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For example, it is NP-hard to evaluate whether a given crease pattern folds into any flat origami. [47] In 2017, Erik Demaine of the Massachusetts Institute of Technology and Tomohiro Tachi of the University of Tokyo published a new universal algorithm that generates practical paper-folding patterns to produce any 3-D structure.
He argued that the toolbox of physics enables a practitioner like Edward Witten to go beyond standard mathematics, in particular the geometry of 4-manifolds. The tools of a physicist are cited as quantum field theory , special relativity , non-abelian gauge theory , spin , chirality , supersymmetry , and the electromagnetic duality .
No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. […] Because of Gödel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them." [48]
Although a common classroom experiment is often explained this way, [442] Bernoulli's principle only applies within a flow field, and the air above and below the paper is in different flow fields. [443] The paper rises because the air follows the curve of the paper and a curved streamline will develop pressure differences perpendicular to the ...
Kawasaki's theorem or Kawasaki–Justin theorem is a theorem in the mathematics of paper folding that describes the crease patterns with a single vertex that may be folded to form a flat figure. It states that the pattern is flat-foldable if and only if alternatingly adding and subtracting the angles of consecutive folds around the vertex gives ...
During this period there was little distinction between physics and mathematics; [18] as an example, Newton regarded geometry as a branch of mechanics. [19] Non-Euclidean geometry, as formulated by Carl Friedrich Gauss, János Bolyai, Nikolai Lobachevsky, and Bernhard Riemann, freed physics from the limitation of a single Euclidean geometry. [20]
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Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical structure; [2]: 6 indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences ...