recruited him to their project, provided him with their algorithm, and Von Derau programmed a computer to do the necessary calculations. McLoud-Mann had already eliminated a number of false positives—mathematically impossible pentagons or repeats of the 14 previously discovered types—when the computer finally turned out one that was the real deal.

According to Mann, the discovery of a 15th tiling pentagon is as major for mathematicians as creating a new atom would be for physicists. A new tiling shape may lead to developments in biochemistry, architecture, materials engineering, and more. With an infinite number of irregular pentagon forms, there could be an infinite number of them that tile the plane. When asked if the team would continue their potentially never-ending quest for more tiling pentagons, McLoud-Mann admitted she simply didn’t know; after all, working through a problem that never ends must take its toll on even the most dedicated researchers. For anyone willing to take up the mantle, so far that’s 15 pentagons down, possibly infinity more to go.

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Dollar Words: The Logophile Game That Has Math Geeks Hooked, Too
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Besides anagrams and palindromes, if there’s one thing wordplay aficionados like to mess around with, it’s the numerical value of the letters of the alphabet. Assigning numbers to letters—A = 1, B =2, C = 3, and so on, all the way through to Z = 26—opens the alphabet up to all kinds of mathematical and numerical games and trivia.

So add the value of ARM (32) to the value of BEND (25) and you get the value of ELBOW (57). Likewise, WHITE (65) plus HOUSE (68) equals GOVERNMENT (133). HAIR (8, 1, 9, 18) is a palindrome in this A to Z number system, as is INSULINS (9, 14, 19, 21, 12, 9, 14, 19). Add up the neighboring letter pairs in CAN (3 + 1, 1 + 14), and you’ll get DO (4, 15). The letters in FOURTEEN DOZEN add up to 14 dozen (168).

One more game that can be played with the numerical values of the alphabet is to search for words that total a specific value—the holy grail of which is precisely 100. Words that total 100 in this A to Z way are affectionately known as “dollar words.” They’re actually not all that rare in English, and a full list of them includes some fairly familiar words:

ANNUALLY BOUNDARY CULTURE DRIZZLE

MITTENS MOODIEST NASTILY OUTSET

PAYPHONE PORTLAND PREVENT PRIMARY

PRINTER SESSION SOURCES STRESS

STYLES SWIMMER TATTOOED THIRTY

TOILETS TURKEY UNDRESS USELESS

WHENEVER WHISKING WHISTLES WEDNESDAY

But given a set total in mind, that raises a couple of questions: What are the shortest and the longest dollar words in the dictionary?

Because 100 is a relatively large total for a short word (and because a lot of the highest value letters at the tail end of the alphabet are hard to find homes for, like V, X, and Z) shorter dollar words are fairly hard to come by. As a result, only a handful of 5-letter dollar words have ever been discovered, including:

BUZZY NUTTY PUSSY

In fact, as proof of just how many seldom-used letters lie at the end of the alphabet, if you were to change the numbers around so that A = 26, B =25, and so on through to Z = 1, the number of five-letter dollar words increases enormously:

ABBEY ACRID BACON BASAL

BEFOG BEGET CATCH CHAIN

CHALK CHINA DODGE ELIDE

FACET HENCE IMAGE LAGAN

LANCE MAGMA MEDAL NAKED

But shortest of all are two 4-letter words: acca, an Australian slang word for an academic, and caca, a childish word for poop.

Oppositely, it can be just as difficult looking for as long a dollar word as possible; the more letters a word has, the higher its total grows. But the relatively high frequency of the letters in the first few places of the alphabet means that there are quite a few lengthy dollar words, including some with as many as 12 letters:

BACKTRACKING COMMANDEERED

DEBAUCHERIES DESEGREGATED

INAPPLICABLE NON-BREAKABLE

Apparently longest of all is the 13-letter word adiabatically, a term from meteorology and thermodynamics referring to any process that occurs without a loss or gain of heat.

But why stop at adding up? Multiplying the numerical values of words leads to some considerably larger numbers—and some considerably higher targets.

Multiply the letters of the word TYPEY together, for instance, and you’ll end up with 1,000,000 (= 20 × 25 × 16 × 5 × 25). TEETHY multiplies to 2,000,000 (= 20 × 5 × 5 × 20 × 8 × 25). And PEYOTE multiplies to 3,000,000 (= 16 × 5 × 25 × 15 × 20 × 5). No word has yet been found that totals precisely 4,000,000 or 5,000,000, but some—like LURING (4,000,752) and JUICING (5,000,940)—have come tantalizingly close.

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15 Riveting Facts About Alan Turing
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More than six decades after his death, Alan Turing’s life remains a point of fascination—even for people who have no interest in his groundbreaking work in computer science. He has been the subject of a play and an opera, and referenced in multiple novels and numerous musical albums. The Benedict Cumberbatch film about his life, The Imitation Game, received eight Oscar nominations. But just who was he in real life? Here are 15 facts you should know about Alan Turing, who was born on this day in 1912.

