How to Fold It: The Mathematics of Linkages, Origami and by Joseph O'Rourke

By Joseph O'Rourke

What do proteins and pop-up playing cards have in universal? How is commencing a grocery bag various from beginning a present field? how are you going to reduce out the letters for an entire notice unexpectedly with one directly scissors lower? what percentage methods are there to flatten a dice? you could solution those questions and extra during the arithmetic of folding and unfolding. From this publication, you'll find new and previous mathematical theorems by means of folding paper and the way to cause towards proofs. With assistance from 2 hundred colour figures, writer Joseph O'Rourke explains those interesting folding difficulties ranging from highschool algebra and geometry and introducing extra complex suggestions in tangible contexts as they come up. He exhibits how diversifications on those simple difficulties lead on to the frontiers of present mathematical examine and gives ten available unsolved difficulties for the enterprising reader. ahead of tackling those, you could try out your abilities on fifty workouts with entire recommendations. The book's site, http://www.howtofoldit.org, has dynamic animations of a number of the foldings and downloadable templates for readers to fold or minimize out.

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Additional resources for How to Fold It: The Mathematics of Linkages, Origami and Polyhedra

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Draw and describe the reachability √ region √of the endpoint v2 of the second link, under three conditions: (1) r ≤ 2/2, (2) 2/2 < r ≤ 1, and (3) r > 1. 10 (Challenge) 2D Angle-Limited Linkages: Two Constraints. Continuing the previous problem, also constrain the v1 joint to only turn within a 90◦ range. To be specific, the angle “v0 v1 v2 is between 90◦ (perpendicular to v0 v1 ) and 180◦ (aligned with v0 v1 ). Again draw and describe the reachability region of the v2 endpoint. Are there critical values of r at which the structure of the reachability region changes?

10, you know that angle constraints greatly complicate the possible motions of the chain. But in many applications, there are angle constraints, so they must be confronted. We consider two applications in this chapter, which are, surprisingly, related: protein folding and a certain pop-up card. Despite the whimsical chapter title, the real focus will be the “maxspan of 90◦ -chains,” the mathematical structure shared between the two applications. Several techniques and ideas from the previous chapters will resurface here, including induction and the triangle inequality.

Second, we will treat the bond angles between adjacent atoms along the backbone as fixed. This is nearly true. Third, we assume the chain permits free dihedral motion about each of its bonds. This is definitely not true, because one bond per amino acid (the so-called “peptide” bond) only permits two dihedral angles, 0◦ and 180◦ . Fourth, we will ignore all chemical and electrostatic forces, leaving only the geometry of dihedral motions and the restriction that the chain cannot pass through itself.

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