|The Universe of Discourse|
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Sun, 12 Nov 2006
An Etch-a-Sketch is a drawing toy invented in 1959 by Arthur Granjean and marketed by the Ohio Art company since shortly afterward. It looks superficially like a flat-screen television with two knobs.
The underside of the screen (that is, the inside surface) is coated with a fine aluminum powder. Also under the screen is a hidden stylus. One knob moves the stylus horizontally, the other vertically. As the stylus moves, it scrapes the aluminum powder off the screen, leaving behind a black line. If you hold the Etch-a-Sketch upside-down and shake it, the powder again coats the screen, erasing the lines.
It is very easy to draw horizontal and vertical lines, but very difficult to draw diagonal lines. (Wikipedia says "Creating a straight diagonal line or smoothly curved line with an Etch A Sketch is notoriously difficult and a true test of coordination.") So although extremely complex drawings can be made with an Etch-a-Sketch:
Most people can only produce something like this:
As a child, I had an Etch-a-Sketch. I did get some entertainment out of it by figuring out how it worked: if you manipulate the knobs to remove the aluminum powder from a large part of the screen, you can see the stylus inside. But for the most part I found it frustrating and disappointing. As an adult, I still find it so.
But it needn't be so. The most frustrating thing about the Etch-a-Sketch, I think, is that its potential has not yet begun to be unlocked.
Consider a forty-five degree line. To draw such a line, one must turn both the horizontal and the vertical knobs at the same time, at exactly the same rate. Suppose, for concreteness, that we're drawing a line from the upper left to the lower right. If you turn the horizontal knob a little too quickly, the diagonal line will bend rightward; if you turn it a little too slowly the diagonal line will bend downward. So in contrast to the mathematically exact vertical and horizontal lines that are easy to draw, it's next to impossible to draw a diagonal line that doesn't wiggle. And when you screw up, you can't fix the mistake without erasing the whole thing and starting over.
But the solution is obvious: If you can link the two knobs somehow, so that they can only turn simultaneously, you can easily draw a diagonal line. As a child, I experimented with rubber bands, trying to get one knob to drive the other. This wasn't successful. Clearly, a better solution is to use gears.
There are plenty of examples of toys that have good-quality cast-plastic gears. (Spirograph is one such.) The knobs on the Etch-a-Sketch could be geared together. If the gears are the same size, the knobs will rotate at the same rate, and the result will be a perfect 45° line.
If you gear the two knobs together directly, they will rotate in opposite directions, so that you can only draw lines with slope -1 (northwest to southeast), not with slope 1 (northeast to southwest). To fix this problem, we need to introduce more gears. There can be an axle peg sticking up from the case of the Etch-a-Sketch, in between the two knobs. Mounting three equal-sized gears on the two knobs and the axle peg gears will force the knobs to rotate in the same direction, at the same rate.
[ The remainder of this article contains a number of very dumb arithmetic errors. For example, you cannot fit three gears of size 1/3 on the knobs and pegs; you need to use three gears of size 1/4 instead. I will correct this on Monday, and provide an illustration to make it clearer what I mean. —MJD ]
[ Addendum 20061116: I have posted the correction, with illustrations. ]
Let's say that the distance between the centers of the two knobs is 1. We can get a line of slope -1 by mounting two gears, each with radius 1/2, on the two knobs; we can get a line of slope +1 by mounting three gears, each with radius 1/3, on the two knobs and on the axle peg. If we want to do both, we had better make the axle peg removable, or else it will interfere with the size-1/2 gears. This is no problem. It can mount into a socket on the front of the Etch-a-Sketch, and be pulled out when not needed.
But why have only one socket? We're including five gears already (two of size 1/2 and three of size 1/3) so we may as well put them to some more use. Throw in a size 1/6 gear, and add another socket for the axle peg, this time 1/3 of the way between the two knobs. Now you can mount a size 1/2 gear on the left knob, a size 1/6 gear on the axle peg, and a size 1/3 gear on the right knob. If the left knob turns at rate r, the middle gear turns at rate -3r and the right knob turns at rate 3r/2. This produces a line with slope 3/2, which is about a 56-degree angle.
Or put in another socket for the axle peg, 1/6 of the way between the knobs, and then mount size 1/2, size 1/3, and size 1/6 gears, in that order. The knobs are now producing a line with slope 3, a 72-degree angle. If you want a line with slope 1/3 (18°) instead, just reverse the order of the gears. (That is, exchange the large and the small ones.)
At this point adding a few more gears expands the repertoire significantly. Add a radius-2/3 gear and another radius-1/6 gear and you can mount [2/3, 1/3] to get lines with slope -1/2, [1/3, 2/3] to get slope -2, [2/3, 1/6, 1/6] to get slope 4, [1/6, 1/6, /23] to get slope 1/4.
Clearly, you can carry this onwards, limited only by the space for the axle holes and the expense of adding in more gears. Spirograph used to deliver fifteen or twenty plastic gears for a reasonable price, so it's clearly not implausible that Ohio Art could have done something like this.
Sometimes I even dare to think that they might have provided cams or elliptical gears. Properly designed cams could gear together the knobs to produce mathematically exact curved lines, squiggles, maybe even circles.
But no, as far as I can tell, it's never been done. Why not?