By Peter Hilton, Jean Pedersen

This easy-to-read ebook demonstrates how an easy geometric thought unearths attention-grabbing connections and leads to quantity conception, the maths of polyhedra, combinatorial geometry, and crew concept. utilizing a scientific paper-folding strategy it really is attainable to build a standard polygon with any variety of facets. This striking set of rules has resulted in fascinating proofs of convinced leads to quantity concept, has been used to respond to combinatorial questions regarding walls of house, and has enabled the authors to acquire the formulation for the quantity of a standard tetrahedron in round 3 steps, utilizing not anything extra complex than easy mathematics and the main easy airplane geometry. All of those principles, and extra, exhibit the wonderful thing about arithmetic and the interconnectedness of its quite a few branches. distinctive directions, together with transparent illustrations, allow the reader to realize hands-on event developing those types and to find for themselves the styles and relationships they unearth.

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Constructing a FAT 10-gon. 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.

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.

We thus say that π7 is the putative angle on this tape. 6(c), (d) show the regular 72 - and 73 -gons that are produced from the D 2 U 1 -tape by executing the FAT algorithm on the crease lines that make angles and 3π , respectively, with an edge of the tape (if the angle needed is at the of 2π 7 7 bottom of the tape, as with 3π , simply turn the tape over so that the required angle 7 appears on the top). 6(c), (d) the FAT algorithm was executed on every other suitable vertex along the edge of the tape so that, in (c), the resulting figure, or its flipped version, could be woven together in a more symmetric way and, in (d), the excess could be folded neatly around the points.