Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
I have a recollection. Years ago, a childhood friend of mine, Boris, shared with me with excitement an unusual experience he had on a visit to the Tretj'yakov Art Gallery in Moscow. He was accompanied by a professional painter, a good acquaintance of his older sister. While Boris was making a round in one of the halls, he observed that the painter remained all that time on the same spot studying a certain picture. Curious, my friend asked the painter what was it about the picture that kept him interested in it for so long. According to Boris, the painter did not reply directly, but, instead, stepped over to the picture and covered a spot on the picture with a palm of his hand. "Have a look at the picture and think of what you see," he requested. After a while, he uncovered the spot, stepped back and asked Boris to have another look.
Well, almost 4 decades later, with the names of the painter and the picture long forgotten, I still vividly remember Boris' excitement when he told me of how entirely different, deeper and more beautiful, the picture appeared to him then.
This recollection is haunting me. In retrospect, I regret to have never arranged with Boris to visit the gallery and learn how to really see the picture. At the time, we were in our early teens and lived in a remote neighborhood on the outskirts of Moscow. We could not travel by ourselves half the way across the big city, and, by the time I knew how to do that, other things, quite naturally, took priority in my life.
I also remember this episode every time I go to an art exhibition. Had I had this lesson on the art of seeing pictures, would I draw more enjoyment from my visit?
I believe I would. It is possible to learn to appreciate painting without being a painter, and the point I want to make is that, by analogy, it is possible to appreciate mathematics without being a mathematician. Appreciation does not require mastery, but it is impossible without adequate knowledge. Knowledge and mastery are labels at the two ends of a scale that stretches from the ability to recognize information through the ability to process it. Potential for appreciation is an increasing function on this scale. (Thus, if differentiable, the function has positive derivative. Furthermore, appreciation feeds on itself and, through a positive feedback loop, seeks to expand knowledge. It then follows that if the appreciation function is sufficiently smooth its second derivative is also positive.)
The appreciation function depends on many parameters: disposition, perseverance, cultural surroundings, educational experience. The latter derives from the mathematical subjects themselves but even more from the manner in which mathematics has been presented.
Even without reference to the current research into knowledge representation, it's obvious that mathematics is not a collection of disparate facts. A reasonable metaphor to use to describe mathematics (it probably also fits any other branch of knowledge) is a network of hierarchical structures - classes, and methods. The appreciation of mathematics grows with the discernment of links between apparently dissimilar pieces of information and the perception of the general in the particular. Facts presented in a manner that conforms to the architectural design of mathematics have a better shot at being appreciated. A worthy goal to strive for!
An elegant theorem was published by Giovanni Ceva in 1678.
Dan Pedoe remarks in his geometry course: The theorems of Ceva and Menelaus naturally go together, since the one gives the conditions for lines through vertices of a triangle to be concurrent, and the other gives the condition for points on the sides of a triangle to be collinear.
Menelaus of Alexandria worked in the 1st century A.D. Giovanni Ceva (1648-1734) was an Italian engineer and geometer who lived some 16 centuries later. Ceva proved his theorem considering centers of gravity and the law of moments [Episodes] - the fundamental tool of Archimedes' Method (3rd century B.C.) How did it happen that the theorem remained unknown until the late 17th century? Given its simplicity, the timing of the theorem is remarkable, but also inspiring for young minds: may there still be simple overlooked facts lying around waiting to be discovered?
The theorem refers to three lines AD, BE, and CF through the vertices of ABC. In Ceva's honor lines that connect a vertex with a point on the opposite side are called Cevians. Altitudes, medians, angle bisectors are all Cevians and, in addition, conform to
Three Cevians AD, BE, and CF are concurrent if and only if
For the mechanical proof, place a system of masses BF·CE, AF·CE, and BF·EA at the vertices A, B, and C, respectively. Then F becomes the center of mass of two points A and B, whereas E becomes the center of mass of points A and C. From here it follows that the center of mass of the three points lies on the intersection of Cevians BE and CF. (1) is then equivalent to the assertion that it also lies on the line AD. For details, see Ross Honsberger's Episodes or an online discussion.
I am grateful to Prof. William A. McWorter Jr for the following, I believe novel, proof and his advice on writing the Java demonstration below. The demonstration is limited to the case when all segments in (1) are rational (and thus can be assumed to be integer) and the statement that the concurrency of the Cevians implies (1).
Create a grid of lines parallel to BE and CF. Refine the grid by dividing every segment parallel to BE into AF smaller segments. Similarly, divide every segment parallel to CF into EA smaller parts. In the refined grid, AK (K being the common point of the three Cevians) serves as a diagonal of the "grid square" - parallelogram with equal number of units (EA·AF) on every side. Therefore, there is a set of diagonals parallel to AD that do not miss a single grid node.
FB is divided into FB·EA small segments by grid lines parallel to CF. Follow those grid lines towards BE and let them refract in BE to the direction of AD. We see that the grid diagonals divide BD into FB·EA small segments. Similarly (starting from CE), the diagonals divide DC into CE·AF parts. Because of the uniformity of the grid,
which is the same as
Thus (1) follows.
Mixing mechanical and geometric ideas, the proof is equivalent to placing masses AF·EA at A, FB·EA at B, and CE·AF at C, and then passing to the isotomic conjugates of the given Cevians. In strictly mechanical terms, we might, as well, place masses FB·EA·CE·AF, AF·EA·CE·AF, and FB·EA·AF·EA at vertices A, B, and C.
Bamboozlement is a term suggested by Greg Frederickson to describe plane dissections, followed by rearrangement of pieces that result in a figure of supposedly different area. One such bamboozlement, found in Frederickson's book, can also be found in a couple of books by Martin Gardner [Puzzles, Magic] and in the classic by W.W. Rouse Ball.
Cut an 8x8 chessboard into two triangles and two quadrilaterals. Then rearrange them as suggested by the right diagram. They will combine in a 5x13 rectangle. Well, it's easy to get fooled or fool somebody else if a real chessboard has been sacrificed for the sake of demonstration. A less expensive way is to use a piece of paper, a ruler and a soft pencil to draw "fat" lines. For an 8x8 puzzle, the computer, however, gives away the secret. When the four pieces are arranged in a 5x13 rectangle a narrow parallelogram is left uncovered. (By Pick's theorem, since the area of that parallelogram is exactly 1, it contains no grid points in its interior.)
The numbers 5, 8, 13 are members of the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... We might have chosen another triple of successive Fibonacci numbers. For the triple 2, 3, 5, a less standard chessboard of 3x3 squares is transformed into a 2x5 rectangle. (In the applet, the only control parameter is the index of the smallest number in the triplet.)
In general, if Fk denotes the k-th Fibonacci number, a FnxFn square can be rearranged into a Fn-1xFn+1 rectangle according to the following formula that is easily proved by induction
Regardless of whether we get surplus area or a deficit, the absolute error is always the same, 1. The relative error which is the ratio of the absolute error to Fn+12 decreases as n grows and becomes virtually indiscernible for
The goal of developing appreciation is often perceived as conflicting with the goal of acquiring knowledge. But to appreciate is to know in the first place. The real difference is in the ability to see a picture as a whole or only as a collection of its constituent parts. As with a bamboozlement puzzle, it possible to see the polygons while missing the relationships that polygons have to each other via their angles and side lengths.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1997-1998 Alexander Bogomolny