Where did the impulse to draw 3D Lissajous figures come originally?
I was looking for a compact three-dimensional curved shape. It should be symmetrical enough to allow for “easy geometrical doubling” of some points. But it should be asymmetrical enough to look surprisingly different from different viewpoints! Finding uncommon ways of looking at 3D structures had been the idea behind my physics-based code art – and it should carry over to traditional drawings!
Lissajous figures are built from harmonic oscillations – sine and cosine functions, the building blocks of nearly anything in physics. When you combine x=cos(t) and y=sin(t) (t being the time) you get the simplest of all Lissajous figures – a circle! The eponymous Monsieur Lissajous has used tuning forks to “overlay” oscillations in different directions; today you would look at an oscilloscope.
I wanted to add a third dimension! Technically, sin(0t) would also count as a Lissajous figure, so I could have drawn just a 2D circle floating in the 3D world.
But it should be a bit more challenging, so I went for:
x = sin(t)
y = sin(2t)
z = sin(3t)
… just three innocuous waves! It turned out to be quite difficult to imagine how this shape would look like. Even rotating a virtual 3D representation on a screen did not cut it for me
In typical engineering-style, I created “models” in the real world! I guess, they would have been rejected at a school science fair project as they were really ugly:
Since then, I have drawn lots of Lissajous figures (e.g. here and here) and created lots with code (e.g. here and here).
I have often been envisaging them rather as a sculpture than a drawing! But even though the sculptural beginnings shown above look promising 🤣 I go for an intermediate version – make the code-generated image look more 3D!
So, I in lieu of a sculpture am rendering the Lissajous curves as metallic tubes!
And I make an earth-shattering change: Most of the Lissajous figures I have drawn with pencils or rendered with code contained only sine functions. But this time I wanted to fill up a volume of space with my virtual metallic tubes.
It turns out that throwing in one cosine function makes it “fatter”!
I like the concept of sets of “concentric” (self-similar) Lissajous figures dancing with each other: I pick the following partners for this dance:
x = cos(t)
y = sin(3t)
z = sin(5t)
x = cos(5t)
y = sin(3t)
z = sin(t)
… and I am zooming in!
Here is the result – let me reveal my new series Oscilloscopic Dance!
Four times the same shapes – just colored differently and viewed from a different position:
These images have been designed specifically to be wrapped around a rather thick wooden canvas frame.
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