Peter Hilton, Jean Pedersen, Sylvie Donmoyer's A Mathematical Tapestry: Demonstrating the Beautiful Unity PDF

By Peter Hilton, Jean Pedersen, Sylvie Donmoyer

ISBN-10:

ISBN-13: 1397805217641

ISBN-10: 0521764106

ISBN-13: 9780521764100

This easy-to-read ebook demonstrates how an easy geometric thought unearths attention-grabbing connections and leads to quantity idea, the math of polyhedra, combinatorial geometry, and team idea. utilizing a scientific paper-folding technique it really is attainable to build a typical polygon with any variety of facets. This amazing set of rules has ended in fascinating proofs of yes leads to quantity thought, has been used to respond to combinatorial questions concerning walls of area, and has enabled the authors to procure the formulation for the amount of a customary tetrahedron in round 3 steps, utilizing not anything extra advanced than easy mathematics and the main straightforward airplane geometry. All of those rules, and extra, exhibit the wonderful thing about arithmetic and the interconnectedness of its quite a few branches. specified directions, together with transparent illustrations, permit the reader to achieve hands-on event developing those versions and to find for themselves the styles and relationships they unearth.

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10. Notice that the tape, which we call U 2 D 2 -tape (or, equivalently, D 2 U 2 -tape) seems to be getting more and more regular – the successive long lines are becoming closer and closer to each other in length, and so are the successive short lines. The smallest angle on the tape seems to be approaching some fixed value. But what is it? 4 Does this idea generalize? 9 A FAT 6-gon. A bigger FAT 6-gon. 10 Folding UP UP DOWN DOWN . . or U 2 D 2 . 4 Does this idea generalize? 11 (b) (a) A long-line 5-gon.

2. 3. 4. 7 Preparing the U 1 D 1 -tape for constructing a hexagon. 10. Notice that the tape, which we call U 2 D 2 -tape (or, equivalently, D 2 U 2 -tape) seems to be getting more and more regular – the successive long lines are becoming closer and closer to each other in length, and so are the successive short lines. The smallest angle on the tape seems to be approaching some fixed value. But what is it? 4 Does this idea generalize? 9 A FAT 6-gon. A bigger FAT 6-gon. 10 Folding UP UP DOWN DOWN .

In fact, we see that 3 = 21 + 1 5 = 22 + 1 9 = 23 + 1. 19 Part of a FAT 9-gon, constructed by performing the FAT algorithm on long lines of the U 3 D 3 -tape. 20 The beginning part of the U 3 D 3 -tape. So now we should suspect that if we fold U n D n , we can use that tape to construct regular (2n + 1)-gons. To see that this is true we can give an error-correction proof analogous to that which we gave in the case n = 1. 20, in the particular but not special case when n = 3. ), then, at point A at the top of the tape we have π9 + + 23 x1 = π, from which it follows that x1 = π9 − 23 .

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A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics by Peter Hilton, Jean Pedersen, Sylvie Donmoyer


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