This art is beautiful and refreshing, as it highlights the beauty of creation. But it isn’t new.
The figures Robert Howsare produced are mathematically known as second-order Lissajous graphs. First-order Lissajous graphs are a combination of vertical and horizontal sine waves with varying amplitudes and a phase difference. They were discovered in the mid-20th century with oscilloscope inputs (see Wikipedia).
The equation for first-order Lissajous graphs is L(x,y) = A*sin(w1t)x + B*cos(w2t+p)y, where A and B are scalar amplitudes, w1 and w2 are the angular frequencies in radians per second, t is time in seconds, and p is phase in radians.
The two frequencies may be equal here. You can produce your own here: math.com/students/wonders/lissajous/lissajous.html.
Howsare’s graphs are second-order in that each turntables push the yoke in two directions instead of one, and the yoke is in effect producing an x-y sum amplified and translated by the additional length of the arm. The equation is L(x,y) = A*sin(w1t)x + B*cos(w2t+p)x + B*cos(w2t+p-pi)y + A*sin(w1t)y, where pi is the familiar number 3.14 etc..
The plots seem unpredictable to the casual observer (or Post writer), but are actually perfectly predictable. The wonder (and it truly is wonderful) of the Fourier series is something engineers learn in their sophomore year and appreciate it, hopefully, all their lives.
Dean Bruckner is an assistant director at the Avionics Engineering Center at OU.
