By Joseph O'Rourke

What do proteins and pop-up playing cards have in universal? How is beginning a grocery bag diversified from starting a present field? how will you lower out the letters for a complete observe unexpectedly with one directly scissors minimize? what number methods are there to flatten a dice? you could solution those questions and extra during the arithmetic of folding and unfolding. From this e-book, you can find new and previous mathematical theorems by way of folding paper and tips to cause towards proofs. With the aid of 2 hundred colour figures, writer Joseph O'Rourke explains those interesting folding difficulties ranging from highschool algebra and geometry and introducing extra complicated strategies in tangible contexts as they come up. He exhibits how diversifications on those easy difficulties lead on to the frontiers of present mathematical study and gives ten obtainable unsolved difficulties for the enterprising reader. earlier than tackling those, you could attempt your talents on fifty workouts with whole ideas. The book's website, http://www.howtofoldit.org, has dynamic animations of some of the foldings and downloadable templates for readers to fold or minimize out.

**Read or Download How to Fold It: The Mathematics of Linkages, Origami and Polyhedra PDF**

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

**Example text**

35. , ﬂat) sections. That the sections are planar is more evident in the overhead view shown in (b) of the ﬁgure: the two end sections {v0 , v1 , v2 , v3 } and {v8 , v9 , v10 , v11 } lie in a plane parallel to the xy-plane, and the middle section {v3 , v4 , v5 , v6 , v7 , v8 } lies in a vertical plane parallel the the z-axis. 1 (Practice) 3-Link MaxSpan. What is the maxspan of the 3-link 90◦ -chain with link lengths (1, 2, 3), if all three links lie in the same plane? 2 (Challenge) 3-Link MaxSpan.

10. Pantographs with different magniﬁcations. (a) A = 3 A : scale factor − → 3− → ×3 (300%). (b) A = 2 A : scale factor × 32 (150%). 2 (Practice) Fivefold Pantograph. Design a pantograph to achieve ﬁvefold magniﬁcation. 3 (Understanding) Two-Thirds Pantograph. Design a pantograph to achieve two-thirds reduction. Another consequence of our analysis is that points x, y, and z always lie on a common line: They are collinear. This follows because z = ky, where k is the scale factor. Suppose k = 3 and, for speciﬁcity, let y = (1, 2) and z = 3y = (3, 6).

13(b) illustrates, this plane intersects Rn in a 2D annulus. We can simply solve the 3D problem within this plane, and use those angles. Thus 3D reduces to 2D. Dynamic Reconﬁguration. Given initial and ﬁnal angles at the joints of an nlink arm, the easiest way to move continuously between the two conﬁgurations is to simply interpolate the angles. We imagine a clock ticking in small increments from t = 0 to t = 1 between initial and ﬁnal angles. If the initial angle at a joint is α and the ﬁnal angle β, as measured, say, counterclockwise from the horizontal, then at time t, the angle is α + t(β − α).