Image by EdPeggJr via Wikimedia Commons
Image by EdPeggJr via Wikimedia Commons

Breakthrough Pentagon Can Tile the Plane

Image by EdPeggJr via Wikimedia Commons
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.

5 Ways You Do Complex Math in Your Head Without Realizing It

The one thing that people who love math and people who hate math tend to agree on is this: You're only really doing math if you sit down and write formal equations. This idea is so widely embraced that to suggest otherwise is "to start a fight," says Maria Droujkova, math educator and founder of Natural Math, a site for kids and parents who want to incorporate math into their daily lives. Mathematicians cherish their formal proofs, considering them the best expression of their profession, while the anti-math don't believe that much of the math they studied in school applies to "real life."

But in reality, "we do an awful lot of things in our daily lives that are profoundly mathematical, but that may not look that way on the surface," Christopher Danielson, a Minnesota-based math educator and author of a number of books, including Common Core Math for Parents for Dummies, tells Mental Floss. Our mathematical thinking includes not just algebra or geometry, but trigonometry, calculus, probability, statistics, and any of the at least 60 types [PDF] of math out there. Here are five examples.


Of all the maths, algebra seems to draw the most ire, with some people even writing entire books on why college students shouldn't have to endure it because, they claim, it holds the students back from graduating. But if you cook, you're likely doing algebra. When preparing a meal, you often have to think proportionally, and "reasoning with proportions is one of the cornerstones of algebraic thinking," Droujkova tells Mental Floss.

You're also thinking algebraically whenever you're adjusting a recipe, whether for a larger crowd or because you have to substitute or reduce ingredients. Say, for example, you want to make pancakes, but you only have two eggs left and the recipe calls for three. How much flour should you use when the original recipe calls for one cup? Since one cup is 8 ounces, you can figure this out using the following algebra equation: n/8 : 2/3.

algebraic equation illustrates adjustment of a recipe
Lucy Quintanilla

However, when thinking proportionally, you can just reason that since you have one-third less eggs, you should just use one-third less flour.

You're also doing that proportional thinking when you consider the cooking times of the various courses of your meal and plan accordingly so all the elements of your dinner are ready at the same time. For example, it will usually take three times as long to cook rice as it will a flattened chicken breast, so starting the rice first makes sense.

"People do mathematics in their own way," Droujkova says, "even if they cannot do it in a very formalized way."


woman enjoys listening to music in headphones

The making of music involves many different types of math, from algebra and geometry to group theory and pattern theory and beyond, and a number of mathematicians (including Pythagoras and Galileo) and musicians have connected the two disciplines (Stravinsky claimed that music is "something like mathematical thinking").

But simply listening to music can make you think mathematically too. When you recognize a piece of music, you are identifying a pattern of sound. Patterns are a fundamental part of math; the branch known as pattern theory is applied to everything from statistics to machine learning.

Danielson, who teaches kids about patterns in his math classes, says figuring out the structure of a pattern is vital for understanding math at higher levels, so music is a great gateway: "If you're thinking about how two songs have similar beats, or time signatures, or you're creating harmonies, you're working on the structure of a pattern and doing some really important mathematical thinking along the way."

So maybe you weren't doing math on paper if you were debating with your friends about whether Tom Petty was right to sue Sam Smith in 2015 over "Stay With Me" sounding a lot like "I Won't Back Down," but you were still thinking mathematically when you compared the songs. And that earworm you can't get out of your head? It follows a pattern: intro, verse, chorus, bridge, end.

When you recognize these kinds of patterns, you're also recognizing symmetry (which in a pop song tends to involve the chorus and the hook, because both repeat). Symmetry [PDF] is the focus of group theory, but it's also key to geometry, algebra, and many other maths.


six steps of crocheting a hyperbolic plane
Cheryl, Flickr // CC BY-SA 2.0

Droujkova, an avid crocheter, she says she is often intrigued by the very mathematical discussions fellow crafters have online about the best patterns for their projects, even if they will often insist they are awful at math or uninterested in it. And yet, such crafts cannot be done without geometric thinking: When you knit or crochet a hat, you're creating a half sphere, which follows a geometric formula.

