Daniel Goldwater: Harmonic Cantilever
Dan Goldwater contributed this piece in 1995 to The Cheapbook: A Compendium of Inexpensive Exhibit Ideas. Dan died in 1999, after many years as an educator and exhibit developer at the Franklin Institute in Philadelphia.
A love of math puzzles goes back to my earliest days. Then when I went to work at the Franklin Institute in 1970 one of my first jobs was to manage the production of a small math exhibit. Given the very modest funding available, we decided that most of the handson activities would be adapted from existing math puzzles. We were pleased (but not surprised) to find that math puzzles are an ideal source of ideas for interactive exhibitry. They are very inexpensive to build and maintenance costs are very low. They have great holding power: I've seen people sticking with puzzles like the famous Soma Cube for 20 minutes or more, and found that they often refuse help when it is offered. Some ask if they can buy a copy of the puzzle. Surely our visitors are enjoying the challenge of trying to solve the puzzle. To me, this is as important as anything that happens in handson science centers; that we provide an environment where people can pursue with pleasure the kind of 'higher level' cerebral processes involved in trying to figure things out for themselves.
Many math puzzles exemplify important mathematics. Such is the case with Harmonic Cantilever. The math behind it—the 'harmonic series'—is of great historical importance and still stands as a paradigm of the property that mathematicians call 'divergence'.
Harmonic Cantilever
Math/Historical Background
The harmonic sequence was first studied about 2,500 years ago by Pythagoras and his followers. Among other things, they were interested in the pitch produced when a plucked string is shortened by 1/2, 1/3, 1/4, 1/5, ... If the mass and the tension of the string is fixed, these lengths correspond to musical relationships now known as the octave, the fifth above the octave, the second octave, and the third above the second octave. If played simultaneously, these notes will sound as a familiar major chord, spread out over two octaves.
The terms of the harmonic sequence get smaller, approaching zero. The sequence has a infinite number of terms. So you can find an individual term of the harmonic sequence that is close to zero as you like: if 1/1000 isn't close enough for you, go on to 1/10,000. Or 1/100,000. The harmonic sequence approaches zero as a limit. But surprisingly, the harmonic series 1/2 + 1/3 + 1/4 + 1/5...etc. has no limit: it diverges. The unusual property of the harmonic cantilever is related to this mathematical property of the harmonic series.
Handson Device
The component consists of a set of hardwood blocks and a shelf on which to build the cantilever. The tabletop beneath is carpeted to reduce noise when the blocks fall. Block number 1 is marked off at half its length. This length (6 inches in our device) is the unit for the series. The next block is marked off at 3 inches (or 1/2 the unit length). Block number 3 is marked off at 2 inches (1/3 of 6 inches) and so on.
We made 15 such blocks, the last marked off at 1/15 of 6 inches or .4 inches. At present only, five blocks are set out for the unsupervised public. The rest are stored and taken out by demonstrators when needed. The blocks are stacked in numerical order, from top to bottom. The stack will stay in balance when the offsets are in the proportion of the harmonic sequence. 1/2 + 1/3 + 1/3 + 1/5 + 1/6 is greater than 1 so with five blocks, the top block can be entirely outside the supporting shelf.
Since the harmonic series diverges, this is the beginning of a structure that could, in theory, extend outward indefinitely with no support except under the bottom block. Alas, only in theory. The harmonic series diverges  it has no limit  but it grows very slowly. In fact, on the scale of our device, a harmonic cantilever extending only 6 feet would require something like 200,00 pieces. To our knowledge, the harmonic cantilever has no practical application.
This description is based on the unpublished Math Exhibit Notebook and is used with permission from the Franklin Institute.
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