Why Beauty is Truth by Ian Stewart Read Free Book Online Page B

work out the logical dependencies among geometrical statements. As a bonus, it may also help us understand the real world, but that certainly wasn’t central to Euclid’s thinking.
    The next comment is related, but it points in a rather different direction. There is no problem finding constructions for trisecting angles approximately. If you want to be accurate to one percent or one thousandth of a percent, it can be done. When the error is one thousandth of the thickness of your pencil line, it really doesn’t matter for technical drawing.The mathematical problem is about ideal trisections. Can an arbitrary angle be trisected exactly? And the answer is “no.”
    It is often said that “you can’t prove a negative.” Mathematicians know this to be rubbish. Moreover, negatives have their own fascination, especially when new methods are needed to prove them. Those methods are often more powerful, and more interesting, than a positive solution. When someone invents a powerful new method to characterize those things that can be constructed using straightedge and compass, and distinguish them from those that cannot, then you have an entirely new way of thinking. And with that come new thoughts, new problems, new solutions—and new mathematical theories and tools.
    No one can use a tool that hasn’t been built. You can’t call a friend on your cell phone if cell phones don’t exist. You can’t eat a spinach soufflé if no one has invented agriculture or discovered fire. So tool-building can be at least as important as problem-solving.

    The ability to divide angles into equal parts is closely related to something much prettier: constructing regular polygons.
    A polygon (Greek for “many angles”) is a closed shape formed from straight lines. Triangles, squares, rectangles, diamonds like this ◊, all are polygons. A circle is not a polygon, because its “side” is a curve, not a series of straight lines. A polygon is regular if all of its sides have the same length and if each pair of consecutive sides meet at the same angle. Here are regular polygons with 3, 4, 5, 6, 7, and 8 sides:

    Regular polygons.
    Their technical names are equilateral triangle, square, (regular) pentagon, hexagon, heptagon, and octagon. Less elegantly, they are referred to as the regular 3-gon, 4-gon, 5-gon, 6-gon, 7-gon, and 8-gon. This terminology may seem ugly, but when it becomes necessary to refer to the regular 17-sided polygon—as it shortly will—then the term “17-gon” is farmore practical than “heptadecagon” or “heptakaidecagon.” As for the 65,537-gon (yes, that too!)—well, you get the picture.
    Euclid and his predecessors must have thought a great deal about which regular polygons can be constructed, because he offers constructions for many of them. This turned out to be a fascinating, and decidedly tricky, question. The Greeks knew how to construct regular polygons when the number of sides is
    3, 4, 5, 6, 8, 10, 12, 15, 16, 20.
    We now know that they cannot be constructed when the number of sides is
    7, 9, 11, 13, 14, 18, 19.
    which leaves one number in this range, 17, as yet unaccounted for. The story of the 17-gon will be told in its rightful place; it is important for more reasons than purely mathematical ones.
    In discussing geometry, there is no substitute for drawing on a sheet of paper with a real straightedge and real compass. It gives you a feel for how the subject fits together. I’m going to take you through my favorite construction, for the regular hexagon. I learned it from a book my uncle gave me in the late 1950s, called Man Must Measure , and it’s lovely:

    How to construct a regular hexagon.
    Fix the radius of the compass throughout, so all circles will be of the same size. (1) Draw a circle. (2) Choose a point on it and draw a circle centered at that point. This crosses the original circle in two new points.