By Peter Hilton, Jean Pedersen, Sylvie Donmoyer
This easy-to-read ebook demonstrates how an easy geometric concept finds attention-grabbing connections and ends up in quantity idea, the maths of polyhedra, combinatorial geometry, and staff concept. utilizing a scientific paper-folding approach it really is attainable to build a customary polygon with any variety of aspects. This amazing set of rules has ended in fascinating proofs of convinced ends up in quantity idea, has been used to respond to combinatorial questions related to walls of house, and has enabled the authors to acquire the formulation for the amount of a typical tetrahedron in round 3 steps, utilizing not anything extra complex than uncomplicated mathematics and the main effortless airplane geometry. All of those principles, and extra, show the wonderful thing about arithmetic and the interconnectedness of its numerous branches. distinct directions, together with transparent illustrations, permit the reader to achieve 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
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.
A brief search of the Web for the word “flexagons” or “hexaflexagons” will turn up the most recent publications, and inventions, in this area. ) of the flexagons in this chapter. We discuss the group theory concerning the 3-6-flexagon in Chapter 13. 1 Should you always follow instructions? All self-respecting human beings, and therefore all our readers, must answer this question with a resounding NO! In the next paragraph we describe two aspects of our paper-folding, and building, instructions where we do advise rather rigid adherence to our specifications.
2 Why does the FAT algorithm work? We used, without explaining why it worked, the FAT algorithm in Chapter 2. 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.
A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics by Peter Hilton, Jean Pedersen, Sylvie Donmoyer