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52 53 Some of the proofs in this collection, especially the dissection ones and those involving infinity, omit a lot of detail and really only try to capture the essence of why something is true. Many of them would only count as full proofs in the eyes of a mathematician if more details were added. Here are a number of infamous fallacies that use arguments that are very similar to some used in this book. aPPendIX III looKs can be deceIvIng Dissection disaster The first proof by dissection shows that 64 65. What goes badly wrong here is that the diagonal cut in the rectangle on the right is not really a line as suggested by the drawing, but a very thin quadrilateral slit of area one. This proof as well as the next is based of the fact that the square of any Fibonacci number differs by one from the product of its two neighbors. In the first proof the square has a smaller area than the oblong rectangle, in the second it is the other way around. Infinite insanity In From Pie to Pi we argued that the regular ngons inscribed in a circle approximate it in shape, and that hence their circumferences approach that of the circle. This is true but requires a proof, as the following fallacy shows. Starting with the large semicircle on the right, its diameter is approximated by strings of ever smaller semicircles. Hence the lengths of the strings approximate that of the diameter. However, every string is clearly exactly as long as the large semicircle. Hence, a semicircle is just as long as its diameter, or p 1. 64 8.8 area square area rectangle 13.5 65 169 13.13 area square area rectangle 21.8 168 Routine risk In Infinite Staircase page 22 we manipulated the following infinite sum as we would manipulate a finite one to prove that it adds up to infinity. We should be careful when doing this. For example, the following similar sum adds up, in a precise sense, to 0.6931... ln 2, the natural logarithm of 2. By just rearranging this sum, we can prove that ln 2 12 ln 2, or 2 1. Even worse, it can be proved that for any real number, there is a rearrangement of this sum into a sum that adds up to this number. Of course, all this does not mean that you cannot do anything meaningful with infinite expressions, only that you have to follow certain rules. Universal uniqueness Many proofs that there are exactly five regular solids start, as we did in our introduction, by showing that there are essentially five possible corners, and end by constructing five solids with these kinds of corners. These proofs are incomplete because they dont show the uniqueness of these solids. Just imagine building a hollow icosahedron such that adjacent rigid faces meet at their edges in hinges. How can you be sure that what you end up with is rigid After all, in isolation every one of the corners can be flexed, so why not the whole shape Here is a related question If you build the skeletons of the regular solids using rigid edges only, such that adjacent edges can be flexed about a common vertex, then which of the skeletons are rigid and which are not
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