By Peter Hilton, Jean Pedersen
This easy-to-read ebook demonstrates how an easy geometric proposal finds interesting connections and leads to quantity conception, the maths of polyhedra, combinatorial geometry, and staff conception. utilizing a scientific paper-folding technique it really is attainable to build a customary polygon with any variety of facets. This outstanding set of rules has ended in attention-grabbing proofs of definite ends up in quantity conception, has been used to reply to combinatorial questions related to walls of area, and has enabled the authors to procure the formulation for the amount of a customary tetrahedron in round 3 steps, utilizing not anything extra advanced than easy mathematics and the main simple airplane geometry. All of those rules, and extra, show the great thing about arithmetic and the interconnectedness of its quite a few branches. distinctive directions, together with transparent illustrations, let the reader to realize hands-on adventure developing those types 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
For the polygons we have constructed so far we didn’t actually need to use the FAT algorithm to obtain the polygon; this was because the geometry of the U n D n -tape allowed us to obtain the regular (2n + 1)-gon if it was folded on successive lines of any fixed length (and there were always n such lengths). So, in those cases, the FAT algorithm just gave us a bonus (2n + 1)-gon. However, we aren’t always going to be so lucky – and that is why the FAT algorithm needed to be invented. Here is the explanation of why it works in a very general setting that produces regular star ab -gons.
Continue folding to make a string of triangles as long as you need. Notice two things. First, the folding process goes UP, DOWN, UP, DOWN . . , and we abbreviate it to U DU DU D . . or U 1 D 1 , and sometimes refer to this folded strip as U 1 D 1 -tape. Second, although the first few triangles may be a bit irregular, the triangles formed always become more and more regular; that is, the angle between the last fold line and the edge of the tape gets closer and closer to π3 . When you use these triangles for constructing models, it is very safe to throw away the first 10 triangles and then to assume the rest of the triangles will be close enough to use for constructing anything that requires equilateral triangles.
4(b). 4(a) also has suitable crease lines that make it possible to use the FAT algorithm to fold a regular convex 4-gon. We leave this as an exercise for the reader and turn to a more challenging construction, the regular convex 7-gon. 3 Constructing a 7-gon Now, since the 7-gon is the first regular polygon that we encounter for which we do not have available a Euclidean construction (nor does anybody else), we are faced with a real difficulty in creating a crease line making an angle of π7 with the top edge of the tape.