April 10, 2014 8:31:26 AM
If you are around mathematicians, you will eventually hear them talk about some proof or finding that is "beautiful." Mathematicians sometimes have a way of perceiving beauty that the rest of us cannot penetrate. There can be, though, a visual beauty to ideas in geometry that can be easily appreciated by the non-mathematician. That's the realm explored within Beautiful Geometry (Princeton University Press) by Eli Maor and Eugen Jost, a good-looking, large-format book suitable for the coffee table, but with lots of mathematical ideas packed in among the colorful illustrations. Maor teaches the history of mathematics at Loyola, and emphasizes in his introduction that rather than being just about numbers, mathematics represents a search for patterns, and it is that search that is shared by artists and mathematicians. Jost is a Swiss artist who describes himself as playing with numbers and patterns when he makes his pictures, which he calls "playgrounds." He points out that many of the pictures here can be viewed in a sense of play, seeing how much the picture reinforces or interprets the text that Maor has written. There are 51 chapters here, each three or four pages, with a main illustration by Jost and often with supplemental diagrams. The text is clear and aimed for anyone including "laypersons who are not afraid of an occasional formula or equation." There's a bit of algebra here, and lots of geometry, but no calculus. The pictures are pretty and often startling, and the range of topics might help non-mathematicians understand what mathematicians mean by beauty.
Euclid shows up on plenty of the pages here, and two chapters are devoted to the Pythagorean Theorem, "the most well-known theorem in all of mathematics." We think of it as a2 + 2 = c2, but the diagrams here remind us that Euclid and the classical Greek mathematicians would have thought of this as a geometric problem, not an algebraic one. The theorem has been proved four hundred times (one proof was by James A. Garfield before he became president), but Euclid's is the one illustrated here because it is the most famous. To read the explanation here is to be dropped into straightedge-and-compass construction, whereas in Euclid the steps to it are taken up gradually; it is his 47th proposition in his first book. Maor praises the proof's "sheer austerity, relying on just a bare minimum of previously established theorems." The classic diagram from Euclid is here, but the color picture by Jost shows a series of right triangles of the same diagram, only the triangles move gradually from having side a of length equal to the hypotenuse c with b therefore being zero (you have to imagine a triangle with one side of zero length), and decreasing the length of side a in four steps until it is equal to side b (nice symmetry here), and then continuing the decrease in four steps until a is the side with length zero. The process, by the way the squares and rectangles upon them are colored in, demonstrates an important step within Euclid's proof.
Euclid covered geometry pretty comprehensively, but one of the later chapters here (the book is more-or-less chronological) is devoted to Morley's Theorem, discovered by Frank Morley, a professor at Johns Hopkins in 1899. If you take any triangle, no matter how squat or how stretched, and trisect its angles, the lines of trisections meet in a way that define an equilateral triangle. No one had expected such a thing until Morley's demonstration. "It shows that even in classical geometry, surprises may still be awaiting us around the corner - or perhaps around the vertex!" (A reason Euclid might never have taken it up is that it involves trisection of angles, an operation that cannot be done with just the compass and straightedge that are the only tools used in his constructions.)
Similar to this is a much older construction on a four-sided figure, a quadrilateral. Take any quadrilateral, no matter how asymmetric, even one that has two sides bent in so it is concave. Mark the midpoints of all four of the lines, and connect them; you will always get a parallelogram with half the area of the original quadrilateral. It's easy to imagine this in the simplest instance, a square: join the midpoints and you get a diamond inside the square and it is obviously half the square's area. But the illustration shows plenty of irregular quadrilaterals and the perfect parallelograms floating within them.
The famous fractals of the Mandelbrot set are not here; that would be too deep a jump into the complex plane, perhaps. But fractals there are. One is one of the first to be studied, the Snowflake Curve that looks like an increasingly fuzzy Star of David, depending on how many times you take a simple step in changing each of the lines of the snowflake. The figure itself doesn't get much bigger; in fact, its area is finite and calculable after an infinite number of changes have been rung on it sides. But the infinite changes make the sides longer and longer without end; there is enough ink to fill the snowflake, but there will never be enough to draw its outline. The Sierpinski Triangle (on the book's cover) is here, too, with a startlingly opposite effect. Infinitely performing the designated changes to this figure perforates it with more and more but tinier and tinier triangles. The infinite reiteration produces triangles the sum of whose perimeters is infinite, but it also produces a shape that is so ridden with holes that it has no more remaining area. Zero area but infinite length: "It shows again that when infinity comes into play, strange things always lurk around the corner."
Pi is here, of course, and the Golden Ratio, and hexagons, and Fibonacci Numbers, and logarithmic spirals, and much more. Even with the chronological chapters, this is not really a survey of geometry, but it does show mathematical interests over time. It is a handsome book for browsing and for some deep thought, and would be a superb gift for anyone (especially a young person) who has interest in mathematics.