By Peter Hilton, Jean Pedersen

This easy-to-read e-book demonstrates how an easy geometric concept finds attention-grabbing connections and ends up in quantity conception, the maths of polyhedra, combinatorial geometry, and crew idea. utilizing a scientific paper-folding process it's attainable to build a customary polygon with any variety of facets. This extraordinary set of rules has ended in attention-grabbing proofs of definite leads to quantity idea, has been used to reply to combinatorial questions concerning walls of house, and has enabled the authors to acquire the formulation for the amount of a standard tetrahedron in round 3 steps, utilizing not anything extra complex than easy mathematics and the main easy aircraft geometry. All of those principles, and extra, demonstrate the great thing about arithmetic and the interconnectedness of its a variety of branches. specific directions, together with transparent illustrations, permit the reader to realize hands-on adventure developing those versions and to find for themselves the styles and relationships they unearth.

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Then pull apart the top of the flexagon which will lie flat again in the shape of a square as shown in the last figure. To repeat this straight flex, as we call it, you must turn the flexagon over. Practice this a few times and draw patterns on all the faces you can find. You are now ready for the more complicated pass-through flex. 13) and make mountain folds along the diagonal lines so that you obtain a 4-petaled arrangement. Then pull 2 opposite petals apart and down. You will then have a square platform above the 2 petals you pulled down.

It is now natural to ask: (1) Can we use the same general approach used for folding a convex 7-gon to fold a convex N -gon with N odd, at least for certain specified values of N ? If so, can we always prove that the actual angles on the tape really converge to the putative angle we originally sought? (2) What happens if we consider general folding procedures perhaps with other periods, such as those represented by D3U 3, D4U 2, or D3U 1D1U 3D1U 1? ) The answer to (1) is yes and we will show, in Chapter 7, an algorithm for determining a folding procedure that produces tape from which you can construct any regular ab -gon, if a and b are odd with a < b2 .

Always feel free to use your ingenuity to avoid an algorithm that is not working for you, or seems to you to be unduly complex. A word to the wise We’ve done a lot of field-testing of the “hands-on” material in this book. Our instructions seem to be, on the whole, quite comprehensible to most readers. However, there are two basic types of error that people seem prone to make in carrying out our instructions. Material error In doing mathematics, it is absurd to specify the quality of paper on which the mathematics should be done.