The Universe of Discourse
           
Thu, 16 Nov 2006

Etch-a-Sketch blue-skying, corrected
In my last article I discussed a scheme for improving the Etch-a-Sketch which contained a serious mechanical error. I was discussing attaching gears to the two knobs of the Etch-a-Sketch to force them to turn at the exact same rate. Supposing that the distance between the knobs is 1 unit, I said, then we can gear the two knobs together by attaching a gear of radius 1/2 to each knob; the two gears will mesh, and the knobs will then turn at the same rate, in opposite directions. This was fine.

Then I went astray, and suggested adding an axle peg midway between the two knobs, and putting gears of radius 1/3 on the peg and on the two knobs. This won't work.

The one person who wrote to me to ask about the problem is a very bright person, but been seriously confused about how I was planning to set up the gears, so I evidently I didn't explain it very well. It needed a picture. So this time I'm going to try to get it right, with pictures. Here is an Etch-a-Sketch:

Recall that the distance between the centers of the two knobs (shown here in gray) is defined to be 1.

Here are some gears, which happen to have radii 1/3, 1/4, and 1/6:

The dark spots are the axle pins, which are at the centers of the gears.

Here's a picture of an Etch-a-Sketch with a radius-1/2 gear mounted on each knob:

Since the two gears are of equal size, the knobs are constrained to turn at the same rate, in opposite directions. This forces the Etch-a-Sketch to draw a line with slope -1, from northwest to southeast or vice-versa.

Here the knobs have been fitted with different-sized gears, one with radius 1/3 and the other with radius 2/3:

The left-hand knob is forced to turn exactly twice as fast as the right-hand knob, producing a line with slope -1/2. To get a line with slope -2, just reverse the positions of the two gears. I got this much right in the original article. (Although it didn't occur to me, before I saw the pictures, just how much of the screen would be occluded by the gears. Better make them transparent.)

Then I suggested that you could drill a little hole in between the two knobs, and use it to mount a third axle and a third gear. If all three gears are the same size, the two knobs are forced to turn at the same rate, this time in the same direction, and you get a line with slope 1, from southeast to northwest:

All fine, except that I said that since all three gears are the same size, they must each have radius 1/3. Wrong. They must have radius 1/4, as the diagram above should make clear: from the center of the left knob to the rightmost edge of its gear is one radius; the width of the middle gear is two radii, not one, and from the edge of the right gear to the center of its knob is the fourth radius. Since the total of four radii is 1, each radius must be 1/4. Oops.

This wrecks the rest of the details of my other article. Since we were already including gears of size 1/2 and 1/3, I reasoned, we can throw in a gear of size 1/6 and get some new behaviors from the 1/2 + 1/3 + 1/6 combination. The corresponding combination for 1/2 and 1/4 is 1/8:

And this gets us lines of slope 2, or, reversing the order of the gears, 1/2.

So what next? The calculations are a bit less obvious than they were back in the happy days when I thought that installing two gears of size p and q left space for one of size 1-(p+q). It's tempting to consider a radius-1/3 gear next, since it's the simplest size I haven't yet installed. But to mount it on the knobs along with a size-1/2 gear, we need to include a size-1/12 gear to go in between:

This produces a 2/3 or 3/2 gear ratio.

Once we have the size-1/12 gear, we can mount it with the size-1/4 and size-1/3 that we already had:

Well, you get the idea. I probably would not have gone on so long, but I was enjoying drawing the diagrams. I used linogram, which rocks. it's almost exactly the diagram-drawing program I've always wanted; expect an article about this next week sometime.


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