#### 1. HE’S THE FATHER OF MODERN COMPUTER SCIENCE.

Turing essentially pioneered the idea of computer memory. In 1936, Turing published a seminal paper called “On Computable Numbers” [PDF], which The Washington Post has called “the founding document of the computer age.” In the philosophical article, he hypothesized that one day, we could build machines that could compute any problem that a human could, using 0s and 1s. Turing proposed single-task machines called Turing machines that would be capable of solving just one type of math problem, but a “universal computer” would be able to tackle any kind of problem thrown at it by storing instructional code in the computer’s memory. Turing’s ideas about memory storage and using a single machine to carry out all tasks laid the foundation for what would become the digital computer.

In 1945, while working for the UK’s National Physical Laboratory, he came up with the Automatic Computing Machine, the first digital computer with stored programs. Previous computers didn’t have electric memory storage, and had to be manually rewired to switch between different programs.

#### 2. HE PLAYED A HUGE ROLE IN WINNING WORLD WAR II.

Turing began working at Bletchley Park, Britain’s secret headquarters for its codebreakers during World War II, in 1939. By one estimate, his work there may have cut the war short by up to two years. He’s credited with saving millions of lives.

Turing immediately got to work designing a codebreaking machine called the Bombe (an update of a previous Polish machine) with the help of his colleague Gordon Welchman. The Bombe shortened the steps required in decoding, and 200 of them were built for British use over the course of the war. They allowed codebreakers to decipher up to 4000 messages a day.

His greatest achievement was cracking the Enigma, a mechanical device used by the German army to encode secure messages. It proved nearly impossible to decrypt without the correct cipher, which the German forces changed every day. Turing worked to decipher German naval communications at a point when German U-boats were sinking ships carrying vital supplies across the Atlantic between Allied nations. In 1941, Turing and his team managed to decode the German Enigma messages, helping to steer Allied ships away from the German submarine attacks. In 1942, he traveled to the U.S. to help the Americans with their own codebreaking work.

#### 3. HE BROKE THE RULES TO WRITE TO CHURCHILL.

Early on, Bletchley Park’s operations were hampered by a lack of resources, but pleas for better staffing were ignored by government officials. So, Alan Turing and several other codebreakers at Bletchley Park went over their heads to write directly to Prime Minister Winston Churchill. One of the codebreakers from Bletchley Park delivered the letter by hand in October 1941.

“Our reason for writing to you direct is that for months we have done everything that we possibly can through the normal channels, and that we despair of any early improvement without your intervention,” they wrote to Churchill [PDF]. “No doubt in the long run these particular requirements will be met, but meanwhile still more precious months will have been wasted, and as our needs are continually expanding we see little hope of ever being adequately staffed.”

In response, Churchill immediately fired off a missive to his chief of staff: “Make sure they have all they want on extreme priority and report to me that this had been done.”

#### 4. HE HAD SOME ODD HABITS.

Like many geniuses, Turing was not without his eccentricities. He wore a gas mask while riding his bike to combat his allergies. Instead of fixing his bike’s faulty chain, he learned exactly when to dismount to secure it in place before it slipped off. He was known around Bletchley Park for chaining his tea mug to a radiator to prevent it from being taken by other staffers.

#### 5. HE RODE HIS BIKE 60 MILES TO GET TO THE FIRST DAY OF SCHOOL.

Though he was considered an average student, Turing was dedicated enough to his schooling that when a general strike prevented him from taking the train to his first day at his new elite boarding school, the 14-year-old rode his bike the 62 miles instead.

#### 6. HE TRIED OUT FOR THE OLYMPICS.

Turing started running as a schoolboy and continued throughout his life, regularly running the 31 miles between Cambridge and Ely while he was a fellow at King’s College. During World War II, he occasionally ran the 40 miles between London and Bletchley Park for meetings.

He almost became an Olympic athlete, too. He came in fifth place at a qualifying marathon for the 1948 Olympics with a 2-hour, 46-minute finish (11 minutes slower than the 1948 Olympic marathon winner). However, a leg injury held back his athletic ambitions that year.

Afterward, he continued running for the Walton Athletic Club, though, and served as its vice president. ”I have such a stressful job that the only way I can get it out of my mind is by running hard,” he once told the club’s secretary. “It's the only way I can get some release."

#### 7. HE WAS PROSECUTED FOR BEING GAY.

In 1952, Turing was arrested after reporting a burglary in his home. In the course of the investigation, the police discovered Turing’s relationship with another man, Arnold Murray. Homosexual relationships were illegal in the UK at the time, and he was charged with “gross indecency.” He pled guilty on the advice of his lawyer, and opted to undergo chemical castration instead of serving time in jail.

