Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
Droodles were invented by Roger Price in 1950s. These were rather abstract drawings of few lines accompanied by an implicit question "What is it?" A punch line the author's answer made the drawings obvious. Everyone I talked to knew one or more droodles, but not everyone had heard the word. I find this strange because of their popularity in the 1950s and 60s. There's still a small industry making a living off droodle party napkins and droodle bibs. Etymologically speaking, the word droodle derives from riddle doodle or, if you prefer, doodle riddle. (A quick search on the web produced a couple of later day interpretations. A dragon doodle is a kind of art with a Celtic motif. An example is also available of a drool doodle which is a unique and an accidental creation of a budding artist who happened to nod in a (math?) class over one of his doodles.)
Droodles found their way into education. Besides their entertainment value, droodles are thought to foster creativity in children (and adults.) The Exploratorium uses droodles to teach children that "It's easiest to remember stuff that makes sense to you and connects with other stuff you know."
The expected punch line is seldom the only sensible answer. In cognitive research, this observation led to an investigation of "young children's grasp of the principle that two or more persons might differently interpret one and the same stimulus event."
Well, that's all good and exciting. Not only in mathematics but in other walks of life ideas conceived in one context become useful in another. But does the foregoing introduction have a punch line? Here it comes. There exist mathematical droodles and using them in instruction might be a valuable educational tool. What is a mathematical droodle? As a preliminary definition, a mathematical droodle is a visual object accompanied by a "What is it?" question to which there is at least one plausible answer with mathematical contents.
The concept might not be altogether new. The much-appreciated Proofs Without Words fit the description. But the class of mathematical droodles is broader by far. A PWW naturally presents a proof. Mathematical droodles may present any mathematical idea, not necessarily a proof. PWWs are about why; mathematical droodles are about what. Here's an example that I hope does not even require a punch line.
(Credits: I had an opportunity to observe a working model of such a device at a math fair organized during ICME-9.)
In Drawing Worlds, R. N. Jackiw creator of The Geometer's Sketchpad argues for the pedagogical value of special-purpose investigation environments. An example is a Mystery World bounded by two invisible mirror-like axes. Parts of shapes (a circle in the diagram on the right) that cross the axes are reflected back into the World's universe. The worlds were presented by the author to teachers participating in the Summer Sketchpad Institute. The task? Toy with the software, observe and try to explain what's going on. In Jackiw's words,
Yes, indeed. Technology, to be useful, must not require real mastery. (One problem I see with using calculators in the classroom is that they must be mastered before a payoff becomes feasible. Meanwhile valuable time is taken away from doing mathematics and spent on mastering calculator functions. Is it always helpful? Is there always a positive payoff? Are all students equally capable of mastering those buttons?) For example, books come with drawings that illustrate and help us grasp mathematical concepts, but the users readers are not expected to be able to produce (or even reproduce) such drawings by themselves. The question does not even arise, in fact, and perhaps there's a moral to learn. Common illustrations as much as the Drawing Worlds demonstrate how technology can be used as a tool and not as a subject of study.
There is more to illustrations than meets the eye. And this is where generation of droodles comes in. How difficult is it to create a droodle? Roger Price is an extremely creative person. Many of his original droodles were unique creations. However, he has also developed a technique that might have put droodle production on an industrial footing. Want to follow in Roger Price's footsteps? Find or draw a picture and cut out of it a rectangular portion. Show the small piece to somebody: "What is it?"
Arguably, Roger Price might have done this better and with more humor, but, as a craft, there you have it your own droodle. Go experiment! As every one knows, much fun may be produced by taking objects out of context.
By their nature, illustrations are always associated with context. In a textbook or a Web page illustrations are added to clarify the surrounding text. Might it make sense to put the sled before the horse, so to speak, and present illustrations outside their context with the question "What is it?"
