Alchemy Islands : --- : Patterning Tutorial

Harmonographs and Lissajous Curves

Patterning can draw the curves produced by harmonographs and their undamped cousins, Lissajous figures.

  • lissajous draws closed curves from one oscillator on each axis, with no damping.
  • harmonograph adds damping so the figure spirals inward.
  • multi-harmonograph sums several oscillators on each axis for richer, multi-pendulum forms.
Pattern 1 
(set-standard-colors)

(lissajous 3 4 300 1
  {:stroke blue :stroke-weight 2})

The simplest Lissajous call takes a single ratio:

(lissajous 3)

This is equivalent to fx=1, fy=3, with other values at their defaults. Only the ratio between the x and y frequencies determines the shape of a Lissajous figure.

For explicit frequencies use:

(lissajous fx fy)

or the full sampling form:

(lissajous fx fy nopoints num-cycles)

Here nopoints is the number of samples drawn per x-axis cycle, and num-cycles is how many of those cycles to trace. One cycle is one period of the slowest x frequency.

(set-standard-colors)

(grid 4
  (map
    (fn [r]
      (lissajous 1 r 300 2
        {:stroke coral :stroke-weight 1.5}))
    [2 3 4 5 5/2 7/3 8/5 9/4]))
Pattern 2

Harmonographs

A harmonograph is the same idea, but each axis can decay over time. Pass separate damping values for x and y:

(harmonograph fx fy damp-x damp-y nopoints num-cycles)

With zero damping, a harmonograph matches the corresponding Lissajous curve.

Pattern 3 
(set-standard-colors)

(harmonograph 3 4 0.015 0.02 400 10
  {:stroke red :stroke-weight 2})

Different damping on each axis distorts the spiral as it contracts, which is often closer to the behaviour of real pendulum machines.

(set-standard-colors)

(grid 3
  (map
    (fn [[dx dy]]
      (harmonograph 5 6 dx dy 420 8
        {:stroke dark-green :stroke-weight 1.5}))
    [[0.01 0.01]
     [0.01 0.03]
     [0.03 0.01]]))
Pattern 4

Multi-pendulum curves

When more than one oscillator drives each axis, use multi-harmonograph with a map of options:

(multi-harmonograph
  {:x [{:freq 1 :amp 0.8 :damp 0.01}
       {:freq 2 :amp 0.4 :damp 0.012}]
   :y [{:freq 3 :amp 0.8 :damp 0.01}
       {:freq 5 :amp 0.3 :damp 0.015}]
   :nopoints 400
   :num-cycles 8
   :style {:stroke blue :stroke-weight 1}})

Each axis value is the sum of its oscillators at each moment in time. The result is still one [x y] point per sample, but the path becomes more complex and organic.

(set-standard-colors)

(multi-harmonograph
  {:x [{:freq 1 :amp 0.8 :damp 0.012}
       {:freq 2 :amp 0.35 :damp 0.015}]
   :y [{:freq 3 :amp 0.8 :damp 0.01}
       {:freq 5 :amp 0.25 :damp 0.018}]
   :nopoints 420
   :num-cycles 10
   :style {:stroke magenta :stroke-weight 1.5}})
Pattern 5

As with trochoids, the simple functions are built on a lower-level point generator. Use harmonograph-points when you want the raw point sequence.

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