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Image by EdPeggJr via Wikimedia Commons

Breakthrough Pentagon Can Tile the Plane

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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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History
When Math Discoveries Led to Banned Numbers
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The literature world has seen more than its share of controversy. The best stories tend to provoke the strongest reactions—both positive and negative—in readers, which is why so many classic books have been banned at one point or another. But even a more objective field like math isn’t immune to conflict. In its new video, TED-Ed rounds up the numbers that caused such a stir when they were introduced that they were banned in math circles.

One of the earliest examples comes from ancient Greece. A mathematician named Hippasus was having trouble solving certain equations with fractions and whole numbers alone, so he came up with irrational numbers to make these values easier to express. The ruling school of thought at the time dictated that everything in nature could be explained elegantly with the numbers that already existed. Threatened by Hippasus’s new notion, his fellow mathematicians rejected the irrational numbers and had him exiled.

Other numbers have been banned for legal reasons. When Arab traders brought their positional number system, which included zero, to Italy in the Middle Ages, Florence banned it from record-keeping fearing that they would be easier to forge than Roman numerals. The Arabic way of counting also led to the rise of negative numbers, which were regarded with disdain by many experts into the 19th century. For more banned numbers, including some that are prohibited today, check out the full story below.

[h/t TED-Ed]

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Live Smarter
An Ex-Google Engineer Just Reinvented the Measuring Cup
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Euclid

Recognizing a problem most people didn’t even know they had, former Google and Facebook software engineer Joshua Redstone has made a bold claim for his recent Kickstarter venture: He’s developed a better measuring cup.

According to the Boston Business Journal, Redstone spent four years tinkering with a solution to something that had long annoyed him as an amateur chef: Traditional measuring cups, which are stocky and not very well tapered, don’t do a great job of accurately measuring their own contents. Redstone believes the shape of a cup determines its success, particularly when a cook overfills a liquid or solid by a tiny amount. The smaller the volume, the more the problem is magnified.

Euclid

Redstone’s cup, Euclid, resolves the issue. According to the Kickstarter page: “With traditional measuring cups, the smaller the amount, the harder it is to measure accurately. The culprit? The shape. Straight sides magnify errors when measuring lower down in the cup. Some have tried to solve this problem with conical measuring cups, but their results fall short of Euclid’s by up to 60 percent. Euclid is the only measuring cup with a mathematically optimal, tapered design for consistent accuracy across amounts.”

Euclid is just about ready to overshoot its $30,000 Kickstarter goal. Backers can pay $24 for the cup now, or wait until it’s available at retail for a slightly higher price to be determined. The cup is scheduled for release in May 2018.

[h/t Boston Business Journal]

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