MacArthur fellow and MIT Media Lab alumnus Karl Sims brings us this great tutorial on how to build your own complex harmonograph (Wikipedia) for making cool…um…”geometric figures?” I’m looking for a 50-cent mathematician’s word (which may or may not exist) for these periodic spirally figures. Can anybody help me out? [Thanks, David!]

Are they fractals? I don’t know if fractals are so spirally…

Spirograph?

Not fractals, and spirograph is a toy (which makes circle-based hypotrochoids and epitrochoids).

Lissajous Figures are what the harmonograph draws.

(Are those enough 50 cent words in one post, Sean?)

Lissajous figures are more properly graphs of pairs of parametric equations in rectilinear coordinate systems – like having two sine wave oscillators of different frequency and amplitude on each trace of a dual-trace ‘scope.

These diagrams are also like polar rosettes, which aren’t strictly Lissajous figures.

I would suggest that these figures – which seem upon glancing at the one photo to depend upon four oscillations (two for the ‘drawing plate’ and one each for the two arms) – are very much like Lissajous figures, and in certain degenerate cases the mechanism may produce Lissajous figures, just as in certain degenerate cases it may produce rosettes.

Let’s call them spirellis

Yes, the term you want is Lissajous Figures. You can also make them with a two-channel oscilloscope in X-Y mode and a pair of function generators. :)

Though normally only 2-parameter figures, these are the result of adding a third input. If you took away the third pendulum, you’d recognize a classic Lissajous figure..

Saying that all figures created by this Harmonograph are Lissjous figures is very much like claiming all fruits are oranges.

That’s not merely a third input – it’s two which share the same period but independent amplitudes. The central pendulum is affixed in such a fashion that it can swing freely in all directions, unlike the other two.

Still, only “sorta” Lissajous figures. They are certainly *related* to Lissajous curves, but are *not* strictly members of the class of curves explored by the mathematician for whom those figures are named, except in limited degenerate cases. A Harmonograph limited to exactly two pendulums (called a “lateral” Harmonograph, apparently) would *only* generate Lissajous curves.

My (admittedly naive) first-order approximation of this three-pendulum harmonograph would be:

x=Asin(at)+Csin(ct), y=Bsin(bt)+Dsin(ct)

Note that while the period of the third pendulum is equal in both directions, the amplitude in each direction is independent (C and D are not necessarily equal, while c is always c)

The period for the third pendulum isn’t necessarily going to be equal in both directions…

If it’s only swinging on one axis, the period for the other would be zero, if it’s swinging in a figure eight type pattern the period for one axis would be twice the other.

So, really, the third pendulum by itself is capable of creating a lissajous figure, or the other two could produce a lissajous. The affect of combining the two is what makes this interesting, like extending the lissajous in another dimension.

The Science Center at the Seattle Space Needle used to have one of these. The massive table/pendulum was free to move in more than one direction, suspended on chains, and the table also oscillated rotationally in the horizontal plane, like the balance wheel in a watch. the marker was free to adjust vertically, but was otherwise fixed…you could run multiple traces on the same paper with different color markers, and with the weights on the table set in varied positions…made some really neat patterns.

George M. Ewing, wa8wte@juno.com

ArenÂ´t these called Guilloches ?

There is a cool online harmonograph from Swantesson Interactive that simulates a harmonograph similar to the one described here. It allows you to create and save all kind of neat pendulum patterns.

http://swantesson.com/harmonograph.html