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Message: <blockquote data-quote="imhotep" data-source="post: 29276596" data-attributes="member: 562115">Everyone has certainly heard about the Möbius strip. The German mathematicians August Ferdinand Möbius and Johann Benedict Listing independently discovered it in 1858. While Möbius got the naming rights, both men were drawn to its peculiar property: its unending surface.<a href="https://imgbox.com/0MwixW4u" target="_blank"><img src="https://thumbs2.imgbox.com/77/df/0MwixW4u_t.jpg" alt="" class="fr-fic fr-dii fr-draggable " style="" /></a>There are some practical uses of this fascinating Möbius strip.  For instance, conveyor belts designed as a Möbius strip distribute wear and tear uniformly, lasting twice as long as conventional conveyor belts. In electronics, Möbius resistors are employed due to their unique electromagnetic properties.  Even the 'Recyclable' symbol is inspired by the Möbius strip.The Möbius Strip has revolutionized the field of topology, which studies the properties of objects that are preserved when moved, bent, stretched or twisted, without cutting or gluing parts together. A coffee mug and a doughnut are, for instance, topologically identical. Both objects have just one hole, which can be deformed through stretching and bending to create one or the other structure.Mug to Torus Morph: Link <a href="https://en.wikipedia.org/wiki/File:Mug_and_Torus_morph.gif" target="_blank">https://en.wikipedia.org/wiki/File:Mug_and_Torus_morph.gif</a>But this simplicity betrays the shape’s mathematical complexity. In 1977, two mathematicians by the name Charles Sidney Weaver and Benjamin Rigler Halpern created the Halpern-Weaver Conjecture, which proposed what the minimum size of that strip of paper needs to be in order to form a Möbius strip.Simple, right? Well, not really. There were a few rules added to this conjecture, but the big one is that the paper strip had to be “embedded” and not “immersed.” This means that the object couldn’t intersect itself at any point. In other words, the embedded paper Möbius couldn’t have any overlaps.While Weaver and Halpern provided a proposed length, they couldn’t prove it definitely—and that’s where Brown University mathematician Richard Evan Schwartz comes in.Schwartz solves the nearly 50-year-old mystery and finally proves the minimum size proposed by the original Halpern-Weaver conjecture—an aspect ratio of 1.73, or √3.In simple terms, the perfect Möbius strip should possess an aspect ratio greater than √3 (approximately 1.73).  Hence a strip that is 1 centimeter long must exceed 1.73 centimeters in width.PS: More mysteries surround the Möbius strip. For example, this conjecture only proves the minimum size of a paper Möbius with one twist—not three or more. For such a simple shape, it’s amazing that it can still remain so perplexing more than 160 years after its initial discovery.</blockquote>

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