Double-Elliptical Pendulum Harmonograph (D.E.P.H.)

In the May, 1965, the Amateur Scientist column in Scientific American, was entitled: Zany mechanical devices that draw figures known as harmonograms . The column was conducted each month by C. L. Strong, who oftentimes published descriptions of experiments conducted by readers. Several kinds of harmonographs were described and illustrated. I was most interested in the Double-Elliptical Pendulum Harmonograph because of the striking complexity of the patterns it could produce.

diagram from Amateur Scientist, Scientific American, May 1965

The following is an except from a letter published in the column from a father-daughter team of Concord, Massachusetts:

Our most recent machine is a version of the elegantly simple double-elliptic pendulum first described in 1907 by British experimenters. It consists of a small wooden platform weighted with 50 pounds of iron [see illustration at lower left]. The platform is suspended through a yoke and a length of iron pipe by ball-bearing gimbals fixed to the ceiling. The figure is drawn on paper that is taped to the platform; the drawing instrument is a pen attached to one end of a counterweight level arm. This simple arrangement generates only circles, ellipses, and straight lines. The interest of these figures can be enhanced by making identical drawings and superposing one on the other with a slight displacement; moiré patterns appear.

The most astonishing increase in the versatility of the machine can be made, however, by suspending a second, less massive pendulum from the bottom of the platform, again using a gimbal arrangement. When it vibrates its own natural period, the shorter pendulum perturbs the motion of the swinging platform in an infinite number of ways. The handsomest patterns are generated when the frequency of the upper pendulum bears a whole-number ratio to that of the lower pendulum - a ratio such as 3:2 or 2:1. The adjustable factors that describe any given figure are then the two amplitudes of the upper pendulum, the phase angle between them, the phase between upper and lower pendulum, the phase and amplitudes of the lower pendulum and the ratios of the frequencies of the upper and lower pendulums. We have scarcely begun to investigate the variety of patterns that can be generated with the double-elliptic pendulum machine.

I wrote to C. L. Strong in 1966 with questions before I built mine, and again later with samples of my patterns.

I adapted my own design from the information and illustrations in the column, and additional feedback from Mr. Strong. I designed it to just clear the concrete basement floor in our house and to be suspended from the floor joists overhead as shown in the column.

Finally, in March, 1967, I had iron pipe cut and threaded. I spent $40 on eight ball-bearing pillow blocks to construct the needed U-joints. I scavenged the 50 pounds of sash weights from Inland Metals in Spokane.

I must have drawn a scale drawing and somehow figured out what lengths of pipe I needed. I'm surprised that I don't seem to have my original plan. It must have been an interesting design and construction project because of the way all the components related to one another.

This could be an expensive project today depending on your access to pipe-cutting and threading tools, etc. For example, I had the ends of eight of the pcs of pipe milled rather than threaded so that they would fit into the bushings of the pillow blocks. I was lucky to have the old-fashioned West Valley hardware store a block way where the man often had time on his hands to do some of these things for me.

After getting it built, I found that the pen apparatus was just as important as the pendulum. I spent quite some time perfecting my control over that design - otherwise, tracing would be botched, or the level of friction would be too high, etc. You must be able to lower and raise the pen gently and cleanly, the pen can't be too massive, or too light.

Because I had a lightweight counter-balance pen arm, it turned out that I needed to add pressure by means of a delicate spring to keep the pen from continuing upwards and leaving the paper at times, resulting in 'skips'. (As the DEPH swings, the pen must be able to move up and down. -- the paper is flat, and the table is swinging in a spherical section.) The lowering, pressure-setting, and raising were all controlled by a single lever.

To store the D.E.P.H. out of the way, I would:

  1. remove the lower pendulum with bar bell weights and set aside
  2. open the top of the writing platform
  3. remove the sash weights and put them away
  4. unscrew the upper pendulum (with platform) from the flange at the top
  5. place the upper and lower pendulums out of the way on a special shelf

Here are more photos of my favorite harmonograms:

The DEPH is currently on loan to the Physics Department at Lewis & Clark College in Portland, Oregon.

Before building the D.E.P.H., I experimented with some other setups described in the column. I started by taking time-lapse exposures in a darkened room -- shooting straight up at a point of light swinging freely in orbits. Since it's virtually impossible to start a freely-suspended object swinging in a perfect circle, the ellipse will rotate and also spiral down due to friction. The locus of this orbit can be quiet beautiful. The suspended flashlight might also twirl or wobble adding variation. The 'point' of light can be a slit, making the width of the light line vary as the flashlist orbits. I had a darkroom, so we could shoot the film, then take it right into the developing process all in the same evening. (We didn't need to print the negative to see the basic results.) This one was made by closing the shutter for a few swings, then re-opening it to capture a few more orbits.

a PhotoPendulation

I also experimented with a crude Twin Pendulum Harmonograph, as illustrated in the Am Sci column, but it was limited in what it could produced by the fixed nature of the two pendulums.

It's always a good idea to keep a dated lab notebook or journal for such projects. I didn't. (Pre iPhone days. LOL.)

See Also

3D harmongrams!


The Amateur Scientist, Scientific American, May 1965, page 130.

The Bibliography from Am Sci was not too useful:

      Harmonic Curves, William F. Rigge, SJ, The Creighton University, 1926.
      Harmonic Vibrations and Vibration Figures, Joseph Goold, Charles E. Benham, Richard Kerr and L.R. Wilberforce, edited by Herbert C. Newton, Newton & Co., 1909.

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