#### 8. THE GOVERNMENT ONLY RECENTLY APOLOGIZED FOR HIS CONVICTION …

In 2009, UK Prime Minister Gordon Brown issued a public apology to Turing on behalf of the British government. “Alan and the many thousands of other gay men who were convicted as he was convicted under homophobic laws were treated terribly,” Brown said. "This recognition of Alan's status as one of Britain's most famous victims of homophobia is another step towards equality and long overdue." Acknowledging Britain’s debt to Turing for his vital contributions to the war effort, he announced, “on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better."

His conviction was not actually pardoned, though, until 2013, when he received a rare royal pardon from the Queen of England.

#### 9. … AND NAMED A LAW AFTER HIM.

Turing was only one of the many men who suffered after being prosecuted for their homosexuality under 19th-century British indecency laws. Homosexuality was decriminalized in the UK in 1967, but the previous convictions were never overturned. Turing’s Law, which went into effect in 2017, posthumously pardoned men who had been convicted for having consensual gay sex before the repeal. According to one of the activists who campaigned for the mass pardons, around 15,000 of the 65,000 gay men convicted under the outdated law are still alive.

#### 10. HE POISONED HIMSELF … MAYBE.

There is still a bit of mystery surrounding Turing’s death at the age of 41. Turing died of cyanide poisoning, in what is widely believed to have been a suicide. Turing’s life had been turned upside down by his arrest. He lost his job and his security clearance. By order of the court, he had to take hormones intended to “cure” his homosexuality, which caused him to grow breasts and made him impotent. But not everyone is convinced that he died by suicide.

In 2012, Jack Copeland, a Turing scholar, argued that the evidence used to declare Turing’s death a suicide in 1954 would not be sufficient to close the case today. The half-eaten apple by his bedside, thought to be the source of his poisoning, was never tested for cyanide. There was still a to-do list on his desk, and his friends told the coroner at the time that he had seemed in good spirits. Turing’s mother, in fact, maintained that he probably accidentally poisoned himself while experimenting with the chemical in his home laboratory. (He was known to taste chemicals while identifying them, and could be careless with safety precautions.)

That line of inquiry is far more tame than some others, including one author’s theory that he was murdered by the FBI to cover up information that would have been damaging to the U.S.

#### 11. HIS FULL GENIUS WASN’T KNOWN IN HIS LIFETIME.

Alan Turing was a well-respected mathematician in his time, but his contemporaries didn’t know the full extent of his contributions to the world. Turing’s work breaking the Enigma machine remained classified long after his death, meaning that his contributions to the war effort and to mathematics were only partially known to the public during his lifetime. It wasn’t until the 1970s that his instrumental role in the Allies' World War II victory became public with the declassification of the Enigma story. The actual techniques Turing used to decrypt the messages weren’t declassified until 2013, when two of his papers from Bletchley Park were released to the British National Archives.

#### 12. THE TURING TEST IS STILL USED TO MEASURE ARTIFICIAL INTELLIGENCE …

Can a machine fool a human into thinking they are chatting with another person? That’s the crux of the Turing test, an idea developed by Turing in 1950 regarding how to measure artificial intelligence. Turing argued in his paper “Computing Machinery and Intelligence” [PDF] that the idea of machines “thinking” is not a useful way to evaluate artificial intelligence. Instead, Turing suggests “the imitation game,” a way to assess how successfully a machine can imitate human behavior. The best measure of artificial intelligence, then, is whether or not a computer can convince a person that it is human.

#### 13. … BUT SOME CONSIDER IT TO BE AN OUTDATED IDEA.

As technology has progressed, some feel the Turing test is no longer a useful way to measure artificial intelligence. It’s cool to think about computers being able to talk just like a person, but new technology is opening up avenues for computers to express intelligence in other, more useful ways. A robot’s intelligence isn’t necessarily defined by whether it can fake being human—self-driving cars or programs that can mimic sounds based on images might not pass the Turing test, but they certainly have intelligence.

#### 14. HE CREATED THE FIRST COMPUTER CHESS PROGRAM.

Inspired by the chess champions he worked with at Bletchley Park, Alan Turing created an algorithm for an early version of computer chess—although at that time, there was no computer to try it out on. Created with paper and pencil, the Turochamp program was designed to think two moves ahead, picking out the best moves possible. In 2012, Russian chess grandmaster Garry Kasparov played against Turing’s algorithm, beating it in 16 moves. “I would compare it to an early caryou might laugh at them but it is still an incredible achievement," Kasparov said in a statement after the match-up.

#### 15. THERE IS ALAN TURING MONOPOLY.

In 2012, Monopoly came out with an Alan Turing edition to celebrate the centennial of his birth. Turing had enjoyed playing Monopoly during his life, and the Turing-themed Monopoly edition was designed based on a hand-drawn board created in 1950 by his friend William Newman. Instead of hotels and houses, it featured huts and blocks inspired by Bletchley Park, and included never-before-published photos of Turing. (It’s hard to find, but there are still a few copies of the game on Amazon.)

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