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Article  Möbius strip problem solved + A tower of conjectures that rests upon a needle

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Mathematicians solve a key Möbius strip problem, after almost 50 years of searching

INTRO: For nearly fifty years, mathematicians have puzzled over a deceptively simple question: how small can you make a Möbius strip without it intersecting itself?

Now, Richard Schwartz, a mathematician at Brown University, has proposed an elegant solution to this problem, which was originally posed by mathematicians Charles Weaver and Benjamin Halpern in 1977.

In their paper, Halpern and Weaver pose a limit for Möbius strips based on the familiar geometry of folded bits of solid paper – that the ratio between the length and width of the paper must be greater than √3, or around 1.73... (MORE - details)

A tower of conjectures that rests upon a needle

EXCERPT: ... these results are possible because “we can look at numbers as waves.” That all these problems link back to Kakeya needle sets “is fascinating,” he added. “You don’t think that so much beauty, difficulty and importance can be hidden in something that can be formulated using line segments.”

INTRO: In mathematics, a simple problem is often not what it seems. Earlier this summer, Quanta reported on one such problem: What is the smallest area that you can sweep out while rotating an infinitely thin needle in all possible directions? Spin it around its center like a dial, and you get a circle. But rotate it more cleverly, and you can cover an arbitrarily small fraction of space. If you don’t require the needle to move in one continuous motion, and instead simply lay down a needle in every direction, you can construct an arrangement of needles that covers no area at all.

Mathematicians call these arrangements Kakeya sets. While they know that such sets can be small in terms of area (or volume, if you’re arranging your needles in three or more dimensions), they believe the sets must always be large if their size is measured by a metric called the Hausdorff dimension.

Mathematicians have yet to prove this statement, known as the Kakeya conjecture. But while it’s ostensibly a simple question about needles, “the geometry of these Kakeya sets underpins a whole wealth of questions in partial differential equations, harmonic analysis and other areas,” said Jonathan Hickman of the University of Edinburgh... (MORE - details)

Kakeya's Needle Problem ...

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