By Peter Hilton, Jean Pedersen, Sylvie Donmoyer

This easy-to-read ebook demonstrates how an easy geometric proposal finds attention-grabbing connections and leads to quantity conception, the maths of polyhedra, combinatorial geometry, and crew conception. utilizing a scientific paper-folding technique it's attainable to build a standard polygon with any variety of aspects. This awesome set of rules has ended in attention-grabbing proofs of sure ends up in quantity thought, has been used to respond to combinatorial questions concerning walls of house, and has enabled the authors to procure the formulation for the quantity of a customary tetrahedron in round 3 steps, utilizing not anything extra complex than easy mathematics and the main hassle-free aircraft geometry. All of those rules, and extra, display the wonderful thing about arithmetic and the interconnectedness of its numerous branches. exact directions, together with transparent illustrations, permit the reader to achieve hands-on event developing those types and to find for themselves the styles and relationships they unearth.

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Assume that we have a straight strip of paper that has certain vertices marked on its top and bottom edges and which also has creases or folds along straight lines emanating from vertices at the top edge of the strip. Further assume that the creases at those vertices labeled Ank , n = 0, 1, 2, . . 2(a)). Suppose further that, for each k, the vertices Ank , n = 0, 1, 2, . . , are equally spaced. 2(c), the direction of the top edge of the tape will be rotated through an angle of 2 aπ . As you know, from Chapter 2, we call this process of Folding And b Twisting the FAT algorithm (see any of [26–30, 33, 35, 38]).

Do this at every top vertex indicated by the arrow as shown in frame 4. 8. An even bigger hexagon can be obtained by increasing the distance between the successive vertices along the top of the tape at which you make your secondary folds. The hexagon is then formed, as before, by performing the FAT algorithm at 6 successive vertices equally spaced along the top of the tape. 9, where secondary lines are folded at every other vertex along the top of the tape, and then the FAT algorithm is executed at those vertices using the secondary fold lines.

And we abbreviate it to U DU DU D . . or U 1 D 1 , and sometimes refer to this folded strip as U 1 D 1 -tape. Second, although the first few triangles may be a bit irregular, the triangles formed always become more and more regular; that is, the angle between the last fold line and the edge of the tape gets closer and closer to π3 . When you use these triangles for constructing models, it is very safe to throw away the first 10 triangles and then to assume the rest of the triangles will be close enough to use for constructing anything that requires equilateral triangles.

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