Download e-book for kindle: A Mathematical Tapestry: Demonstrating the Beautiful Unity by Peter Hilton, Jean Pedersen, Sylvie Donmoyer

By Peter Hilton, Jean Pedersen, Sylvie Donmoyer

ISBN-10:

ISBN-13: 1397805217641

ISBN-10: 0521764106

ISBN-13: 9780521764100

This easy-to-read booklet demonstrates how an easy geometric notion finds attention-grabbing connections and ends up in quantity idea, the maths of polyhedra, combinatorial geometry, and workforce idea. utilizing a scientific paper-folding approach it's attainable to build a typical polygon with any variety of aspects. This awesome set of rules has resulted in fascinating proofs of yes leads to quantity concept, has been used to reply to combinatorial questions concerning walls of house, and has enabled the authors to procure the formulation for the quantity of a typical tetrahedron in round 3 steps, utilizing not anything extra advanced than simple mathematics and the main effortless airplane geometry. All of those rules, and extra, display the wonderful thing about arithmetic and the interconnectedness of its a variety of branches. specified directions, together with transparent illustrations, allow the reader to achieve hands-on adventure developing those versions and to find for themselves the styles and relationships they unearth.

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Extra resources for A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics

Example text

The crease lines on this tape are called the primary crease lines. The really interested paper-folder should, before reading further, get a piece of . 6(a) as described above, throw away the first 10 triangles, and see if you can tell that the first angle you get between the top edge of the tape and the adjacent crease line is not π7 . 6(b). You will then believe that the D 2 U 1 -folding procedure produces tape on which the smallest angle does, indeed, approach π7 , actually rather rapidly. You might also try executing the FAT algorithm at every other vertex along the top of this tape to produce a regular 72 -gon.

9 (2 ) Note that this verifies that every time a correct fold is made the error is cut in half (count the number of fold lines before the pattern repeats to see why you have the factor of 23 appearing in the denominator of the error term), and every time you complete the folding on one edge of the tape the error on the other edge of the tape will change sign. 21 A regular 92 -gon, formed by performing the FAT algorithm on medium lines of the U 3 D 3 -tape. 5 Some bonuses As we have discovered in other contexts, mathematics is generous, often giving us much more than we originally asked for.

14 Tying a pentagon. 4 Does this idea generalize? 1 Loooking for a general pattern. By folding tape and executing the FAT algorithm at equally spaced intervals along the top edge of the folded tape, we obtain a regular polygon having U 1D1 U 2D2 U 3D3 .. U nDn 3 sides 5 sides ? sides (make a guess) .. ? 15 The beginning of a U 3 D 3 -tape. Is there a general pattern to all this? So far in this chapter we have discussed a systematic folding procedure, where we make the same number of folds at the top of the tape as at the bottom of the tape.

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


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