1. |
Imagine a cork bobbing on a wave, not an ocean wave, but an idealized mathematical wave (a sinusoid). As the wave travels by, the horizontal position (x) of the cork does not change but its vertical position (y) does. The cork moves up and down along a vertical line segment. |

2. |
Now imagine the wave traveling along a vertical line (the y-axis) instead of a horizontal one (the x-axis). If you turn your head sideways you can still envision a cork bopping on the wave, this time following a horizontal line segment. It may be easier to dispense with any physical analogy and simply say that this wave drives the horizontal position of a pen. |

3. |
Suppose the two waves discussed thus far are acting simultaneously on a single pen. The wave traveling along the horizontal (x-axis) controls the vertical position (y) of the pen, while that traveling along the vertical (y-axis) controls the horizontal position (x). The pen now traces a diagonal line. |

4. |
So far the paths traced by the pen are not very interesting. In an attempt to obtain a more interesting result, let the two waves differ. One way in which they can differ is phase – the relative time at which they peak. For example, suppose that instead of peaking at the same time, one wave peaks when the other crosses its axis. The waves are then said to be 90 degrees out of phase. The pen now draws a circle. Certainly there are simpler ways to draw a circle. A little more patience is needed before this approach begins to bear fruit. |

5. |
Another wave characteristic that can be altered is amplitude, the height of the peaks and depth of the valleys. When the wave amplitudes are not the same, an ellipse is drawn instead of a circle. |

6. |
In addition to changing the phase and amplitude of a wave let’s change its period, the distance between adjacent peaks. For example, if the period of the wave traveling along the y-axis is half that of the wave traveling along the x-axis, the pen traces a figure eight. Another way of describing this relationship is to say that the frequency of the wave traveling along the y-axis is twice that of the wave traveling along the x-axis (the frequency being inversely proportional to the period). |

7. |
Now that the results are looking a little more promising, let’s eliminate the amplitude difference, maintain the 90-degree phase difference and play with the frequency ratio. For example if this relationship is changed from 2:1 to 4:3 the result is a Lissajous figure. Lissajous figures take their name from Jules Antoine Lissajous, a French physicist who studied these patterns in the context of sound waves produced by tuning forks. |

8. |
In addition to altering a wave’s phase, amplitude or frequency, a constant offset can be added. Offsetting waves merely serves to shift the location of the pattern drawn. For example, if both waves are reduced in amplitude and offset to ride above their respective axes a small circle is drawn to the upper right. |

9. |
The results get more interesting if instead of a constant offset, another sinusoid is added. For example, one can add to the original sinusoid, a second sinusoid with a relatively small amplitude and high frequency. This produces a small ripple riding atop a larger wave. With such a wave on each axis the pen loops its way around the large circle. |

10. |
In the previous example the ripple frequency was an integral multiple (6 for the waves shown) of the base wave frequency. If the ripple to base frequency ratio is a fraction rather than an integer, the pen must travel around the large circle multiple times before ending up back at its starting point. The result will probably look familiar to anyone who has played with a Spirograph^{TM}.
By varying the relative amplitudes and frequencies of the base wave and ripple, any classic Spirograph pattern can be produced. |

11. |
Lissajous figures and Spirograph patterns are interesting but there’s no reason to stop a that. Another avenue of variation is the wave shape. For example a pair of triangular waves, 90 degrees out of phase, will draw a diamond. |

12. |
Substituting triangular waves for sinusoids while maintaining the frequency ratio that yielded the Lissajous figure produces a diagonal lattice. |

13. |
Obviously, triangular waves can be substituted for sinusoidal waves in the Spirograph pattern as well. |

14. |
Time to switch gears. Recall that driving each coordinate (x and y) with a sum of two waves produced a Spirograph pattern. Now consider using products of waves. In particular, multiply a sinusoidal wave by a relatively low frequency triangular wave. Such wave multiplication is referred to as amplitude modulation, because the lower frequency wave slowly varies, or modulates, the amplitude of the higher frequency wave. With amplitude-modulated waves on each axis comprised of sinusoids 90 degrees out of phase and triangular modulating waves in phase, the pen spirals out from the center and then back in. |

15. |
Taking the same waves as above but with the triangular waves out of phase produces a very different pattern. |

16. |
The last two patterns were produced by variations of amplitude modulation. Another way to combine two waves is phase modulation. Here one wave varies the phase of another. With phase modulation on each axis the resulting pattern is harder to describe, but still interesting. In general, patterns produced by phase modulation are less symmetric (although the pattern produced here does have one axis of symmetry) and in many cases look messy. |

17. |
So far we have constructed patterns from sinusoidal waves, triangular waves and various combinations thereof. While many basic wave shapes could be considered, we will limit attention to just one more, the square wave. Not surprisingly, a square wave on each axis and a 90 degree phase difference, causes the pen to draw a square.
It’s worth noting that sums of sinusoids, with suitable amplitude, frequencies and phases, can produce any periodic wave shape, so in theory, sinusoidal waves alone are sufficient to produce all the patterns shown here. However, an infinite series of sinusoids must be summed to obtain the sharp features exhibited by triangular and square waves. It is therefore more convenient to treat these as distinct fundamental wave types for pattern generation. |

18. |
One again we’ve managed to draw a mundane shape (a square) in a rather complicated manner. One way to get a better payoff (in terms of visually interesting patterns) from square waves is to use them to replicate other patterns at various offsets. For example if each axis has a sinusoid plus a square wave with a relative frequency of 1/4, four interlocking circles can be drawn. While watching this pattern being drawn you may have noticed that when the pen moved from one circle to another it stopped drawing. What’s happening here is that when the pen encounters a wave discontinuity, i.e. a place where the wave instantly jumps from one value to another, it’s being lifted. |

19. |
A finite series of square waves can be summed to produce waves resembling staircases. Like their square wave components the staircase waves by themselves don’t produce particularly interesting patterns. |

20. |
A fruitful way to use staircase waves is drive discreet jumps in parameters of other waves. For example, discrete jumps in sinusoid amplitude can be used to draw a series of concentric circles. |

21. |
Many of the patterns produced thus far resulted from x and y-axis waves that were quite similar, often differing in only a single parameter, such as phase. Introducing more dissimilarities between the x and y-axis waves can yield interesting results. For example, a Lissajous figure variation can be drawn with a sinusoid on one axis and a triangular wave on the other. |

22. |
Along similar lines, a basic sinusoid on one axis and an amplitude-modulated sinusoid on the other produces a another unique pattern. Clearly the possibilities are virtually limitless. |