Droujkova isn't the only math lover who has made the connection between geometry and crocheting. Cornell mathematician Daina Taimina found crocheting to be the perfect way to illustrate the geometry of a hyperbolic plane, or a surface that has a constant negative curvature, like a lettuce leaf. Hyperbolic geometry is also used in navigation apps, and explains why flat maps distort the size of landforms, making Greenland, for example, look far larger on most maps than it actually is.


people playing pool

If you play billiards, pool, or snooker, it's very likely that you are using trigonometric reasoning. Sinking a ball into a pocket by using another ball involves understanding not just how to measure angles by sight but triangulation, which is the cornerstone of trigonometry. (Triangulation is a surprisingly accurate way to measure distance. Long before powered flight was possible, surveyors used triangulation to measure the heights of mountains from their bases and were off by only a matter of feet.)

In a 2010 paper [PDF], Louisiana mathematician Rick Mabry studied the trigonometry (and basic calculus) of pool, focusing on the straight-in shot. In a bar in Shreveport, Louisiana, he scribbled equations on napkins for each shot, and he calculated the most difficult straight-in shot of all. Most experienced pool players would say it’s one where the target ball is halfway between the pocket and the cue ball. But that, according to Mabry’s equations, turned out not to be true. The hardest shot of all had a surprising feature: The distance from the cue ball to the pocket was exactly 1.618 times the distance from the target ball to the pocket. That number is the golden ratio, which is found everywhere in nature—and, apparently, on pool tables.

Do you need to consider the golden ratio when deciding where to place the cue ball? Nope, unless you want to prove a point, or set someone else up to lose. You're doing the trig automatically. The pool sharks at the bar must have known this, because someone threw away Mabry's math napkins.


tiled bathroom with shower stall

Many students don't get to calculus in high school, or even in college, but a cornerstone of that branch of math is optimization—or figuring out how to get the most precise use of a space or chunk of time.

Consider a home improvement project where you're confronted with tiling around something whose shape doesn't fit a geometric formula like a circle or rectangle, such as the asymmetric base of a toilet or freestanding sink. This is where the fundamental theorem of calculus—which can be used to calculate the precise area of an irregular object—comes in handy. When thinking about how those tiles will best fit around the curve of that sink or toilet, and how much of each tile needs to be cut off or added, you're employing the kind of reasoning done in a Riemann sum.

Riemann sums (named after a 19th-century German mathematician) are crucial to explaining integration in calculus, as tangible introductions to the more precise fundamental theorem. A graph of a Riemann sum shows how the area of a curve can be found by building rectangles along the x, or horizontal axis, first up to the curve, and then over it, and then averaging the distance between the over- and underlap to get a more precise measurement. 

The Math Puzzle That’s Driving the Internet Bananas

Here at Mental Floss, we love a good brain teaser—and clearly we’re not alone. From cows and chickens to river crossings, we’ve never met a riddle we didn’t want to solve—even if it was originally meant for a 5-year-old. The Bananas, Clock, Hexagon Viral Logic Puzzle, which math puzzle enthusiast Presh Talwalkar posted to his Mind Your Decisions blog, is the latest riddle to have us admittedly stumped.

According to Talwalkar, 99 percent of the people who attempt to solve the problem fail, leaving the remaining one percent to be dubbed geniuses for figuring it out. Which side will you land on?

The key to solving this puzzle is to look closely. We’ll give you a minute to do just that (or you can start the video below—it will give you a little time before giving anything away).

Now it’s time to come up with your answer. We’ll give you another minute …


So what did you come up with?

If you answered 38, congratulations—you might be a genius.

If you answered something else, don’t worry, you’re not alone.

Where most people seem to be going wrong with the problem is by not looking at the images closely enough when attempting to assign a numerical value. Specifically: In the last line of the problem, all of the images—the clock, the bananas, and the hexagon—are all slightly different than the images shown in the previous lines. If you noticed this, then you probably realized that the bunch of bananas in lines one, two, and three have a different value than the fruit seen on line four. Same goes for the clock and the hexagons. Which makes this as much a visual puzzle as it a math problem.

Finish watching the video above for Talwalkar’s detailed explanation of how to solve the problem. Then stump your friends!

[h/t: Mind Your Decisions]


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