By Peter Hilton, Jean Pedersen
This easy-to-read ebook demonstrates how an easy geometric inspiration finds attention-grabbing connections and ends up in quantity conception, the math of polyhedra, combinatorial geometry, and crew idea. utilizing a scientific paper-folding technique it truly is attainable to build a customary polygon with any variety of facets. This notable set of rules has resulted in fascinating proofs of definite leads to quantity conception, has been used to reply to combinatorial questions related to walls of house, and has enabled the authors to procure the formulation for the quantity of a customary tetrahedron in round 3 steps, utilizing not anything extra advanced than simple mathematics and the main common airplane geometry. All of those principles, and extra, demonstrate the wonderful thing about arithmetic and the interconnectedness of its a number of branches. special 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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6 Three FAT 7-gons. 5(a) might not have been precisely 2π . 5(b) only approximately π7 ; let us call them π7 + (where may be either positive or negative). Consequently the angle to the right of this crease, at the bottom of the tape, would measure 6π − . 5(c), resulting acute angles nearest the bottom of the tape, labeled 3π 7 − , forcing the angle to the right of this crease line would in fact measure 3π 7 2 4π at the top of the tape to have measure 7 + 2 . When this last angle is bisected twice by folding the tape down, the two acute angles nearest the top edge of the tape will each measure π7 + 23 .
18 A short-line 9-gon. Now, since we are heavily involved with bisecting angles we might suspect these numbers have something to do with the powers of 2. Ahah! The numbers 2, 4, and 8 are closely related to the sequence 3, 5, and 9. 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.
So after 10 correct fold lines the error must be smaller than 1024 ! 5 Folding triangles. ) 23 24 Another thread Constructing bigger triangles Suppose you want a bigger triangle. 6. 6. We advise you to master this systematic algorithm, since it is used later in the construction of other regular polygons. 2. Constructing hexagons We can, of course, construct a hexagon, as we did in Chapter 1, by using the fact that a regular hexagon may be decomposed into 6 equilateral triangles. But there are other ways to construct the hexagon from the strip of equilateral triangles that are useful to know later on (when we want to construct regular N-gons of the form 2n N0 , where N0 is any odd number).