3D Lissajous Figures

x frequency

1

y frequency

1

z frequency

1

relative phase

π

4

length

100

tube thickness

show as tube

show box

These 3D Lissajous figures are parametric curves, where each of x(t), y(t), and z(t) is a sine function. If the ratios of their frequencies are rational, the curve will always eventually close. If one or two of the ratios is irrational, the curve will never close and eventually will fill a 2D or 3D region. Click and drag to rotate the 3D graphic.