In fact, I was asked to do just that on more than a few occasions. The requests came from math teachers. My site features about 200 Java applets. The overwhelming majority serve as illustrations for the surrounding text. Shifting the viewpoint, the text around an illustration serves to explain what is the object that is presented by the illustration. The correspondents argued that students would benefit greatly by first trying to toy with the applets before having a chance to see the explanations. (This was also my plan for this column when it started 3 years ago. But later I slipped into a more conventional format.) Only recently I connected this idea with Roger Price's technique of droodle generation.
Interactive illustrations leave room for experimentation and for observing more problem cases than is possible with traditional illustrations. Thus they may be more suggestive of the mathematical statement they were brought to illustrate. There are additional benefits in using them as droodles.
First, as with Drawing Worlds, interactive illustrations may offer "an in-depth experience of the geometric context established by the curriculum developer, without requiring any significant amount of intimacy with the general purpose software environment."
Second, mathematical droodles fall into the category of "open-ended" problems. No single right answer is suggested immediately. Droodles naturally provide students with a focus for discussion and a playing field for sharing ideas.
Third, droodles are in fact ubiquitous. It's easy to create a coherent curriculum that leaves enough leeway for students' unstructured activities.
There is at least one drawback to all this, which is shared with other kinds of dynamic software. Besides figuring out a mathematical idea presented by a droodle, the user must also struggle with idiosyncrasies of the implementation style of the droodle's creator. There is a mitigating factor, though. Droodles are very limited in scope. Nothing like a general purpose calculator or a software package. Just 2-3 kinds of controls to move things around. Provided the curriculum developer paid attention to consistency in user interface, the upfront effort needed to achieve comfort with a sequence of droodles is much smaller than that required to master a general purpose device or a software package.
Well, try this one. To help you out, there are two circles each defined by a couple of points: the center and a boundary point. Each can be dragged when a cursor is in its interior. There's one additional point "between" the two circles which I would start doodling with.
Here's another one.
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 © 1996-2000 Alexander Bogomolny
Between two circles there is a chain of circles, each tangent to its immediate neighbors and the two given circles. The last circle in the chain may or may not be tangent to the first circle. They cross if they do not touch. The moveable point in-between the circles serves to move the chain between the given circles. It appears that if the first and the last circles in the chain were tangent for one position of that point, they remain tangent for any other position of the point.
This statement is known as Steiner's Porism [Geometry Revisited, p. 124]: Given two circles one inside the other. Pick up a point in-between and draw a circle tangent to the given two. Then draw a circle tangent to the new circle and the original two. Continue building a chain of circles each touching the two given circles and its predecessor in the chain. It may happen that, for some n, the n-th circle will touch the first circle in the chain. Then this will happen regardless of the position of the starting point.
There is an easy proof that depends on existence of a transformation T of the plane that has the following properties:
(Such a transformation indeed exists. It belongs to the class of inversions and its existence is established in Inversive Geometry after just a few definitions.)
For concentric circles, i.e., circles with the same center, the statement is quite obvious. The radii of the given circles and n must stand in a certain relationship for the chain to close on itself. Whether it does or not does not depend on the position of the starting point.
Two polygons with the same number of vertices inscribed into a fixed circle. Vertices of the polygons may be moved along the circle. The sides of the polygons are paired in such a way that paired sides always remain parallel. This is true if the number of vertices is even. If the number of vertices is odd, this is true for all but 1 pair of the sides. The two sides in the that special pair have equal lengths.
Let two n-gons be inscribed into the same circle. Assume that (n-1) sides of one polygon are successively parallel to (n-1) one sides of the second polygon. Then, the remaining two sides are parallel if n is even, and are equal if n is odd [FGM, p. 302]
There's a proof by induction. For n = 3, the theorem is obvious. Two pairs of parallel sides form equal inscribed angles that are bound to subtend equal chords the third sides.
Let n = 4, and two polygons A1A2A3A4 and B1B2B3B4 be given such that
The last paragraph also demonstrates one half of the inductive step: if the statement holds for an odd
Assume now that the statement holds for n = k, where k is even. Let two polygons A1A2...AkAk+1 and B1B2...BkBk+1 have the first k sides parallel. Then by the just established first half of the statement,