By Peter Hilton, Jean Pedersen, Sylvie Donmoyer
This easy-to-read e-book demonstrates how an easy geometric suggestion finds interesting connections and ends up in quantity concept, the maths of polyhedra, combinatorial geometry, and crew thought. utilizing a scientific paper-folding technique it really is attainable to build a standard polygon with any variety of aspects. This amazing set of rules has resulted in fascinating proofs of convinced ends up in quantity concept, has been used to respond to combinatorial questions concerning walls of area, and has enabled the authors to procure the formulation for the quantity of a customary 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, display the great thing about arithmetic and the interconnectedness of its numerous branches. certain directions, together with transparent illustrations, allow 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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Additional info for A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics
4 Does this idea generalize? Having had so much success with folding triangles it is very natural to ask what would happen if you folded twice at each vertex. 4 Does this idea generalize? 6 Constructing a FAT triangle. 25 26 Another thread 1. 2. 3. 4. 7 Preparing the U 1 D 1 -tape for constructing a hexagon. 10. Notice that the tape, which we call U 2 D 2 -tape (or, equivalently, D 2 U 2 -tape) seems to be getting more and more regular – the successive long lines are becoming closer and closer to each other in length, and so are the successive short lines.
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. Why do we get equilateral triangles?
As we implied when constructing the flexagons in Chapter 1, there are at least 2 ways to turn a piece of paper over. 3. Assume the square is a transparent square of plastic. 1 Should you always follow instructions? 3 Two different ways to flip. 4 Showing the result of a move. square; whereas in (b) the orientation of the symbol tells you to flip the square over a vertical axis along the right-hand side of the square. 4 the heavy right-pointing arrow indicates that by performing the move on the left-hand figure (rotating the entire figure 90◦ in a clockwise direction about the right angle), we obtain the right-hand figure.
A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics by Peter Hilton, Jean Pedersen, Sylvie Donmoyer