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

By Peter Hilton, Jean Pedersen, Sylvie Donmoyer


ISBN-13: 1397805217641

ISBN-10: 0521764106

ISBN-13: 9780521764100

This easy-to-read booklet demonstrates how an easy geometric thought finds attention-grabbing connections and ends up in quantity idea, the math of polyhedra, combinatorial geometry, and workforce concept. utilizing a scientific paper-folding method it truly is attainable to build a standard polygon with any variety of aspects. This notable set of rules has ended in attention-grabbing proofs of convinced ends up in quantity concept, has been used to reply to combinatorial questions related to walls of area, and has enabled the authors to acquire the formulation for the quantity of a standard tetrahedron in round 3 steps, utilizing not anything extra advanced than uncomplicated mathematics and the main user-friendly airplane geometry. All of those principles, and extra, display the great thing about arithmetic and the interconnectedness of its a variety of branches. targeted directions, together with transparent illustrations, let 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 info for A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics

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14 15 Executing the reverse pass-through flex. 15 Pattern piece for the 8-8-flexagon. this algorithmic sequence of flexes, it is not true that each face will appear on the top (as happens with the Tuckerman traverse for hexaflexagons) nor that each face will appear on the bottom. Can you find Jennifer’s algorithm? We’ll give you a hint. Her algorithm involves only the straight flex and the reverse-through flex. But this is not surprising, since, as you may verify, the pass-through flex has the same effect as the following sequence of three flexes: reverse pass-through flex – straight flex – reverse pass-through flex.

This makes it clear that, every time we repeat a D 2 U 1 -folding on the tape, the error is reduced by a factor of 23 . We see that our optimistic strategy has paid off – by blandly assuming we have an angle of 2π at the top of the tape to begin with, and folding accordingly, we get 7 what we want – successive angles at the top of the tape which, as we fold, rapidly get closer and closer to π7 , whatever angle we had, in fact, started with! We thus say that π7 is the putative angle on this tape.

The first 10 triangles, and play with it. Try folding it on successive long lines. Then try folding it on successive short lines. 11. From the geometry of the situation we can figure out what the smallest angle on this U 2 D 2 -tape is approaching. 11(a) the base angles are equal. Let us call these angles α. Then, since the we know that 2α + 3π = π, from which it interior angle of a regular 5-gon is 3π 5 5 follows that α = π5 . There’s more! 12. By inserting secondary fold lines, just as we did with the U 1 D 1 -tape to produce FAT 6-gons, we can insert secondary fold lines on the U 2 D 2 -tape to enable us to produce FAT 10-gons.

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

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